This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
1 , F(rei" ) converges all the way from the origin to a point outside the unit circle and hence provides an analytic continuation of f(z) across the arc of the unit circle near ei" . This result enables us to say something about the location of the singularities of f (z) and its analytic continuation. Firstly, f(z) has no singularities in lzl < 1 which entails Eq. (3.2.8) being true for any a and ro < 1. Consequently, the 'Y in Theorem 3.2.1 could be brought down to 1. Secondly, if
with M < 1, f(z) can be continued across the arc at eia and any singularity on the ray is no closer than ei"/M. An application to cp(z) = e2 which satisfies lcp(zt) l < exp(tr cos a) shows that f(z) can be continued analytically in any region in which cos a :::; 0; for cos a > 0 it can be continued up to cos a = 1. I n other words analytic continuation i s available for R(z) < 1 . Actually the continuation has a single pole at z = 1 since it is 1/(1 - z). In contrast knowledge of the singularities of f(z) and its analytic continuation gives some information about the growth of cp. For, if r0eia is a singularity of f (z), Eq. (3.2.8) must fail otherwise Eq. (3.2.7) would be convergent. Again, if there are no singularities in R(z) < 1 , lcp(zt) l < K exp(tr cow) must be valid to guarantee the right convergence of Eq. (3.2.7) . It remains to discuss am in the light of Eq. (3.2.4). For future purposes it is convenient to deal with a slightly more general condition. Theorem 3.2.2 Let m be unlimited. If am+I am
a + �-1. = l +
mq
Series
Chapter 3
52
then, if 0 :S q < 1 with a = 0 if q = 0, and, if q = 1 ,
am = exp{(a + 1-1 ) ln m}.
Proof. Let m be a positive integer and n an unlimited integer. Then
=
Since p is unlimited (a + 1-1 ) /Pq is infinitesimal (a = 0 if q right-hand side is n+m- 1 n+m- 1 a + 1-1. 2: -- = (a + 1-1 ) 2: -q1 by Theorem
pq
p=n
A.5.10. From Theorem A.5.11
[+m dt
-
w! and d' is limited. d', - ln (1 + .!..) - 1 / 2 for t 2: w'. Integrate the inequality from w' to n when n 2: w' to obtain n + d' n/w� - (w + n + �) ln(1 + njw) < - � (n - w' ) + w' + d'w'jw! - (w + w' + � ) ln(1 + w' jw) < -w'n/4w on employing Eq.(3.3.3) and noting that any positive terms are dominated by w'nj4w. Insertion of the inequality into Eq.(3.3.2) leads to "" exp(1 - w'2 j4w) L I w < 1 - exp( -w' j4w) < w318 exp( -w< /2) 0. (3. 3.4) Consider now the terms in which n < w'. It is straightforward to verify that then the derivative must be less than
Hence
(-z ) n l bn n=w'
1
c::!.
( - �z ) w];'-1 bn (�z )n bo - bw'- 1 (:;z ) w' - w];'- 2(bn - bn+I ) (�z )n+l 1
=
b0
55
Partial sums
Section 3.3
Here 1 by definition and the next term is infinitesimal by what has just been demonstrated. So =
( - W-z ) w'L-1 bn (W-z ) n + 1-1. - w'L-2(bn - bn+I ) (W-) n+I n=O n=O Z
(3 3 ) Eq.(3.3.5) ' o(b - b1 ) � - (bw'- 2 - bw'- 1 ) (�r - I w'-3 n+ - � (bn - 2bn+I + bn+2) ( :;z ) 2(3.3.6) Eq.(3.3.6) w 2 (a + 1-1. ) 0 bo - bl w + 1 - w+ 1 Eq.(3.2.4). lz/wl < 1 + d/w� Eq.(3.3.6). Eq (3.2.4) , n < w' . bn - 2bn+I + bn+2 = 1-i.bn/w 'i . Eq.(3.3.2) Eq.(3.3.3) "E J (bn - 2bn+I + bn+2) (!..) n+2 1 :S 1-1. wt3 \ (� n2 ) (3.3.7) W n=O n=O W 2 W 2 1/w� 1-1. . w1L-3 exp(nd'h - n2 h2 /4). n=O 1
=
1
The plan now is to establish that the series on the right of infinitesimal. To this end observe that
The second term on the right of shown already. As regards the first term =
1 __
. .5
is
is infinitesimal by what has been
1
�
on utilising Since the first term is infinitesimal also. It remains to consider the last term of Since that it may be verified readily, by means of 1
Substitute from
with
incorporated; then exp
for some infinitesimal
Put h
=
-
4w
so that the series becomes
h
Since h is infinitesimal, comparison with Definition A.5.1 reveals that this series represents an integral which is bounded by
[" exp (d't - t2 /4)dt. The value of the integral is limited and so the right-hand side of Eq. ( 3. 3. 7) is infinitesimal. Hence Eq. ( 3. 3 . 6 ) implies that
Series
Chapter 3
56
'w -1 12 - z/wl ::=: 8( z )0n+1 8 n=OL (bn - bn+l) - 11
Accordingly, provided that
>
with
=
W
and it follows from Eq. (3.3.5) that
standard,
� bn (�r (1 11)/(1 -z/w). =
+
/w l z l :::; w +lzldw�
The upper limit can be altered to oo on account of Eq. ( 3.3.4) and limited. The validity of the theorem has been affirmed for d limited. The set of integers n E N for which
� Zm - 1 :::; 1 l w--'--!(1aw--zwz/w) -'--'- m=w I a,.,.
-
L..J l m.
(lzl -w)/w�
being with
n
and < n includes all the standard integers by what has just been proved. There is no set which consists solely of the standard integers. As a consequence there must be an unlimited integer � for which these relations hold with � in place of n. Since � is infinitesimal there is nothing more to prove. • Theorem 3.3.1 approximates the sum of the higher order terms in the series for the entire function when lies inside a certain circle. Broadly speaking, an explanation of how the formula originates is the following. To go from one term to the next when n is large multiply by Successive terms diminish is small and the dominant behaviour is dictated by the first of rapidly when them, namely As increases successive terms become more important until is about the same as when the terms resemble those of a geometric grows still larger successive terms increase and series with ratio When it is the last one retained which is dominant. Thus, one can expect a theorem, analogous to Theorem 3.3.1 , for the contribution of the lower order terms in when is outside a suitable circle; it was proved originally by Izumi ( 1927) in a classical manner. Theorem 3.3.2 If is an unlimited positive integer and a,.,. satisfies the same condition as in Theorem 3. 3. 1
1/ z
cp(z)
lzl
cp(z)
z/n.
lawz z l w jw!. l z l zjw. lzwl
z
w
'£ m 1 + 11 aw w m=O m! z z/w - 1 . w! z z/wl 2: 8 0 with 8 standard. for l z l 2: w provided that 1 Proof. 1)!w 1 --'-- w"- 1 -am zm - w"- 1 ( z )m- w+1 (w ----'--aw- 1 Z - m=O ml m=O w a,.,.
Set
=
1-
L..J
>
•
L..J
Cm
-
57
Partial sums
Section 3.3 where
(w - 1)!am wm -w+I . m.a' w - ! When m is a standard integer utilise Stirling's formula for (w - 1)! and Theorem 3.2.2 for aw - l · The modulus of (z/w) m-w+ I does not exceed 1 since m < w - 1. Hence a I exp{(m + !) lnw - w + 2(lal + 11 )w2} z m-w + I :::; Ir:;; Cm ( ;:;) Cm
I
I
=
1
lam l exp(-w/2)/m! -::= 0 since � is limited for standard m. Theorem A.l.3 warrants the assertion that m' L cm (z/w) m -w+ ! -::= 0 m=O for every standard integer m!. By Robinson's Lemma there IS an unlimited integer M such that M /w 0 and (3.3.8) L:=O em (z ;w ) m- w+! o. m For m > M, � can be approximated by Theorem 3 . 2.2 and m! by Stirling's ::S
-::=
M
-==
formula with the net result
Cm = exp [ (m + !) ln : + m - w + 2(a + l1){m! - (w - 1) ! } + 11 ] . (3.3.9)
The precise value of M is unknown; to be on the safe side it will be assumed that M < w" where w" is the next integer above w718. If the value of M were known and was greater than w" it might be possible to dispense with some of the subsequent steps. The function
1 ) ln -W + 1 - W- + 2 lal {(w - 1)2 - t > }/t f(t) = ( 1 + t 2t t I
I
lal (1 + f'(t) = t2� 1 - !__W - W2 which is positive for t :::; w - w' where w' i s the next integer above w518. Hence, for M :::; t :::; w", f(t) :::; f(w " ) < -w/2w" (3.3.10) whereas, for t :::; w - w', (3.3.11) f(t) :::; f(w - w' ) < -(w' /2w?. has a derivative
{
11)}
58
Series
Chapter 3
Therefore, from Eq.(3.3.10) ,
w" L \ em (z/w)m-w+ l \ m=M+ l
< < �
w" L exp (- mw / 'U..v " ) m=M+ l exp{ - (M + 1)w/2w"} - exp (-w) 1 - exp ( -w / 'U..v" ) 0.
On the other hand, Eq.(3.3.11) supplies
w ' f l em (z/w) m-w+I I m=w" + I
< < �
w-w' L exp{ -m(w' /w ? } m=w" + I exp{ -w"(w' /21..v) 2 } - exp { - (w - w')(w' /21..v ) 2 } 1 - exp{ - (w' /2w) 2 } 0
since w"(w' /w) 2 > w 118 and w(w' jw) 2 > w 114 . Consequently, the only part of the series which may not be infinitesimal is
w-1 w1- 2 w m + l = e (z/w) L Cw- 1 n (w/zt. L m n=O m =w-w' + I
Now (1 - wjz) and
w'- 2 L Cw- - n (w/zt n=O 1
=
1+
(3.3.12)
w1- 2 11 + L (cw-1- n - Cw- n) (w/zt n=l
(3.3.13)
w1-2 - Cw-n )(w/zt n=l w'- 2 (cw-2 - Cw_ I )wjz + 11 + L (cw- 1 -n - 2cw- n + Cw- n+ I )(w/zt . n=2 (3.3.14)
(1 - w/z)( L (cw- 1-n
The first term on the right of Eq. ( 3.3.14) is infinitesimal and
Cw - 1 -n - 2Cw- n + Cw- n+ I = 11 Cw- I - n /W2 because n/w < w - 318. Also, from Eq. (3.3.9) , (w ) n l :::; 11 wL'-2 exp 4 \ a \ 1n - 4wn2 1 w'- 2 1 1 L Cw- 1 -n Z W 2 n=2 W2 W2 n=2 I
(
)
.
Section 3.4
59
Asymptotic behaviour of an entire function
As in Theorem 3.3.1 the right-hand side is bounded by a convergent integral and so the right-hand side of Eq.(3.3.14) is infinitesimal. Hence, so long as j1 - z/wl 2: 8 > 0,
w'-2 L (cw- - n - Cw- n )(w/zt 11 n= 1 1 =
whence, from Eq. (3.3. 12)and Eq.(3.3.13) ,
w- 1 :L em (z/w) m -w+ 1 = (1 + 11)/(1 - w/z). =w-w' m +l Accordingly, collecting together all that has been proved, we have
w-1 :L em (z/w) m- w+ I = (1 + 11)/(1 - w/z). m=O The statement of the theorem follows now after invocation of Eq. (3.2.4). • 3.4
Asymptotic behaviour of an entire function
The region where z = w has been excluded from both Theorems 3.3. 1 and 3.3.2. Indeed, the two theorems create the illusion that there is a pole at z = w. That no singularity can be present is evident from cp(z) being an entire function. If both theorems were valid for z close to w their combination would supply a formula for cp(z), namely
cp(z) = 11 aw zw /{w!(1 - z/w)} after two large quantities cancel one another. When z/w -:::= 1 this suggests that (3.4.1)
Of course, this discussion does not constitute a proof of Eq.(3.4.1) but, because Eq.(3.4.1) holds when an = 1 and cp(z) = ez , indicates that it could be worth seeing whether or not a proof could be devised. After all, the example could be exceptional and engender a spurious feeling of confidence. The relevant theorem will be demonstrated now; it differs slightly from Eq. (3.4.1 ) . Theorem 3.4.1 Under the same condition on am as in Theorem 3. 3. 1
for
z = w + d with w
an unlimited positive integer and d limited.
60
Series
Chapter 3
Proof. In the proof of Theorem 3.3.1 the condition \1 - z /w \ � 8 has not been used up to Eq.(3.3.4) and, in fact, is not employed until well after Eq.(3.3.7). Therefore, Eq.(3.3.4) continues to hold and
(3.4.2) L:noo=O bn (-z ) n w'L:- 1 ( z ) n Taking advantage of the smallness of n/w , we can expand terms in Eq.(3.3. 1) to obtain wi:l !_ n wtl { bn ( ) = n=O exp (a+ 11) 7/2 - 2n2 + 11 } since z/w 1 + d/w and d is limited. The sum is estimated by an integral as in Theorem 3.3.1 and leads to consideration of ' fw1f2 w > J: exp { (a + 11)t- �e + 11 }dt. The integrand differs infinitesimally from exp(at - �t2) for limited t. It I S bounded by exp{(a + 1)t- �t2 + 1 } for any t. Hence, by Theorem A.5.5, n w � (1 + 11 ) [exp(at - �t2 )dt (3.4.3) �\ !_ n ) ( n=O W
n=O
bn W
W
W
n=O
=
�
W
1
W
=
0
and the upper limit of summation can be replaced by oo on account of Eq.(3.4.2). For the series involving Cn in Theorem 3.3.2 note that Eq.(3.3.13) is reached without invoking \ 1 - z/w \ � 8. Treat the series in Eq.(3.3.13) in the same way as that with the but starting from Eq.(3.3.9) . There results
bn
w- 1 L cm ( z /w) m-w+ I w > (1 m� =
1
+ 11) Jrnoo exp( -at - �t2 )dt. 0
(3.4.4)
Combining Eq.(3.4.3) and Eq.(3.4.4) we have
cp ( z)
since waw - I
=
(1
=
+ 11)awz. The evaluation of the integral is immediate and
On substituting Stirling's formula for the factorial the requisite expression is derived and the theorem is proved. •
Section 3.4
61
Asymptotic behaviour of an entire function
It is convenient to abuse notation a little and denote by a the quantity z obtained by replacing m by z in the definition of llm · When z = w + d then az (1 + 11 )aw by virtue of Theorem 3.2.2. With this convention Theorem 3.4.1 can be stated as (3.4.5 ) =
for unlimited z in the neighbourhood of the real axis. Thus, Eq. (3.4.5) furnishes the asymptotic performance for such z for an entire function whose coefficients have the appropriate behaviour. When cp(z) = ez , an = 1 and a 0; there is no disagreement with Eq. (3.4.5 ) . As another check on Eq. (3.4.5) , suppose that cp(z) is the confluent hypergeo metric function M (b, c, z) defined by =
b z + b(b + 1) z2 + M(b, c, z) = 1 + -1 c 1 . c (c + 1) 2.1
Here
· · · .
(b + m - 1)!(c - 1)!/{(b - 1)!(c + m - 1)!} 1 + (1 + 11)(b - c)/m when m is unlimited but b and c are
am =
so that am+I /Ilm limited. Consequently, the conditions of Theorem 3.4.1 are met with a Taking advantage of Stirling's formula we have, from Eq. (3.4.5 ) , =
=
0.
M(b, c, z) (c - 1)!zb- cez( l + 11)/(b - 1)! for unlimited z close to the real axis when b and c are limited. =
This agrees with the usual asymptotic behaviour quoted for the confluent hypergeometric function. Since Eq. (3.4.5 ) reproduces the correct formula for the confluent hypergeo metric function it will do the same for any function defined in terms of M. For instance, for the modified Bessel function lv (x) it gives
lv (x) for unlimited positive
'Y(b, x)
(�xr e-"' M(� + 1 + 2v, 2x)/v!
(1 +
v,
11)e"' /(27rx) !
x and limited
v.
Likewise, we have
[ tb- l e-tdt = xb e-x M(1, 1 + b, x)/b
(b - 1)!(1 + 11 ) for unlimited positive x and limited b. Both these asymptotic formulae are in conformity with standard results, though that for 'Y(b, x) is an immediate
consequence of the integral representation without any need for the intervention of M.
62
Series
Chapter 3
The sums of other series can be deduced from the foregoing. From Eq. (3.4.5)
Both Theorems 3.3.1 and 3.3.2 are valid when z =
-w. Hence
Addition and subtraction of these two equations supplies
H1 + 11)aw exp(w + a2 /2), 00
L a:m. 1w2m+ I /(2m + 1 )! m=O + These formulae can be rewritten in a somewhat more convenient form by adopt ing the convention of Eq.(3.4.5) . Theorem 3.4.2 If for unlimited
m then 00
L A.nx2m /(2m)!
00
m=O
L A.nx2m+ l /(2m + 1)!
m=O
!(1 + 11)A �., exp(x + a2 /4), �(1 + 11)A! <x- I ) exp(x + a2 /4)
for unlimited positive x. As an application consider again the modified Bessel function the series representation
lv (x). It has
lv (x) = L Ox) v+2m /{m!(v + m)!}. m=O After extraction of the factor (!xt the series is in the form of Theorem 3.4.2 with Am = m!(v(2m)! + m)!22m Clearly Am satisfies the condition of Theorem 3.4.2 and, indeed, it can be checked that Am = (1 + 11)/rr!mv+ ! when m is unlimited while v is limited. 00
Hence
63
Asymptotic behaviour of an entire function
Section 3.4
in harmony with w hat has been derived earlier. Example 3.4.1 The Airy function
The Airy functions have the representations
-
f(z) g( z) , {f(z) + g(z)}3 112
Ai(z) Bi(z) where
(
f(z) g(z )
=
)
z3 1.4 6 1. 4. 7 9 1 1 a 1+ 9·, z + · · · , 6 ., z + 3., + (-3)!3a 1 2 4 2.5 7 2.5.8 10 .. z+2 ! ,. z + 7,· z + 4 10 ·' z + . . . ( -3)!3a
(
)
At first sight these series do not app ear to conform to our earlier pattern. However, after the substitution x (3w/2)213 , f(x) becomes a series of even p owers of w in which (m - V !(2m)!33m t Am (3m) !7r22m+I on observing that (- �)!(- �)! = 27r/3 L For unlimited m =
-
_
Am
=
( 1 + 11 ) ;32/37r 1/2m1/6
so that Theorem 3.4.2 can be applied with a = 0. Hence (1 + 11) exp(2x � /3) 2(37r) � x �
(3.4.6)
for unlimited p ositive x. A similar procedure may be followed for g(x)/x and it is found that g(x) is given by Eq.(3.4.6) also. These results are not much help in determining the asymptotic behaviour of Ai(x) because the dominant terms cancel. This is not too surprising since it has been shown already in Section 3.1 that Ai(x) is exponentially damped (see Eq.(3.1. 7)). In contrast, Eq.(3.4.6) is useful for Bi(x) and Bi(x)
=
1 + 11
---y--y-
3
7r 2 X4 exp(2x2 /3)
for unlimited positive x. It must be born e in mind that the formulae which have been established are based on w being unlimited. When it comes to numerical values with w in the region of 10 it is n ot so clear that a theoretically infinitesimal quantity will be truly negligible. We can expect the theorems of this section to be tolerably
64
Chapter 3
Series
accurate but it is not so obvious that we can have the same confidence in Theorems 3.3. 1 and 3.3.2 as z approaches w because of the consequent small denominator. Therefore, it is of interest to investigate how Theorems 3.3. 1 and 3.3.2 fare numerically as the phase and magnitude of z vary. Multiply the equations in those theorems by factors so that the right-hand side becomes 1 + 11 . Then the left-hand side should be about 1 for all z and w satisfying the conditions of the theorems. As a test values of the ratio for cp(z) = e' were calculated for w = 10 and w = 20. These values are pretty close to the margin when regarded as unlimited numbers and form a severe trial for the theorems. Table 3.4. 1 gives some results for positive real z. As can be seen Theorem 3.3.1 is better than 10% accurate when z is less than w/2 but the performance deteriorates rapidly for larger z. In contrast , the accuracy of Theorem 3.3.2 improves as z increases, being better than 10% when z exceeds 2w . To put it another way, for these relatively low values of w, z must be a reasonable distance away from w to secure accuracy. Table 3.4.1 Ratio for real z Th. 3.3.1 Th. 3.3.2 z w 10 w = 20 z w = 10 w = 20 2 0.97404 0.992 19 11 0.28524 15 0.71850 5 0.87764 0.98027 9 0.34794 0.94174 21 0.87284 0.22158 15 0.74612 25 0.91022 0.64321 19 0.25379 30 0.93541 0.81550 40 0.95942 0.91826 =
The degeneration in performance is caused by z approaching w with insuffi cient terms in Theorem 3.3.2 to offset 1 - z/w being nearly zero. The situation is much better when there is no possibility of z coalescing with w. In Table 3.4.2 the same ratio is calculated for a purely imaginary z. Now the performance is highly satisfactory even when l z l = w. Table 3.4.2 Ratio for imaginary z Th. 3.3.1 w = 10 w = 20 z 2i 1.00580 - 0.01683i 1.00073 - 0.00488i 5i 1.02894 - 0.02743i 1.00495 - 0.01026i 10i 1.05547 - 0.00398i 1.01533 - 0.01294i 15i 1.02350 - 0.008 13i 20i 1.02633 - 0.00078i
65
Partial sums of integrals
Section 3.5
z l Oi 15i 20i 25i 30i
Th. 3.3.2
w = 20 10 1.05497 - 0.00387i 1.04758 + 0.02018i 1.03376 + 0.02740i 1.02633 - 0.00078i 1.02400 + 0.02766i 1.02521 + 0.00540i 1.01763 + 0.025941i 1.02240 + 0.00942i w =
Functions which are not entire are not covered by Theorems 3.3.1 and 3.3.2. Notwithstanding, there is a theorem similar to Theorem 3.3.2. Theorem 3.4.3 If izi > 1 + 8, where 8 is positive and standard,
w-1 m L amz = (1 + 11 )aw zw /(z - 1) m=O
for unlimited w. Proof. Write
- 1 m w-1 m- w 1 wL: d z +! w aw- ! z - ! m=O amz = mL: =O m with dm = �/aw - l · When m is standard lzl m-w+ I is sufficiently small to make the term infinitesimal. Hence, there is an unlimited integer M possessing the properties M /w 0 and M L dm zm-w + ! 0. m=O Also w - M -2 w-1 L dmzm-w+ ! = L dw- 1-nZ- n . n=O m=M+ l For limited n, dw - n- ! 1; in addition, for any n, (1 + n ldw - 1 -nz-n l � 1 + 118 ) Since (1 + 11)/(1 + 8) < 1 this is a convergent sequence. Hence, by Theorem A.5.9. w-�-2 ) (1 + 11 )z . -n � ( 1 w-1_ L-, dw -!- n Z M ! Z 1 1 Z-1 / z n=O -=-
-
�
�
�
_
_
_
_
_ _
The proof can be completed now in an obvious manner. • 3.5
Partial sums of integrals
The theorems which have just been established permit estimates of the partial sums of
f (z) =
00
L a mzm .
m=O
66
Chapter 3
Series
The early terms have been dealt with in Theorem 3.4.3 so that the late terms are of concern here. Theorem 3.5.1 If f(z) can be continued analytically across the arc of its circle of converyence at ei {a standard and 0 < a < 21r) there is M' > 1 such that, for lzl :S M' and ph z a, =
f(z)
=
w- 1 a L amzm + (1 + 11 ) � zw J(w, -z/w) W. m=ll
where
J (J.t, z) =
and w is unlimited.
looo -t�'e--t dt o
1 + zt
The theorem can be stated alternatively as
(3.5.1) which complements the result in Theorem 3.4.3. The restriction on a is to ensure that J does not have a singularity on the interval of integration. Note, however, that, if the condition in Eq.(3.5.2) below holds for a = 0, it is valid for any a because all the terms in cp(z) have the same sign when a = 0. Proof. The discussion of Section 3.2 indicates that the assumption on the ana lytic continuation of f is equivalent to assuming that (3.5.2) for t ;::: 0 and M
1 with ph z = a - f3 where 0 < a < 27r and 1!31 < �7r - 8,
rooe'�
lo
e - t 1 + li) . ( ) 1-z Choose li < M' and then
n 1oo e-t mL00=n am (Et)mdt anEn 1oo (11+-11Et)t e-tdt + 100 e-tj(Et)dt. =
0
0
M'/1-t) dt = E m!amEm>. + an En>. (l + 1-1. )J(n, -E>-) Jo m=O for n E N , stating any conditions imposed on a. 6. If b 2: 0 prove that
roo
roo
t�"e-zt dt lo 1 + bt
1 (�-t, b/z) = zP-+J
when z is a positive real number. Deduce by analytic continuation that the relation holds for !Ph z l � 7r/2 - 6 with 6 standard and positive. If the upper limit is changed to ooeia with l a l � 7r - 6 show that the right hand side is unaltered provided that Ia + ph zl � 1r/2 - 6 and I ph zl � 7r - 6. 7. If a m is as in Exercise 1 and -1 and Show that, i f E i s infinitesimal with ph E = a ,
for n E N where
11
2: 1 .
Exercises on Chapter 3
Section 3.7
77
Is there a generalisation to an integral of the type in Theorem 3.1.6? 10. If l f (tei"') l < AtP + B for t 2: 0, 0 < a < 27r and E is infinitesimal show that, for n E N ,
where ph E
=a-
fJ and
oo+i!'l eJJ-t- t2/2 dt. 1 + Eet -
K (fJ, f.L, E) = J oo+i!'l
--
Chapter 4 UNIFORM ASYMPTOTICS
4.1
A simple example
To introduce the topic of this chapter we consider I (z) As l z l
---> oo
with Iph z l �
but, as R (z )
--->
- oo
=
l e -ztdt = (1 - e-z )jz.
7r/2 - 6
,
I (z)
I (z )
�
(4. 1 . 1 )
1/z
(4. 1.2)
- e- z jz.
(4.1.3)
�
The two asymptotic forms are quite different and neither is valid when z tends to infinity along the imaginary axis. Here the full expression of Eq.(4.1 . 1 ) must be retained because both parts are bounded and of the same order of magnitude.
Thus, the asymptotics of the function I ( z ) depend crucially upon the value of ph z. As ph z increases through 7r/2, the approximation for I ( z ) switches from Eq. (4.1.2) to Eq. (4. 1.3) after passing through Eq.(4.1.1). This is an illustration of a Stokes ' phenomenon since Stokes was the first to observe such behaviour. The imaginary axis , where the switch from one form to another takes place, is known as a Stokes line.
A formula like that of Eq.(4.1.1) which is valid on both sides of a Stokes line is said to be a uniform asymptotic representation, being uniformly valid as ph z varies. Usually, the transition across a Stokes line will be more complicated than in Eq. (4.1.1) but Eq.(4.1.1) serves as a reminder that behaviour in the neighbourhood of a Stokes line can be simple.
4.2
The function
J
The function
79
80
Uniform Asymptotics
Chapter 4
has occurred in several places in Chapter
3
and the approximation
(4. 2 . 1 )
J(f.L, E) = f.L!(1 + 111 )/(1 + f.LE)
has been deployed when E is not near the negative real axis. When E crosses the negative real axis J changes discontinuously so that the negative real axis constitutes a branch line of is specified by
J
and its analytic continuation. The discontinuity
J(f.L, E) - J(f.L, E e- 2"'i ) = 27rie"'iJJ-e 1 1• jEil-+ 1
or, more generally, by
27ri ..�"'i sin kf.L7r 1 1, · J( f.L, E) - J(f.L, Ee_ 2k"'i ) = --e�. -- e fll-+ 1 Slll f.L7r for integer
(4.2.2)
k.
Sometimes it is more convenient to handle
Jo(f.L, Z) =
laoo tll-e-t dt --
t+z
=
J(f.L, 1/z)/z.
(4.2. 3)
From Eq. ( 4.2.2 ) J.
o (f.L, Z) - J.o (f.L, ze- 2k?ri ) = -27rie -
Jl-k'lri sin kf.L7r ll. --z e . Slll f.L7r z
(4.2.4)
There are no restrictions on Eq. ( 4.2.4 ) so long as z = 0 is avoided. These relations show that Eq. ( 4.2.1 ) is not an adequate description of J near
the negative real axis. To cover this neighbourhood a formula is needed for which is uniformly valid as E varies near the negative real axis. In fact, it is more convenient to discuss
J1 (f.L, z) =
lao tll-e-t dt = Jo(f.L, ze"''.) = -J(f.L, e -"''. /z)/z 0
-
t-z
J
(4.2.5 )
in which the uniform behaviour near the positive real axis is desired. If ]1 is determined for !Ph zl < 1r then Jo(f.L, z) is known on the sheet 0 < ph z < 27r which includes the negative real axis. For applications in asymptotics f.L is generally large so the restriction f.L > 0
will be imposed.
Make the substitution
(4.2.6 ) To determine r ( t) the square root of the left-hand side has to be specified. The left-hand side increases from -oo to 0 as t goes from 0 to f.L and then decreases
The function J
Section 4.2
81
t increases to infinity. Therefore, take r(t) < 0 when t < f.L r(t) > 0 when t > f.L · With that convention the expansion of r (t) around t to -oo as
IS
r(t)
and
=
f.L
t - f.L - (t - f.L) 2/3f.L + 7(t - f.L)3 /36f.J,2 - 73(t - f.L)4 /540f.J,3 (4.2.7) - 1331(t - f.L)5 /12960f.L4 + O { (t - f.L)6} .
Then
where
r has been
written for
r(t)
and
(4.2.8) The pole at
t=z
can be accommodated by writing
1 dt t - z dr
B + C + k(r) r - r (z)
( 4.2.9)
where k(r) is a regular function of r such that k(O) = 0. The determination of r(z) from Eq. (4.2.6) is more elaborate than that of r(t)
because z is complex. We wish to arrange that r(z) ---> r(t) as z ---> t so that r(z) is suitably continuous. The formula of Eq. (4.2.7) discloses that the imaginary part of r(z) has the same sign as the imaginary part of z when z is near f.L ·
If this is true for general z then the rule for fixing r(z) is known-the sign of has to be the same as the sign of I(z). Suppose that ph z = (} and r(z) = r1 + ir2 where r 1 and r2 are real. From
I{r(z)}
Eq. (4.2.6)
lzl cos (} - f.L - f.L ln(lzl /f.L), f.L(Izl sin (} - f.L8).
(4.2.10) (4.2.11)
As (} ---> 0 , r1r2 ---> 0 and r� - r� i s positive s o long as lzl =f. f.L · Therefore r2 -+ 0 i .e. r2 is small when IBI is small. Indeed, for small positive 8, r1r2 has the same sign as lzl - f.L· To satisfy r(z) -+ r(t) as z -+ t we must have r 1 > 0 when lzl > f.L and r1 < 0 when lzl < f.L· Consequently, r2 is positive for small positive 8. In other words, r2 has the same sign as I(z) when z is just above the positive real axis. A repetition of the argument with (} negative reveals that this is still valid for z just below.
Now allow (} to grow to 1r; r2 must remain positive unless it passes through a zero. Eq.(4.2. 1 1 ) shows that this is impossible if lzl < f.L· When lzl 2: f.L, the right-hand side of Eq.(4.2.11) increases as (} increases until it reaches a maximum at (}
=
Bm where cos Bm
= f.L/ lzl
and then decreases to a negative value at
Chapter 4
82 (}
=
Any zero occurs where (} >
1r.
Bm.
But the right-hand side of
is a diminishing function of (} and is negative at (}
1"2
cannot possibly vanish for (} >
sign has been se cured.
z is below the
Bm
Uniform Asymptotics
= Bm
since
and confirmation that
A similar argument comes to
r2
lzl
Eq. (4.2.10) f.L· Hence
>
does not change
the same conclusion when
real axis.
r(z) is that its imaginary part should have the I(z). The constant B i n Eq.(4.2.9) must ensure that the singularity at r = r ( z ) balances that at t = z. In terms of t Thus , the rule for choosing
same sign as
[r 2 - {r(z) } 2] /2f.L = f.L ln (z/t) + t - z 1"::1 (t - z )(1 - f.L/z) so
that
B = 1.
r - r(z)
1"::1
By putting
f.L(t - z)(1 - f.L/z)jr. Since dt/dr = rt/f.L(t - f.L) r = 0 (t = f.L) we see that
we
infer that
1 1 C = - + -- . r(z) f.L - z For
I{r(z)} < 0
e-r2 /2JJ 100 --dr -
oo 1" - r(z)
100 e-r2/21J- roo eiy{r-r(z)}dydr Jo -oo e-i(27rf.L)� lao iyr(z ) - JJ-Y2f 2 dy -i
after interchanging the order of integration. The last integral can be expressed in terms of the complementary error function erfc (w )
=
2 7[" 2
1
[ e-y2dy w
joo --e- r2f2JJdr = - 7rie -{r (z)}2 /21l- erfc{ir(z)/(2f.L) � } .
with the result that
- oo 1" - r(z)
For
I{r(z)}
>
0
change the sign of
i
throughout and use the fact that
erfc( - z) = 2 - erfc(z). The net effect is that
(4.2. 1 2) where
H(x)
is the Heaviside step function introduced in Section
3.5.
Section
The function J
4.2
It follows from
Eqs. (4.2.6)-(4.2.9)
and
83
Eq.(4.2.12)
that
J1 (f.L, Z) = 7rizl' e -z [2H{I(z)} - erfc{ir(z)/(2f.L) � } ] 1 1 2 + ev oo k (r)e- r f 21L dr. + (27rf.L) � ev - + 1' (z ) f.L - z (4.2. 13)
{
--
}
j
- 00
The final integral of Eq. (4.2.13) is expected to rnake a smaller contribution than the pole which has been accounted for already. An assessment of its influence can be arrived at by noting that VI a
k (r) is regular and can be expanded
k (r) = L AprP. p=l
I: k (r)e-r2 f21Ldr = �(p - �)! (2f.L)P+ � A2P .
Hence
The coefficients
Ap can be determined in a number of ways from Eq. (4.2.9).
Derivatives with respect to 1' may be taken and then 1' made zero. The deriva tives can be found analytically or by taking advantage of the symbolic facilities of MATHEMATICA or MAPLE. Another way of reaching the structure of the
coefficients is to observe that there is no singularity in Eq. (4.2.9) as 1' and r(z) vary. In particular, there is none when both 1' and r(z) are zero. But the nth derivative of the term containing B will involve {1/r(z) } n+I when 1' = 0. A singularity would be created when r(z) = 0 unless it was cancelled by a cor responding singularity from the left-hand side. Hence the left-hand side must produce the singular part of the expansion of { 1/r(z) } n + l in powers of f.L - z. Whichever method is adopted it is found that
1---: { _T'(z)1_ } 3 + (f.L 1 z)3 - f.L(f.L -1 z)2 + -:-::-' -;:-:2 (f.L-- - z) 12f.L { r(z)1 }5 + (f.L -1 z) - 3f.L(f.L5- z)
(4.2.14)
5 4 1 25 1 (4.2. 1 5) + + -,:-------;---; "'C"' 36f.L2 (f.L - z)3 36f.L3(f.L - z) 2 864f.L4 (f.L - z) ' 1 77 1 7 7 + + 2 r(z) (f.L - z)1 3f.L(f.L - z) 6 4f.L (f.L - z) 5 180f.L3(f.L - z)4 49 1 139 (4· 2 · 16) + 4320f.L4 (f.L - z)3 - 4320f.L5 (f.L - z) 2 777600f.L6(f.L - z) ·
{
}7
When z is near f.L the forms of Eqs.(4.2.14)-(4.2.16) and the formula for C are not very suitable for computation because of the numerous cancellations
Uniform Asyrnptotics
Chapter 4
84
which take place. Expansions for z near p, are 1 1 -+ r(z) p, - z
RJ
--
A2
RJ
A4
RJ
As
RJ
353 1 23 z - fJ (z p,) 3 (z p,) 2 + 12960p,4 3p, - 12p,2 540p,3 81083(z - p,)5 7783(z - p,)6 589 + (z - p,) 4 + 653184p,7 ' 5443200p,6 30240p,5 (4.2.17) 1 z - p, 23(z - p,) 2 3733(z - p,)3 - 540p,3 - 28 8p,4 + 6048p,5 - 1088640 6 ' (4· 2 · 18) p, 25 139(z - p,) 259(z - p,) 2 7717(z - p,) 3 + + 155520p,6 466560p,7 22394880p,8 ' 18144p,5 (4.2. 19) 101 2016373(z - p,) 2 571 (z - p,) + 5542732800p,9 2332800p,7 37324800p,8 194036993(z - p,)3 (4.2.20) + 4655895552000p,10 0
Notice that these formulae imply that A2 is of the order of 1/p,2p+ I whether P p, - z is small or not. In so far as p,-dependence is concerned, the terms in the integral of k(r) diminish like 1/p,P+ t and so the series can be expected to furnish suitable asymptotic behaviour for large p,. The final result is
lt (fJ, z) (4.2.21) The purpose in deriving Eq. (4. 2. 21) was to provide an expression for J1 which held uniformly for z near the positive real axis especially when p, is large. Ac tuallr, it is valid over a much wider region of complex z. Suppose that I(z) < 0 and l r(z)/(2p,) � » 1. Then the asymptotic expansion
�
erfc w
�
e�w2 1r 2 w
{
1+
E (p -
p=I
P!(-)P }
7r 2 w2P
(lph wl
�
7r/2 - 6)
(4.2.22)
together with Eqs. (4.2.14)-(4.2. 16) shows that all the terms involving 1/r(z) cancel and
Section 4.2
J
The function
85
for large f.L· By means of Stirling's formula
which is the same as would be deduced from Eq.(4.2.1) and Eq. (4.2.5) when z is not close to the positive real axis. The same conclusion can be drawn when I(z) > 0 by using erfc( -w) = 2 - erfc(w) . Pertinent to the meaning of z�'- in Eq.(4.2.21) is that it has been assumed that Jph zJ : (p,) f' (p,) + + + + 216p,2 72p, 72p,3 18144p,5 864p,4 f(o)(p,) + {973/(p,) - 1500p,f' (p,) + 630p,2f'' (p,) + 120 +5040p,3!"' (p,) + 31500p,4f(4> (p,) + 15120p,5 f(5> (p,) -
}
{
+1512p,6f(6)(p,) }
B6
�
�
��
(4.2.28) 10 8 p,6 ' {707f(p,) - 2919p,f' (p,) + 1890p,2!" (p,) + 30870p,3f111 (p,) +291060p,4f(4) (p,) + 238140p,5f (5) (p,) + 52920p,6f (6> (p,) +3240p,7f(7>(p,) } /16329600p,7 + {3997f(p,) + 11312p,f' (p,) -23352p,2 !"(p,) + 10080p,3/111 (p,) + 123480p,4f(4> (p,) +931392p,5 f (o) (p,) + 635040p,6/ (6) (p,) + 120960p,7 !(7) (p,) +6480p,8f(8l (p,)} (z - p,)/261273600p,8 , (4.2.29) f" (p,) f(p,) f(p,) f'(p,) (z p,) + + + f'(p,) + 3p, 2 12p,2 3p, 23/(p,) !' (p,) !" (p,) !111 (p,) 2 (z - p,) . (4.2.30) + + + 6540p,3 12p,2 � -
{-
{
}
}
_
The formula analogous to Eq. (4.2.21) is
l"" f (t) -dt t�' e-t 0
t-z
�
7rif(z)z�'e-z [2H{I(z)} - erfc{ir(z) / (2p,) � } ] + (27rp,) ! ev
{ fr(z)(z) + p,f(p,)- z } +
+ ev L(p - ! ) !(2p,)P ! B2p · p=l
(4.2.31)
Sometimes it is more convenient to replace the erfc by another representation, namely erfc(i ) = where
w _3,.ew'+,.if4F(we"if4) 7[" 2
(4.2.32)
87
Pole near a saddle point
Section 4.3 The function
F( w)
has the property
F( - w) = 7r � eiw2 -tri/4 - F(w) and, when
lwl
7r/2 > ph w > - 1r, i 1 1 F(w) = - - + -3 + 0 -5 2w 4w lwl
is large with
(
(4.2.33)
)
.
(4.2.34)
Pole near a saddle point
4.3
The asymptotic behaviour of an integral with a saddle point has been considered in Section
2.4.
When the integrand contains a pole, which may be near the
saddle point , further examination is necessary in order to acquire an answer which is uniformly valid.
In Section
2.4
the saddle point lies on a complex
contour and the essence of the method there is to convert the integral into one along the real axis. Here it will be assumed that any such transformation has been accomplished so that consideration can be limited to integration along the real axis.
Of course, the contribution of any pole captured during the
deformation must be included in the final result. The integral to be considered is
where
c
and
b
are real.
It will be supposed that
x
is an unlimited positive
h( t) has a saddle point at t = a where a is not near c or b. Further, h(t) - h(a) will be taken to be positive when t =/= a and bounded away from zero at the endpoints c and b. The function f(t) will b e assumed to be regular and the pole t = p does not lie in the interval ( c, b) though it may be close to a. Our goal will be confined to finding out how the contribution of the number and that
saddle point is affected by the presence of the pole. Make the change of variable
h(t) - h(a) = �Au2 A = h"(a) , is positive and not infinitesimal. Take u > 0 when t > a and u < 0 when t < a so that u Ri t - a when t is near a. Then , as in Section 2.4,
where
the dominant contribution to
I � e- xh(a)
I is
joooo (t J-(t)Au exp(-1Axu2)du p)h'(t) 2 -
Chapter 4
88
Uniform Asymptotics
where here � signifies that smaller terms are being neglected. As in Section 4. 2 put
f(t)Au (t - p) h' (t )
=
_
B_
U - Up
+ C + k(u)
(4 .3. 1 )
where
�Au!
=
h(p) - h(a)
on the understanding that Up RJ p - a when p is close to a. This makes I(Up ) have the same sign as I(p) when p is near a but, unlike Section 4. 2 , we cannot infer that this is true for other positions of p without more detailed information on h(t). From Eq. (4.3.1) B
= J (p) , C
=
J(p) + f(a) . a -p Up
The function k (u) will be ignored from now on because the analysis of Section 4. 2 indicates that it will be of lesser importance than the terms which are retained. Hence
(4.3.2) or
in the notation of Eq.(4.2.32). When p is not near a, uP is not small. Therefore, the largeness of Ax entails, via Eq.(4.2.34) and Eq. (4.2.33) ,
I�
( )
f(a) 27f ! e -xh(a) _ a - p Ax
This is the same result as in Example 2.4.2 of Chapter 2 for the contribution of a saddle point when there is no pole nearby. Thus, as p moves about Eq. (4.3.2) and Eq. (4.3.3) provide a smooth transition from the behaviour of a saddle point alone to the combined effect of saddle point and pole i.e. it is uniformly valid.
Section 4.4
4.4
Saddle point near an endpoint
89
Saddle point near an endpoint
The discussion of the saddle point in Section 2.4 was based on the saddle point being either at an endpoint of the range of integration or well away from the endpoint. It is time now to consider what happens when the saddle point is in the vicinity of an endpoint. A typical case is the integral
where x is positive and unlimited. It will be assumed that f.L > - 1 and that f is differentiable as many times as desired. The function h(t) is supposed to be defined for negative values of t as well as positive and to possess a single minimum at t = a where h"(a) > 0 but is not infinitesimal. The value of a may be positive or negative to allow for the saddle point being inside or outside the interval of integration. In addition, the condition that h(t) -+ oo as t ---> oo will be imposed. Make the transformation
h(t) - h(a) = ! (u - b? so that u = b corresponds to the saddle point t = a. For t > a choose u > while, for t < a, u < b. Then
b
u - b RJ {h" (a)} ! (t - a) when t is near a. With regard to b it is selected to have the same sign as a and to make u = 0 correspond to t = 0. Hence
b = ±[2{h(O) - h(a)}]! according as a � 0. After the transformation
where
go (u) = f(t)
( t )IJ. dt .
; du The target now is to expand go (u) near the origin and express quantities in terms of where
90
Uniform Asyrnptotics
Chapter 4
VJL
The function can be rewritten as a parabolic cylinder function. There are two notations for the parabolic cylinder function and (Abramowitz & Stegun in terms of which
1965)
U(v,x) Dv(x) JL(x) - f.J,.e _,2 /4D-JL- I (- x) - f.L.e-x2f4U(f.L+ !2 , -x) .
v.
-
I
-
To enable the integral to be expressed via
I
� put
(4.4. 1)
go(u) = a0 + (30(u - b) + u(u- b)G0(u).
Then
ao go(b) = f(a) (�Y 1{ h" (a)}� , l f3o = {go(b) -go(o) }fb = [go(b) - J(o) { - h'�o) r+ ]j b. go(O) b h' (O) Ie"'2h(a) = xJL+1 �' (bx) + Av' x�'+2 �' (bx) + lrooo uJL+l (u - b)Go (u) e- �"'2(u-b)2(du.4.4.2) looco u�'+1 (u- b)G0(u)e - 2"'1 2 (u-b)2 du = -x21 loo00 u�'g1 (u)e- 2"'1 2(u-b)2du g1(u) = (f.L + 1)Go(u) + uG�(u). g1 g0 g1 (u) = a1 + f3t (u - b) +u(u-b)Gt (u). a1 jx2 ao f31/x2 (30• 1/x2 V�(bx). VJL(bx) .(4.4. ) I � {� (4.4.3) xJL+2 JL' (bx) } e -"'2h(a) xJL+I v.JL (bx) + Av
There is no difficulty over the interpretation of opposite sign to
because
always has the
on account of the conventions adopted. Now
� v;
Integration by parts yields
with
Treat
in the same way
as
i .e. write
Clearly, the effect is to add to and to The procedure can be repeated and generates an asymptotic series in powers of multiplying and a similar one as a factor of Evidently, the dominant part of the asymptotic expansion is given by the first two terms of Eq
2 i.e.
to a first approximation. This approximation varies continuously as the saddle point moves through the endpoint and its performance should be checked against
Saddle point near an endpoint
Section 4.4
91
earlier results for the saddle point in coincidence with the endpoint and well away from it. When the saddle point coincides with the endpoint a = 0 and b = 0. Either directly from the integral for VJL or from Eq. (4.4.1) and the known values for parabolic cylinder functions (Abramowitz & Stegun 1965)
V�' (O) = (�JL - �)!2t�'- � , V�(O) = (�JL)! (-2��') on deploying Jl! = ( � Jl)!( � Jl - �)!2�'j1r� . Hence I OJL - �)!2��'- � ao e_ "'2 h(O) jxP-+1 � (JL+I) -x2 h(O) 2 f(O) (! Jl ! ) ·1 -e 2 2 2 h" (O)x2
}
{
_
(4.4.4)
since bja tends to {h" (O)} � in the limit. Agreement with Eq. (2.3.1) of Chapter 2 is confirmed on making the appropriate changes of parameters there. For the saddle point away from the endpoint the separation will be regarded as sufficient for l bxl » 1. Asymptotic formulae for � can be introduced then. They are
(4.4.5)
as x
---->
as x ---->
oo, and
- oo .
VJL (x) � JL! e- �"'2 (-x)_�'_ \ V�(x) � JL! e- �"'2 (-x)-�' When a > 0, b > 0 and Eq. (4.4.5) is relevant; thus I � ao (27r) t ife_"'2 h(a) � X
(4.4.6)
1
} 2 f(a) a�' e-x2h(a) {� h" (a) X
which is consistent with the result of Section 2.4 for the contribution of an interior saddle point (cf. Eq. (2.4.5) and Example 2.4.2 there ) . When the saddle point is outside the interval of integration both a and b are negative; then Eq. (4.4.6) is pertinent and I �
JL!(ao - bf3o ) e - x2h(O) x2�'+2( - b) JL+I 1- JL+l e -x2 h(O) Jl!f(O) h'(O)x2
{
}
which is in harmony with Eq.(2.3.1) of Chapter 2 when there is no saddle point in the range of integration. Consequently, as the saddle point moves from outside to inside the interval of integration, Eq. (4.4.3) provides a continuous transition between values that have been secured before. Once again, a uniformly valid result has been achieved.
Chapter 4
92 4.5
Uniform Asymptotics
Coalescing saddle points
As already remarked the contribution of an isolated saddle point has been dealt with in Section 2.4. The presence of other saddle points which may not be far away produces a more complicated situation. A typical example is furnished by
J f(w)e-xh(w,u)dw in which x is unlimited positive and h depends upon the parameter u. As u varies a saddle point moves about and, on departing from one position to another, its influence on the integral may wax or wane. The possibility that it may pass through another saddle point must be taken into account. Suppose, for instance, that (4.5. 1) If u =/= 0 there are two saddle points w = ±u and they are of the second order. If lui is large enough the contribution of each can be assessed separately by means of Section 2.4. However, as lui reduces, they get closer and closer together until, when u = 0, they coalesce into a single saddle point at w = 0 which is of the third order. Since second and third order saddle points generate quite different asymptotic contributions the need for a suitable transitional formula is evident. More generally, let h(w, u) possess two saddle points of the second order which coalesce into a single saddle point of the third order when u = 0. To be specific let the solutions of 8h(w, u)/8w = 0 be Wt (u) and w2(u) where w1(u) = w2 (u) when u = 0 but otherwise Wt and w2 differ. The form of Eq. (4.5. 1) models this kind of conduct which suggests the introduction of a new variable u via (4.5.2) h(w, u) = iu3 - ((u)u + g(u) where Hh(w1 , u) + h(w2, u)} , Hh(w2, u) - h(w� , u) } . The insertion of u = (! (u) into Eq.(4.5.2) discloses that the corresponding value of w is w1(u). Likewise, w = w2(u) corresponds to u = -(! (u) . It was shown by Chester, Friedman & Ursell ( 1957 ) that the transformation of Eq.(4.5.2) makes u uniformly regular for small w and u. The relation between u and w can be regarded as one-to-one for u in the circle lui � � and u in the circle Jul � U which contains the image of the circle lwl � W. Remark that these properties are totally independent of x; they are attributes of the mapping of Eq. (4.5.2) alone.
Section 4.5
93
Coalescing saddle points
On account of these properties, f (w)dw/du can be expanded in a power series of u near u = 0. Rather than deploying a direct expansion we shall follow the procedure of Section 4.4 and write f(w)
dw du
=
go(u, u) = ao (u) + u,Bo ( u ) + {u2 - ((u) } G0(u , u) .
(4.5.3)
The last term in Eq.(4.5.3) vanishes at the saddle points and so is expected to offer a less significant contribution than the other two terms. The quantities ao (u) and ,Bo(u) are fixed by choosing u = ±(� with the result a0(u) f3o(u)
Hgo (( � , u) + go(-( ! , u) } , Hgo ( ( � , u) - go (- ( ! , u) } / ( � .
Then G0 is known and will have an expansion in powers of u near u = 0. Note that in the calculation of go
when u = ( � and u f' 0; for u = - (� change the sign of (� and replace w1 by W2 . When u = 0 when U = 0. The mapping of Eq.(4.5.2) creates a new contour of integration. To fix ideas we will assume that it can be made the path of integration in Fig. 3.1.1. Then
J f(w)e- xh(w,u)dw
[
) = 21rie -xg(u) ao � Ai(x � () + ,Bov;) Ai '(x � () +e- xg(u)
j (u2 - ()G0 (u, ()e- x(!u3 -(u)du. X3
X3
]
(4.5.4)
The integral can be converted by an integration by parts into one of the type already considered and the process can be repeated. Since the integration by parts divides by x it is transparent that the major contribution to the asymptotic development comes from the first two terms of Eq. (4.5.4). The definition of Cl2 (u) does not specify ( � (u) uniquely; three possibilities are open to it. Each of the choices can be expected to lead to a different contour of integration in the u-plane. Only one of these should be the contour selected in the preceding paragraph. To identify the corresponding ( � ( u) the argument runs as follows. Let u = 0 entail w = w0 in Eq.(4.5.2). Let the point w near w0 map into u neighbouring the origin. Then, if the line segment from w0 to w
94
Chapter 4
Uniform Asymptotics
points at the original path of integration we want the segment from the origin to u to point at the selected contour i.e. 27r/3 < 2n7r + ph u < 47r/3 for some integer n. But, in a conformal mapping, a line segment is rotated positively through an angle ph dujdw. Hence, from Eq. (4.5.2) 7r 1 8h( Wo, u) 27r - 3 > ph ( 2 ( u) - 21 ph (w - Wo ) - 21 ph 8w + n1r > - 3
(4.5.5)
for some integer n. These inequalities are sufficient to fix the phase of (! ( u). An interpretation of Eq.(4.5.5) is that, if the original path were deformed to pass through w = w0, the contour in the u-plane would go through u = 0 and a tangent there would lie in the sectors (27r/3, 47r /3) and ( -7r /3, 7r/3). Example 4.5. 1 Consider where
h(w, u)
=
-i { cos w + (w - �1r) sin u}
with 0 < 6 � a � 7r/2. The equation for the saddle points is sin w = sin u with solutions w = u or 1r - u. Then �(312(u) =
i
{ cos u + (u - �1r) sin u} .
Since the quantity in { } is positive the choices for ph (! are 7r /6, 57r/6 and 37r/2. Now w0 7r/2 and =
8h(w, u) . . = Z (1 - sm u) aw when w w0. In implementing Eq.(4.5.5) w can be taken real and greater than 7r/2 so that 1 7r 57r - > ph ( 2 (u) + n7r > 12 12 =
which can be satisfied only by ph ( ! (u) = 57r/6 with n = - 1 . Thus (! (u) = e5"i/6[3 { cos a + (u - �1r) sin u} /2]113. For another disposition of the saddle points take h(w, u) =
-i { cos w + (w - �1r) cosh a}
Section 4.5
Exercises
on Chapter 4
95
with u � 0. In this case there are saddle points at w
=
� 7r ± iu and
� (312(u) = -{u cosh u - sinh u}. Again the quantity in { } is positive so that ph (�(u) can be 7r/3, 7r or 57r/3. Here ahj8w = -i(cosh u - 1) when w = 7r/2. Consequently Eq.(4.5.5) becomes -
1 77r > ph ( > (u) 12
+ n7r >
-
117r
12
which is satisfied by ph (� (u) = 7r/3 with n = -1. Accordingly (� (u) = e"if3[3(u cosh u - sinh u)/2]113. When there are more than two saddle points u should be replaced by the vector u and m saddle points are permitted. The obvious analogue of Eq. (4.5.2) is the mapping 1 2 h( w, u ) = -- um + l + x1 u + x2u + + xm -!Um- 1 + g (tr ) . (4.5.6) m+1 Clearly, the analysis will end up with integrals in which the exponent consists of the right-hand side of Eq.(4.5.6) with g ( u ) absent. Properties of such integrals are hard to come by except when m = 3 (when they are called Pearcey integrals) and when m = 4 (swallowtail integrals). Therefore the matter of multiple coalescence of saddle points will not be pursued further. · · ·
Exercises
1 . Since
on
Chapter 4
100
a t�'e-t dt - Jl (f.L, z) = az 0 ( t - z) 2 when z is not positive real, a derivative of Eq.(4.2.18) could supply an asymp totic formula for the integral on the right. Can the formula be justified? 2. Estimate the asymptotic behaviour of
1oo -t�' ln t e tdt.
0 t-z 3. Show that, to a first approximation,
Uniform Asymptotics
Chapter 4
96
where f.L = M - 1 - A. 4. Show that, if 0 : oo so that, to be of use, it has to be truncated at some value of n, say no, where the remainder Rr,0 (z) can be estimated or shown to be negligibly small. Usually an optimal n0 is derived by minimising Rn (z) for fixed z with the consequence that no is a function of z generally. Chapter 3 had examples where the optimal remainder turned out to be exponentially small. It was suggested by Stieltjes (1886) that the accuracy of the estimate for f(z) could be improved further by expanding Rr,0(z) itself in an asymptotic series. Then no n f(z) = L amrpm (z) + L alm'Plm (z) + Rln (z) m=l m=l where the asymptotic sequence { tp 1m (z)} need not be the same as { 'Pm (z)} . The additional series is truncated at n1 where R1n, (z) is optimal and expected to be exponentially small compared with Rr,0 (z) as lzl -> oo . Such a possibility was considered formally by Dingle (1973). Of course, having carried out the process once we can repeat it i.e. expand R1n1 (z) in an asymptotic sequence up to its optimal remainder R2n2 (z) and so on. The continual repetition of this procedure was dubbed hyperasymptotics by Berry & Howls (1990, 1991 ) and is intended to supply steadily more accurate estimates of f(z). The subject has been developed by several authors; see, for example, Berry (1991), Boyd (1990, 1993, 1994), Howls (1992) , McLeod (1992) , Olde Daalhuis (1992, 1993), Olde Daalhuis & Olver (1994), Olver (1991a, 1991b, 1994), Paris (1992a, 1992b), Paris & Wood (1992, 1995). 99
100
Chapter
5
Hyperasymptotics
The question of determining the optimal R,.0 (z) has been discussed in Chap ter 3. Finding the optimal remainder of a remainder is a new topic and generally involves quite complicated manipulation. The following section illustrates this point. 5.2
A Laplace integral
The integral to be considered in this section is
where x is an unlimited positive number and the limited JL > -1. It will be assumed that j possesses as many derivatives as are desired and that, for un limited n, (5.2.1) f(nl (t) -< (b + n)!K n 1 (t >- 0)
l
l (p + t) +
where K , b and p are standard constants independent of n with p > 0 . Subse quently, K will be used as a generic standard constant, not necessarily the same in all places. According to the theory already described for the asymptotics of a Laplace integral the dominant contribution is dictated by the behaviour of j(t) near the origin. Therefore, make the expansion
Then
I(x) = }3 m(JLl x+�"+mm +) l1 j <ml (o) + Ro(n' x) m=O
where
Ro(n, x) =
1= e-xtt�"+nfi(t)dt.
(5.2.2)
The first objective is to choose n ( no ) so that Ro is exponentially small in Previous experience in Chapter 3 indicates that this will require no to be of the order of x and unlimited. To fix no necessitates some knowledge of the properties of j1 (t). The usual formula for the remainder in a Maclaurin series is =
x.
1 f1 (t) = (n -1 1)! J[o (1 - ut- l f(nl (ut)dt.
Therefore
1 ur u IJ1(r) (t) l - (b(n+ n- +1 )!r) l K J[o1 (p(1 +- tuu))nn-+r+ 1d
(J.t + no + nt ) jt�. The algebraic dependence on x is the same as in Ro (no, x) since both no and n1 essentially contain x as a factor. As regards exponential behaviour that of Eq. (5.2.15) is smaller if
- x(t2 - p) + (J.t + no ) ln(t 2/tt ) < 0. Since
ln(1 + y)
0 the left-hand side is less than
t - t (t - tt) 2 - x(t2 p) + (J.t + no) 2 1 _ 22t t tl l2 2 < - (J.t + no)(t2 - tt) /2t tt2 < - 2 (J.t + no)/3 _
{
(5.2.16)
}
104
Hyperasymptotics
Chapter 5
on account of Eq. (5.2.6) and Eq.(5.2.8). Thus R1 (n1 , x) does display a satisfac tory smaller exponential behaviour than Ro(no, x). In fact, if J.l + no is replaced by px as a rough approximation in Eq. (5.2.15), the exponent is
px( -3 + ln 3) = -1 .90px which is nearly double that in Ro(no, x). Note also that, with the same approx imation, n1 is about 4(J.l + no)/3 so that a third more terms are needed to reach R1 from Ro than are needed for Ro . Moreover, these terms involve the calcu lation of integrals which are no longer straightforward factorials. The price of increased accuracy is a substantial extra computational effort. The next step is the expansion of R1 (n� , x) which entails expanding h(t) about t = t 2 . However, rather than do that, we will proceed to the general step under assumptions which are consistent with what has been found earlier. Then induction will validate the process. Take
Rm (n, x) = where
100 e -xtti"+no (t - t1 )n1
• • •
(t - tm- l tm-1(t - tm ) n fm+ I(t)dt (5.2.17) (5.2.18)
and
I J!;:l (t) l -< (no + . . . + nm- 1 + r)!pno (p + t 1 )n1 . . -. (p1 + tm- 1 )nm-1 (p + tY+ 1 . (b + no + n 1 + . . . + nm + r)!r!K
(5.2.19)
Then, from Eq. (5.2.3)
n1 . . . (p + tm )n (p + W+ 1 IJm,: sin 6 (5.4.6)
by virtue of Eq.(5.4.2). For unlimited v, Stirling's formula can be inserted in Eq.(5.4.6). Then the right-hand side is a minimum when v = p(1 - a) to within an infinitesimal. Therefore, in Eq.(5.4.3) , take n = no where
no = [p(1 - a)]
(5.4. 7)
and the remainder is exponentially small. To refine Ro note that Eq.(5.4.6) stems from a neighbourhood of u where -p(1 - a) + (J.l + v)ju1 = 0.
= u1
In view of the choice of v this means that
U1 = 1 if an infinitesimal is neglected. Consequently, the denominator in expanded about u = u1 with the result
(5.4.8)
Ro (no , z) is
Ro (no, z) = (-to rf'e- ino8 (- � l { 00 f(pu)u�'+no (u - u1)m e- pudu + RI(n, z) m=O U1 + e m+ Jo
x{'f (
}
(5.4.9)
where
n [ f(pu)u�'+ no n pu R1(n, z) - (U1(-) + e )n o u + e (u - u1) e- du. i8
i8
(5.4.10)
K . r = u!J.+ no lu - ul l v e- p( l - u)udu . IRI ( v, z )l s lui + eiBI v sm 6 lo The integrand is stationary at u = � where v J.l + no + --(5.4.11) U2 U2 - U1 - p (1 - a) = 0 so long as u2 > U1 as will be checked shortly. Then the exponential behaviour of IR1(v, z)l is essentially (J.l + no ) ln � + v ln(u2 - u1) - p(1 - a)u2 - v ln l ui + eiB I Thus
112
Chapter 5
which is stationary for
Hyperasymptotics
u2 - u1 I u1 + eiO I . =
(5.4.12)
u2 given by Eq. (5.4.12) the number of terms in Eq.(5.4.9) is fixed by n1 where n1 = [ ] being determined by Eq. (5.4. 1 1 ) . Now the denominator in R1 (nl l z) i s expanded about u = � to n2 terms where n2 = [ ] with 11- + no n1 -- + -U3 - U2 - p(1 - u ) = 0 U3 U3 - U1 + -and U3 - u2 = l u2 + eiO I ·
With n=
v , v
v
v
Clearly, the process can be continued.
and n.
=
[v] where
In
general,
Us+ l - Us = I Us + eio l
(5.4.13)
s- 1 n,. 11- + no + (5.4.14) -L.:; Us+ l - Ur + Us+ l - Us - p(1 - u) = 0. Us+ I r=l There is an alternative to Eq. (5.4.13) for determining U8• From Eq. (5.4. 13) u. + eio + Ius + eiO I 2 l us + eiO I eicf>,/2 cos(¢./2) where ¢. is the phase of U8 + eiO. Evidently ¢•+I = ¢./2. Since it is clear that u1 + ei = 2ei0f2 cos(0/2) , ¢1 = 0/2 and so ¢s = 0/2". Now Eq. (5.4.13) implies v
that
0
2 1us - 1 + eiO I cos(¢.- I /2)
0 2• cos !!.... cos --I . . . cos � (5.4.15) 2•2• 2 on repeated application. Thus, with u1 = 1 , u2 , u3 , . . . can be found from Eq.(5.4.15) . Naturally, for numerical purposes, often it will be more efficient to employ Eq. (5.4.13). Remark The condition of Eq. (5.4.2) forces f(t) to be bounded at the origin. Such a restriction is unnecessary. It is sufficient for f(t)t�' to be integrable in a neighbourhood (0, E ) of the origin and to satisfy Eq.(5.4.2) outside. The change affects Eq. (5.4.6) only infinitesimally so that the succeeding argument is unaltered. Example 5.4.1 Choose f(t) = Ko(t) where Ko is the modified Bessel function and 11- = -�. Then it can be shown that
Section 5.5
113
Stokes' phenomenon
According to the remark above the preceding theory can be applied with u = -1 since K0(t) decays exponentially for large enough t. Now F(5i) = 0.0192710158058762 - 0.78538578788912lli to 16 places of decimal and the approximations are
0.0192752428015537 -0. 7853904081417598i 0.0192710242449071 -0. 7853857917780257i 0.0192710158078479 -0. 7853857878949455i
Correct-Approx -4.227 x w-6 +4.620 x w -6 i -8.439 x w -9 +3.889 x w - 9 i - 1.972 x w - 12 + 5.824 x w 12i
no = lO
-
Notice that, in contrast to the Laplace integral, the value of nm does not necessarily increase with m. As z approaches the negative real axis the integral tends to become singular and the approximations are likely to become less accurate. An indication of this happening can be seen by taking z = 5(i - 1)/.J2. Then F( z) = -0.5547417804 - 0.5779236647i and the approximations are
-0.5547518491 -0 .5779268407i -0.5547419221 -0.5779242095i -0.5547419520 -0.5779233259i
5.5
Correct- Approx 0.0000101 +3. 176 x w 6 i 1 .417 X 10- 7 +5.448 x w -7i 1 .716 x w - 7 -3.388 x w - 7i
no = lO
-
Stokes' phenomenon
The expansion of F(z) in Section 5.4 was limited to the sector lph zl < 1r. One reason is that, if the contour of integration cannot be moved from the real axis, the integrand of the Stieltjes transform becomes singular when z is on the negative real axis. However, if f is suitably regular in a sector including the positive real axis it is possible to shift the contour and so supply an analytic continuation of F(z) across the negative real axis. Of course, the deformation effectively captures a pole when z is near the negative real axis (compare Section
114
Hyperasymptotics
Chapter 5
4.2) and this extra contribution, although exponentially small, can be of the same order of magnitude as the remainders considered in Section 5.4. Moreover, as the phase of z increases from 7r (or decreases from - 1r , the exponential decay becomes less pronounced and the extra term may well have a dominating influence on the asymptotic expansion of F(z). In other words, the negative real axis will be a Stokes line. The result is that a more complicated analysis than that of Section 5.4 is necessary to accommodate the asymptotic behaviour of F(z) for z in a neighbourhood of the negative real axis. The starting point is the expansion of Eq.(5.4.3), namely
)
(5.5.1) where Ro(n , z)
=
(
)n 1"" J(t)tl-'+n e-tdt. 0 z t+z
-=--n
(5.5.2)
The extra term referred to above is contained in Ro and, to take care of it, we shall make certain assumptions about It will be supposed that, for a2 � ph � a1 where ?r/2 - 6 � a2 > 0 > a 1 � -?r/2 + 6 (6 being a small standard positive number), = (5.5.3)
f(t). f(t) fo(t)e"t
t
with u < 1. The function fo is regular in the specified sector except possibly at the origin and satisfies 2
I o(m) I .S
f (t) (b('R-1 t)+m)! +m K2 +m K1 + ((b'Rt)+f3m)!
(5.5.4)
where 0 < f3 .S a < 1 . The term in K1 is to allow for a possible singularity at the origin whereas that in K2 governs what happens at infinity. It will be seen later that more freedom can be permitted to f3 than has been prescribed so far. Now rewrite Eq.(5.5.2) as Ro ( n , z)
=
(5.5.5) The second integral can be expressed in terms of the function J0(p,, z) discussed in Section 4.2 and defined by Jo(p,, z) =
1"" t�-'e-t dt. 0 t+z -
115
Stokes' phenomenon
Section 5.5
As for z it wi ll be sufficient to examine what occurs when ph z passes through since the case when ph z is near -7r can be dealt with in a similar manner. Therefore, take z = pei(1r - ¢>) where a2 :2': -¢ :2': a . Then
1r
1 -i e (cf>) t fo(p Ro (n , z) = zn ( 1 - u)l-'+n J0{J.t + n , (1 - u)z} + r0 (n , z)
(5.5.6)
where (5.5.7) and
g(t) =
fo(pt) - fo(pe- icf>) . t - e-•cf>
(5.5.8)
Since g (t) is regular at t = e-icf> the formula of Eq.(5.5.6) supplies an analytic continuation of Ro(n , z) covering the sector 1r + a2 :2': ph z :2': 1r + a1• In order to have an estimate of the magnitude of ro it is of assistance to bound a certain integral. If c > 0 , a > 0, n > 0 , m :2': 0, 0 < 1 < 1 the substitution
) -c_,_ ( 1_- v 'u = --,c(1 - v)...,.+.---' av
gives
[ 1 (1 - ut- J umdu Jo { c(1 - u) + au }'Y+m+n = c'Y+n-J1 a'Y+m Jof 1 vn-J (1 - v) m {c(1 - v) + avp- J dv (n_---1 ),_---' ! (,_m_+.:..._l --1-'-:-)!_ 0 as � -+ a1 ensure that W(O is close to e{ as � -+ a 1 ; they enforce A = 1 , B = 0. Consequently
e(�) = � [{ (1 - e2v - 2{)'1/J(v){1 + e(v) }dv Jo ,
(6.2.1)
Section 6.2
121
An error bound for the WKB approximation
provides a solution of Eq. (6.1.6) which is near to e{ when � is in the neighbour hood of a1. Solving the integral equation in Eq. (6.2.1) is accomplished by iteration in which successive estimates are related by (6.2.2) If en tends to a suitable limit as n -> oo then the scheme of Eq. (6.2.2) will solve Eq. (6.2.1). The iteration is started with eo(O = o so that (6.2.3) Then
en
can be estimated via n en (�) = L {em (O - em-1 (�)} m=l
(6.2.4)
where (6.2.5) from Eq. (6.2.2). Bounding the integrals in Eqs.(6.2.3) and (6.2.5) is not quite straightforward because a 1 , � and 'ljJ may be complex in general. Identify points on the path of integration by the arc length s measured from a 1 . Let s = s0 when v = � - Then the integral in Eq. (6 . 2.3) can be expressed as
ro (1 - e2v-2{ ) '1/J (v) dvds ds.
lo
j
1
If n(v) does not decrease on the contour n(v - 0 ::; 0 and 1 - e2V- 2{ ::; 2. Accordingly (6.2.6) 1'1/J(v) l ds. leJ (�) I :S Now suppose that, for some
m,
{o
m = 1 by virtue of Eq.(6.2.6). From Eq.(6.2.5) lem ! (� ) - em (� ) l < lo 1'1/J (v) l { l i 'I/J (v) l ds r ds1 m! + l < {{0 1 '1/J (v) l ds r+ / (m + 1 ) !
which is true for
(6.2.7)
Differential Equations
Chapter 6
122
m. It follows from Eq.(6.2.4) that 1:1 {fo 11/J(v) l ds r 1 m! (6. 2.8) exp { fo 11/J(v)l ds } - 1 .
and induction verifies Eq.(6.2.7) for all
ien (�) l <
oo;
(- - -) 1 r
the terms in
1
p
p
+
8+1 4p
disappear and
l e(z) l :S exp(1/4 lzl 2 ) - 1. This is a smaller bound than that derived for Fig. 6.3.1 and so offers a preferable path. The bound is applicable for 0 :S ph z :S 1r/4.
p
>
Figure 6.3.2 Another possible path If 'R(�) < 0 a path analogous to Fig. 6.3.2 is furnished by drawing a line vertically upwards from � until it meets the circle of radius p and then following the circumference of the circle as in Fig. 6.3.2. The details are left as an exercise.
126
Differential Equations
Chapter 6
6.4
Solutions in series
The WKB approximation gives a good idea of asymptotic performance. With more detailed information on it is possible to extend the WKB approxi mation to an asymptotic expansion. The differential equation to be discussed
g(z) d2w = g(z)w dz2
IS
where
(6.4. 1)
g(z) can be expanded in a convergent power series of the form g(z) = m=O L gm/Zm 00
for
Jzl
> R. For large
Jz l
1
(6.4.2)
1
g>1 (z) gJ + g1 j2gJ z �
so that the WKB approximation contains an exponential factor exp
{gJ z + (gJ/2gn ln z} .
This suggests that a solution of Eq. (6.4.1) should be sought in the form 00
w = ekzP m=O L am/Zm
a0
(6.4.3)
with -j. 0. If Eq.(6.4.3) is substituted in Eq. (6.4.1) and the factor constant terms cancel if
>.2 = go
since ao -j. 0. The cancellation of the terms in
e>-zzJJ. eliminated the (6.4.4)
1/z entails (6.4.5)
with implementation of Eq.(6.4.4) . From the remaining powers we have
- 2>. (n + 1)a,..+1 + {n(n + 1) - (2n + 1)�-t + �-t2 - g2 }a,.. = g3 0,.. - 1 + g4an- 2 + · · · + gn+2llo
(6.4.6)
for n 2": 0, the right-hand side being zero when n = 0. Evidently Eq.(6.4.6) permits the recursive determination of a coefficient when its predecessors are known provided that -j. 0. Hence special consideration is required when either is zero. The various possibilities when is zero and non-zero or are now discussed separately.
g0 g1
>.
an+ J g0
Section
6.4
Solutions in
127
series
(i} 9o = 0, 91 =J 0 In this case return to Eq. (6.4.1) and make the change of variable followed by the substitution w = d w The net result is
{
z
t2
}
d2W 3 = 4t29(t2 ) + 2 w 4t dt2
The factor of W contains the constant term 491 and we have reverted to the case with a non-zero constant term. Therefore, this case can be subsumed under the treatment when 9o =J 0. (ii} 9o = 0, 9I = 0 In this event both Eq.(6.4.4) and Eq.(6.4.5) are satisfied by >. = 0. Then Eq. (6.4.6) with n = 0 gives (6.4.7) JJ-2 - /)- = 92 · Insertion of Eq.(6.4.7) into Eq. (6.4.6) furnishes n(n + 1 - 2JJ-) an = 93 0, - 1 + · · · + 9n+2 ao
(6.4.8)
for n 2': 1. The an can be determined recursively from Eq.(6.4.8) so long as 2/)- - 1 is not a positive integer. Let /J-1 and JJ-2 be the two roots of Eq. (6.4.7) with JJ-1 chosen so that its real part satisfies 'R(JJ-I ) .::::; 'R(JJ-2)· Since
/J-1 + /J-2
=
1
.S � so that 2/)- I - 1 cannot be a positive integer. Therefore Eq. (6.4.8) always fixes an when JJ- = JJ-1 . Problems arise only for JJ- /J-2 · If JJ-1 = /J-2 (which can occur only when both are � and 92 = -1/4) both lead to the same an so that only one solution of the differential equation is generated. Hence the problem cases are (6.4.9) 2/)-2 = k + 1 , 2/)-1 = 1 - k
'R(JJ-1)
=
for k = 0, 1, . . . . For the moment these possibilities will be left on one side. When /)- = /J-1 2(1 - JJ-da1 = 93 11o
from Eq. (6.4.8). Because of the assumed convergence of the expansion for 9 (z) there is some M such that l9m l .S M Rm - 2 ; there is no loss of generality in taking M 2=: 1. Since 'R(2JJ-1)
., say >.1 and >.2 are obtained as solutions of Eq. (6.4.4). The corresponding values /-Ll and /-L2 of J.L follow from Eq.(6.4.5) . The coefficient of an+ l in Eq. (6.4.6) never vanishes now and all terms in Eq. (6.4.2) can be derived. Unfortunately, the analysis of (ii) cannot be repeated normally to prove that the resulting series is convergent. A notion of why this is so can be attained by supposing that we had succeeded in showing that the right-hand side of Eq.(6.4.6) was bounded by l ao! n(M R t . In general this bound will be insignificant compared with n2 an when n is large. The conclusion is that an+ I is pretty much the same as nan /2>. for large n. Thus, in general, the most that can be hoped for is that the series is an asymptotic representation of a solution of the differential equation. Only in special circumstances where the right-hand side of Eq. (6.4.6) makes a substantial cancellation in the a,.. term is there any chance of securing convergence.
6.5
An error bound for the series
In the case (iii) of the preceding section it was observed that the series stemming from the assumed form of the solution was unlikely to be convergent. Conse quently, it is necessary to have an idea of the size of the remainder after n terms have been calculated. It is possible then to decide whether or not the series is asymptotic. The purpose of this section is to confirm that the series does , indeed, provide an asymptotic representation of a solution of the differential equation. Suppose that the differential equation is
d2 w = h(t)w dt2 where h(t) = ho + h i /t + for t outside some circle centred on the origin and ho =f. 0 to fit (iii). Make the substitution t xjA. Then �w (6.5.1) = g(z)w dz2 where g(z) = h(z/Vho)jho. In view of the assumption on h g g (z) 1 + -l + (6.5.2) z for lzl > R, say. It is convenient to discuss the solution of Eq.(6.5. 1) subject to · · ·
=
=
Eq.(6.5.2).
· · ·
130
Chapter 6
Differential Equations
In this event Eq.(6.4.4) reduces to >..2 = 1 . Choose >.. 1 = -1 and >..2 = 1 . Then, from Eq. (6.4.5), /-LI = - M = - g1 /2. There i s n o loss of generality in selecting ao = 1 . The an resulting from Eq.(6.4.6) with >.. = ).. 1 and J.L = /-LI will be denoted by �1• Similarly an2 is the coefficient when >.. = >..2 and J.L = /-L2· Now consider the determination of e1 (z) such that
n- 1 w1 (z) = e - zz�'1 L u.mi /zm + e1 (z) m =O is a solution of Eq.(6.5.1). Then
rfle1 = g(z)e1 - e-z z�'1r(n, z) dz2
(6.5.3)
where the last term originates from the series. Since the satisfaction ofEq.(6.4.6) removes the term in 1 /zn+2 it follows that there is a En such that (6.5.4) for lzl > R. Add -e1 to both sides of Eq. (6.5.3) and then convert to an integral equation as in Section 6.2. The aim is to find a solution which is recessive as z -> +oo so put e 1 (z) = e - z E(z) with the result
E(z) =
� 100 (1 - e2z-2t) [{g(t) - 1 }E(t) - t�'1r(n, t)]dt.
(6.5.5)
The next step is to try to solve Eq. (6.5.5) by iteration with
E1(z) and
= -� LX) (1 - e2z -2t}ti'1r(n, t)dt
(6.5.6)
When 0 :S ph z :S; 1r/2 pick the path of integration to be the same as in Fig. 6.3.2. On it 'R(t - z) does not decrease and
1 1 - e2z-2t l :S 2.
Furthermore i t"1 l :S; Jtjl'r exp (1r 1141 /2) with that n > J.Lr · Hence Eq.(6.5.6) gives
/-L l
= J.Lr + iJ.L;. It can be arranged
Section 6.5 Now there is some
131
An error bound for the series
B such that lg(t) - 1 1
'R(2p,1) + q the right-hand side of Eq. (6.6.2) q- 1 e"i1J.2 L.:: (p + 2p,2 - 1 - m) ! (-)m a.n2 /2"+2JJ.2 - m +O(p") + 0{ (p + 2p,2 - 1 - q) ! }
in view of the bound on �On combining these results we have
e27rip2 [C1 � (p + 2p,2 - 1 - m) ! (-) m a.,2 2m-p- 2p2 (-)� 21ft
m=O +O{(p + 2p,2 - 1 - q) ! }]
(6.6.3)
when p is large enough. Similarly
q- 1 [C (p + 2p,1 - 1 - m)! a.,12m - p- 21J.1 -21ft. 2 mL =O +O{(p + 2p,1 - 1 - q) ! }] . 1
(6.6.4)
Once a sufficient number of the coefficients a.,1 and am 2 has been calculated Eq.(6.6.3) and Eq.(6.6.4) offer approximations to the constants in the connection formulae. In general, it will be necessary for q and p to be quite large to secure a suitable accuracy. For then
q- 1 c2 = -27riap2 /{ L.:: (p + 2p,1 - 1 - m)!a.,l2m -p- 21J.l } + 0 (1/pq) m=O
(6.6.5)
which indicates that a minimal requirement is likely to be that p and q should both exceed 10. Example 6.6.1 To illustrate the above theory a simple example will be con sidered in which it is easy to evaluate the coefficients. Normally, recourse to symbolic manipulation and numerical technique will be necessary for more com plicated differential equations (see Olde Daalhuis & Olver 1995). The differential equation to be discussed is _
Then P,1
=
-p,2 =
{
}
d2w 2v v(v - 1) 1+ + w. 2 dz2 z z -v and Eq.(6.4.6) provides an+!, 1 = - (n + 2v)ani /2
for
135
Hyperasymptotics
Section 6.7
n = 0, 1, . . . . With ao1 = 1 the solution of the recurrence relation is (2v + p - 1)!(-)" (6.6.6) a"1 = (2v - 1)!21'
On the other hand
2(n + 1)an+1,2 = n(n + 1 - 2v)an2 which shows that ap2 0 for p 2: 1 . On account of the form of ap2 we can take q as large as we like in Eq. (6.6.3) without affecting the sum in Eq. (6.6.3) . Therefore the error term can be dropped and =
or by virtue of Eq. (6.6.6) . It is transparent that
6.7
c2 = 0.
Hyperasymptotics
The hyperasymptotic expansion of the remainder R1(n, z) is the subject to be studied in this section. With z outside the circle of radius p introduced in the last section 1 1
1 27ri
tn - n - a 1pe-pe"' -dt = 0 tm "' t - m:E=O _!!!}_ Z
} tn- 1 et 1pe"'. { -w1 (t) - R1(n, t) -dt = 0.
since there are no singularities inside the circle. Hence
1 -. 2m
��
�
t-z
The contour of the integral containing R1 is deformed now as in the preceding section. In the process the pole at t = z is captured. There results
21rizn-J R1 (n, z)
=
-
1pe-"' e wl (t) dt pe "' t�"'. (t - z)oo tn- 1 -pt 1 t
+( - te -"''"' C1
P
--
t+z
. e- tw 2 (te"')dt . (6.7.1)
The first integral of Eq.(6.7.1) is estimated easily as l zl -+ oo because lzl will be much larger than p. As regards the second integral it is a Stieltjes transform and amenable to the methods described in Sections 5.4 and 5.5. If Theorem 6.5.1 is cited a series of J0(p,, z) together with an error term is obtained. Such a version could be convenient for dealing with Stokes' phenomenon.
6.8
Differential Equations
Chapter 6
136 Parameter with zero
The treatment of a differential equation with a parameter in Section 6.3 is unsatisfactory if g(z) possesses a zero in the domain of interest. A canonical problem to cope with the presence of a zero is the differential equation
d2w dz2
=
{ k 2 zm + '1/J(z)}w
(6.8.1 )
where m i s a positive integer and is a solution of
'1/J i s a regular function. Suppose that W (z) d2 W - zmw. (6. 8 . 2) dz2 Then, with l k l large, a first approximation to a solution of Eq.(6.8.1) is W( ,..z ) where ,_.m+ 2 = k2 . This suggests trying to solve Eq. (6.8. 1) by w = A(z)W(,..z ). Then
z,_.W'( ,..z )A' (z) + W ( ,..z )A"(z) = '1/J (z)W( ,..z )A(z) . The dominant term as lkl increases is the first one on the left-hand side. Therefore, the differential equation will not be satisfied unless A'(z) = 0 i.e. A(z) =constant. But that returns w to the first approximation and precludes any correction for the presence of '1/J. Consequently, a more elaborate form must be selected for w. Try instead
w = A(z)W(,..z ) + ,..B (z)W'(,..z)jk2
(6.8.3)
in Eq.(6.8.1). The consequent equation is
,_.W'( ,..z )(2A' + B'' jk2 ) + W(,..z )(A" + 2zm B ' + mzm-J B) 'ljJAW (,..z ) + 'ljJB,..W' (,..z )jk 2 . Since W and cordingly
W' are linearly independent their coefficients must vanish. Ac 2A' 2zm B' + mzm- J B
('1/JB - B" )/k 2 , 'ljJA - A" .
(6.8.4) (6.8.5)
The occurrence of the factor 1 jk 2 in Eq.(6.8.4) floats the notion of introducing the expansions
A(z) = 1 + L Ap (z)jk 2" , B(z) Fl
=
L Bp (z)jk2" . p�
(6.8.6)
137
Exercises on Chapter 6
Section 6.8
Equating the coefficients of the powers of 1 /k and integrating we derive
Ap+ I (z)
-�B�(z) +
r 1/J(t)Bv(t)dt,
(6.8.7)
:,. { t2/2 {?jJ(t)Ap (t) - A; (t)}dt.
Bv (z)
(6.8.8)
2 12
Bp is determined by Ap from Eq.(6.8.8) and then Ap+l follows from Eq. (6.8.7). Thus, all the terms can be found recursively, starting from Ao = 1. As a con sequence Eq. (6.8.7) and Eq.(6.8.8) furnish a formal solution of Eq.(6.8.4) and Eq.(6.8.5) so that Eq.(6.8.6) leads to a w which satisfies formally the differential equation in Eq.(6.8.1). It is by no means obvious that Eq. (6.8.7) and Eq. (6.8.8) provide functions Ap (z) and Bp (z) which are regular at the origin. Bp would have been certainly singular if the lower limit of integration in Eq.(6.8.8) had been chosen to be other than zero. More can be said. Suppose 1/J(z) contains a term whose variation with z is zn where n is a non-negative integer since 1/J is regular. Clearly, Bo(z) will be singular unless n 2': m - 1. In other words, 1jJ must have a zero of at least m - 1 at the origin if our procedure is to succeed. In fact, B0(z) is a multiple of zn- m+ I so that A1 (z) contains (n m + 1)zn- m and z2n- m+2 . Evidently, A1 (z) is singular also unless n 2': m - 1 . However, B1(z) is singular when n = m - 1 except for m = 1 . The other cases when B1 (z) is not singular for n < 2m + 1 are n = m , m + 1 but it turns out that n = m+ 1 is excluded by B2 (z). Continuing step by step we discover that the only permitted form for n is m + q (m + 2) with q a non-negative integer. This restricts 1/J(z) to being of the type C + zm ¢(zm+2 ) with ¢(z) a regular function and C absent if m =/= 1 . Only subject to this constraint does our proc;edure solve the differential equation. Theorem 6.8.1 The differential equation -
-
d2 w 2 zm + c + zm ¢(zm+2 )}w dz2 {k can be solved by the procedure described above provided that m is a positive integer, ¢(z) is regular and C = 0 if m =/= 1 . Differential equations which possess a zero i n the factor of k 2 but whose structure is more complicated than zm have to be transformed into the canonical form of Eq. (6.8.1) before Theorem 6.8.1 can be applied. Often the ensuing 1/J is -
=
far from simple. Exercises on Chapter 6 1. Find a solution of
Differential Equations
Chapter 6
138
which is recessive as z -> +oo and bound the error. 2. In Example 6.3.2 show that the error is bounded by exp(11/8 IW - 1 for 7r/2 < ph � S 1r and by exp(7r/4 IRW - 1 for 1r < ph � < 37r/2. What are the corresponding statements in terms of z? 3. Obtain WKB approximations for
when x is positive and not near 1 . Show that, as x -> oo , ekx jx4 k is an approx imation to a solution. 4. Find asymptotic solutions in series for large l zl of
d2w dz2
=
(z + 1) ! w z
calculating the first three terms. For what ranges of ph z are these series valid? 5. Find connection formulae for the differential equation
d2w = ( 1 + � - _2_) w. dz2 z 4z2
6. Investigate the possibility of finding connection formulae for the solutions of the differential equation of Exercise 4. 7. Obtain the formula analogous to Eq.(6.7.1) for R2 (n, z) . 8. Show that can be solved by the method of Section 6.8, taking W as a known solution of
d2 W dz2
=
(zm + 2 ) W c
z
9. In the differential equation
{
change x to ( where � (3/2 (x2 - x) ! - ln x ! + (x - 1) 4 �3 ( - ( ) 312 = cos- 1 x4 - (x - x 2 ) ! I and put w = (dxjd()> W Show that =
d2 W
d(2
}
= (k 2 ( + '1/J)W
(x � 1, ( ;:::: 0), (0 < X S 1 , ( S 0)
Section 6.8 where
Exercises on Chapter 6 'ljJ = ( (3 - 8x)
16x(x - 1) 3 Deduce that a first approximation to w is
4 / 1 ( ) _5_ x- 1
+
139
_5_ . 16(2
Ai (k213 () .
Does this bear any resemblance to the approximations of Exercise 3 when x is large and when x is small? What is the second approximation to w?
Appendix INTRODUCTION TO NONSTANDARD ANALYSIS
A.l
Basic ideas
Many who deal with the practical applications of mathematics do not bother much about the foundations but are probably aware that conventional mathe matics can be developed from set theory. Usually mathematicians will have met an informal discussion of sets at some stage in their training without worrying about the axioms on which they are based. The aim of this appendix is to provide a similar informal discussion of nonstandard analysis. It is not intended to cover all the subtleties of the subject but to provide sufficient information for a working mathematician to be able to use nonstandard analysis whenever it is helpful. One formal system of axioms for set theory is the Zermelo-Fraenkel system. Without going into the details of the axioms the essence of the system is that, given a set E and a property P, there is always a set F which is part of E and consists of those elements x E E which have the property P. In 1977 Nelson showed that by adding a prescription for when a set is standard the theory of nonstandard analysis, which had been introduced by Robinson in 1966 via mathematical logic, could be obtained. Thus, every set is either standard or nonstandard. So long as the definition of standard encompasses conventional mathematics there is no change to conventional mathematics but some new (nonstandard) entities become available. In fact, in Nelson's article, there is a theorem which states that any conventional mathematics established by non standard analysis could be proved also by Zermelo-Fraenkel theory alone. This does not mean that nonstandard analysis is useless but that it offers an alterna tive method of attack which may be more powerful in suitable circumstances. Indeed, Bernstein & Robinson (1966) demonstrated that nonstandard analysis could effect the resolution of unsolved questions. To decide whether or not a set is standard three rules are furnished. As will be seen the rules result in an implicit definition rather than an explicit one so that experience and usage are necessary to gain confidence in making an 141
142
Appendix
Nonstandard Analysis
identification. The notation to be employed for two frequently occurring sets is ¢ for the empty set and N for the set of natural integers i.e. N = {0, 1 , 2, . . . }. The set N can be constructed from ¢ by taking 0 to be ¢ and then 1 = {¢} , 2 = {0, 1 } , 3 = {0, 1 , 2}, . . . . Two other notations are R for real numbers and C for complex numbers. The construction of N might suggest to you that a standard set consists of standard elements but this inference would be false. A standard set can contain nonstandard elements in the same way that a finite set can contain infinite elements. For instance, the set { N , R} is finite having j ust 2 elements but neither N nor R is finite. Subsequently, attaching the adjective conventional to a statement or relation will mean that the statement or relation does not contain any reference to the term standard either explicitly or implicitly. The first rule for identifying standard sets concerns any conventional relation, denoted by R(x, y), between x and y such as x < y or x E y. Idealisation Rule The statement that there is an x such that R(x, y) holds for all standard y is equivalent to the statement that, for every standard and
finite set F, there is an x (which can depend on F) such that R( x, y) holds for all y E F. Suppose that R(x, y) signifies that x and y are natural integers such that x > y. When F is a finite set of integers there is n E N such that 0 .:S: y .:S: n. Hence x = n + 1 ensures that R(x, y) holds for all y E F. By the Idealisation Rule we deduce that there is an integer x E N such that x > y for all standard integers y E N . Consequently, nonstandard integers exist. An integer x such that x > y for all standard integers y will be called un limited. If x is an unlimited integer so is x + 1 since x + 1 > x > y for all standard integers y and, generally, x + n for n E N is unlimited. Thus, there is a profusion of unlimited integers. The notion of unlimited can be extended to finite numbers in R or C by saying that x is an unlimited number when there is an unlimited integer n such that lxl ?. n. Remark that an unlimited number is finite. The second rule for standard sets is Standardisation Rule Let E be a standard set and P a property which the
elements of E may possess. Then there is a standard A, a subset of E, whose standard elements are the standard elements of E which possess the property
P.
Section A.l
Basic ideas
143
Note that , in this rule, the property P does not need to be conventional. Note also that the rule refers only to the standard elements of A. There may be nonstandard elements of A which do not possess the property P. Furthermore, there may be nonstandard elements of E which possess the property P but which are absent from A. The third rule is Transfer Rule Let F(x, a, b, . . . ) be a conventional statement relating x, a, b, . . .
in which the parameters a, b, . . . are standard. If the statement holds for every standard x then it holds for all x.
In this rule parameter means a quantity capable of free adjustment. For instance, in the statement
I Y I < 1 + a for all y E E a and E are parameters but not y. The number 1 can be regarded as a constant or as a standard number according to one's preference. If E were standard and the statement held for every positive standard a the Transfer Rule would apply and the statement would be true for every positive a. The Transfer Rule has an important consequence. If a statement fails for all standard x then the Transfer Rule implies that it fails for all x. Hence, if a statement is true for some x it must be valid for at least one standard x. This principle is used so often that it will be set out separately and referred to as (T) subsequently. (T) If there is some x such that the conventional f(x, a, b, . . . ) holds for standard a, b, . . . then there is a standard x for which f(x, a, b, . . . ) is true. When it happens that f(x, a, b, . . . ) defines x uniquely it follows immedi ately that x is standard. Hence all quantities defined uniquely in conventional mathematics are standard. In particular, N , R, C, e , 1r and so on are stan dard. So is the interval [a, b] if a and b are standard elements of R. Similarly, x = E U F, x = E n F imply that E U F and E n F are standard when both E and F are standard. Likewise, the set of all subsets of E is standard when E is standard. If E1 is a subset of E2 then x E E1 implies that x E E2 . This is a conventional statement so that, if E1 and E2 are standard, and the statement holds for every standard x E E1 then it is valid for all x by the Transfer Rule. Thus, to verify that E1 is a subset of E2 when E1 and E2 are standard it is sufficient to confirm that the standard elements of E1 are in E2 . In particular, two standard sets are equal when they have the same standard elements. The last sentence shows that the set A of the Standardisation Rule is unique. This has some interesting implications. Let B be the subset of N which contains the integers from 0 to the unlimited w. Now seek the set A of the Standardisation Rule which has the property x E B. A must contain all the standard integers
Appendix
144
Nonstandard Analysis
since they are in B because w is unlimited. But A and N are standard sets with the same standard elements and so must coincide. Hence, in this case, A = N. Thus, A can be larger than the set which provides the defining property. On the other hand, if B contained only nonstandard integers A ¢ be cause both have the same standard elements. In other words the size of A in the Standardisation Rule is unrelated to the size of the set giving the defining property. Theorem A.l.l All elements of a set E are standard if, and only if, E is =
standard and finite. Proof. The statement that a set E is not contained in a standard finite set G is equivalent to saying that, for every standard finite set F, there is an x E E such that x -f. y for all y E F. By the Idealisation Rule this is equivalent to saying that there is an x E E such that x -f. y for all standard y. In other words, the statement that E is contained in a standard finite set G is equivalent to saying that all elements of E are standard. If E is standard and finite choose G E and the if part of the theorem =
follows. If, on the other hand, all elements of E are standard E is contained in a standard finite set G. Thus E is finite. Moreover, the set of all subsets of G is finite and standard. Since E is an element of this set it is standard by what has been proved already. The proof of the theorem is complete. • An obvious conclusion from Theorem A.l.1 is that an infinite standard set must contain nonstandard elements. A function f, defined for all elements of E, is said to be standard when f (x) is standard for all standard x E E. When x E E is nonstandard a standard f may n return standard or nonstandard values. For example, the function f(n) for n E N returns nonstandard values for nonstandard n whereas the function f(n) 1 for n E N always returns a standard value. If f, g are both standard and f(x) g(x) for all standard x E E then the functions are equal by the Transfer Rule. This indicates that once the definition of a standard function has been fixed for the standard elements of E its definition for the nonstandard elements is settled already. For, if f and g differed only on the nonstandard elements of E, their difference would be zero by the first sentence of this paragraph. The Transfer Rule is called upon frequently in nonstandard analysis. You may be misled by the notation into thinking that x must be a scalar but that is not the intention. For instance, a result may have been proved for every standard x 1 , x2 , X3 . By regarding the standard triple (x1 , x2 , x3 ) as x the Transfer Rule extends the result to all x1 , x2 , x3 in the set under consideration. What the Transfer Rule enables one to do is, on the one hand, to extend results proved for standard x to all x and, on the other hand, infer the existence =
=
=
Section A.l
145
Basic ideas
of a standard x satisfying a formula when some x does. Perhaps the generality of the Standardisation Rule should be stressed. For example, if you have standard functions which are continuous on an interval, you can form a standard set of functions whose standard elements are continuous standard functions. Results proved for continuous standard functions may carry over to other members of this standard set if transfer is applicable. A complex number x is said to be limited when there is a standard real number y such that lxl .:S y. If lxl < y for every standard positive y then x is called infinitesimal. The notation x '::::'. y will signify that x - y is infinitesimal. It is obvious that, if x is unlimited, 1 /x is infinitesimal. Also, if x '::::'. 0, xy '::::'. 0 when y is limited. When y is unlimited no conclusion can be drawn in general; for example, with x > 0, y 1/x! , 1/x, 1 /x2 makes the product infinitesimal, limited, unlimited respectively. Theorem A.l.2 There is only one standard infinitesimal and that is 0. Proof. If 1-1 is standard and l 1-1 l < x for all standard x > 0 the Transfer Rule implies that 1 1-1 1 < x for all x > 0. By choosing x 1 1-1 1 /2 we see that 1-1 0 is the only possibility. • Theorem A.l.3 If an '::::'. bn for n 1 , 2, . . . , N then =
=
=
=
N
L(an - bn )/N '::::'. 0. n=l Proof. If E is any positive standard number ian - bnl < E and
which proves the theorem. • Remark that N could be unlimited in Theorem A.l.3. However, if limited, multiplication by N does not affect the formula and
N is
N
L(Un - bn ) '::::'. 0 (N limited). n=l Other results can be obtained with more assumptions (see Sections A.2 and A.5 later). Theorem A.1.4 If x is limited there is a unique standard number st(x) such that st(x) '::::'. x. Proof. Consider firstly that x is a positive real number.. Since x is limited there is a standard M such that x .::; M. Let A., be the standard set which contains all the standard y satisfying 0 � y � x; A., exists by the Standardisation Rule. If standard y E A., then clearly y E [0, M]. Since A., and [0, M] are both standard, A., is contained in [0, M]. Consequently, A., is bounded above. Take st(x) as the upper bound of A.,; it is standard by (T).
146
Nonstandard Analysis
Appendix
Select any standard E > 0. If st(x) + E < x then st(x) + E is in A., but greater than its upper bound which is impossible. If st(x) - E > x then st(x) - E would bound A., above which is not possible with st(x) as the upper bound. Hence lst(x) - xl < E and st(x) � x. If there were another standard number X1 such that X1 � x then necessarily st(x) - x1 � 0. Theorem A. 1.2 forces x1 = st(x) and the theorem has been demonstrated when x is positive. When x is negative take st(x) = - st( -x) . If z is the limited complex number x + iy st(z) = st(x) + i st(y). The proof of the theorem is complete. • There is a natural extension to a limited vector
with n standard of st(x) = (st(x1 ) , st(x2 ) , . . . , st(xn)). Observe that, if 1-1 � 0 and 1-1 > 0, then st( �-1 ) inferred only st(x) 2 0. It is transparent that st(xy)
=
=
0. Thus, from x
>
0 can be
st(x) st(y)
when both x and y are limited.
A.2
Sequences
Suppose that there is a rule which associates with each standard integer n a standard element a.. of a given standard set E. The function f(n) = an is standard by the definition of a standard function given earlier. As pointed out previously no other standard function can have the same property and, in addition, there is no choice about the behaviour of f for nonstandard n. Hence the sequence {an} which takes the constructed standard values for standard n is unique. Such a sequence will be referred to briefly as a standard sequence i.e. in a standard sequence {a..} the an is a standard element of a standard set E when n is a standard integer. Theorem A.2.1 If {a..} is a standard sequence and lirnn _, oo an = a then a is
standard. Proof The existence of a implies the statement: there is an a such that, for every standard E > 0 , the set n E N with ian - al 2 E is finite.
Section A.2
147
Sequences
Here {an} and € are standard. Hence, by ( T ) , there must be a standard b with the same property as a. But a limit is unique and so b = a. The theorem is proved. • Having established that a is standard replace 'is finite' in the statement quoted in the proof by 'contains N(t:) elements'. Then, by ( T ) , N(t:) can be taken as standard. In other words for every standard € > 0, there is a standard N(t:) such that for every n _2 N(�:). Theorem A.2.2 If {an } (i) limn-> oo an = a,
ian - ai < €
is a standard sequence, the statements
(ii) a is standard and an � a for all unlimited n E N , (iii) st (an ) = a for all unlimited n E N are equivalent. Proof. By what has just been shown a is standard and n _2 N(t:), when n is unlimited; so an � a and (i ) implies (ii ) . Since a is limited it follows from Theorem A.1.4 that (ii ) and (iii ) are equiv alent. When (ii ) holds choose a definite unlimited
N. Then the statement there is N such that ian - ai < € for n > N is true for every standard € > 0. Treat N as a constant in this statement. The parameters are standard. The Transfer Rule then asserts that it is true for all € > 0. Hence (i ) holds and the theorem is proved. • Corollary A.2.2 If f is standard, the statements (i) lim.,_,o f(x) = a
(ii) a is standard and f(x) � a for all infinitesimal x, (iii) st (f(x)) = a for all infinitesimal x are equivalent. Proof. Assume ( i ) and that standard x0 is in the domain in which the limit is taken. Then {f(x0jn)} is a standard sequence with limit a whence Theorem A.2.2 makes a standard. Also, by the definition of limit, for every standard E > 0, there is 8 > 0 such that if(x) - ai < E for ixi < 8. Since f, E , a are
standard (T ) informs us that the statement holds for a positive standard 8. Thus (ii) is verified and thereafter the proof goes along the lines of Theorem A.2.2. • The theorem can be applied to limits at other standard points by a simple change of variable. Robinson's Lemma If {an } is any sequence then (a) if an = 0 for all standard n, there is an unlimited v E N such that an = 0
for all n :::; v,
148
Appendix
Nonstandard Analysis
(b) if a.. � 0 for all standard n, there is an unlimited v E N such that a.. � 0 for all n ::; v. Proof. (a) Let m E N be such that an 0 for all n ::; m. The set of m contains all standard m in N by hypothesis. Hence, by Theorem A . l . 1 the set contains =
a nonstandard v. (b) We cannot proceed as in (a) because an � 0 is not a conventional state ment. However, let m E N be such that Jan l < 1/m for all n :S m. When m is standard and positive so is 1/m ; accordingly the set of m contains all standard m in N. Hence the set contains a nonstandard v such that Janl < 1/v for all n :S v. Since v is unlimited the required result follows. • There are some slight variants on the above theorems which are sometimes useful. Let k be a fixed standard integer and define bn tln +k with {an} standard. The sequence {bn } i s standard so that Theorem A.2.2 can be applied. Thus , altering a standard number of an , with n standard, has no effect on the conclusions. Even if {a.,} is not standard the same argument reveals that Robinson's Lemma is still valid if a standard number of the an are not (a) zero, (b) infinitesimal for standard n. Another result concerns the persistence of a formula from one unlimited integer to others. Corollary A.2.2a If there is an unlimited integer w such that, in the standard sequence {an}, an � 0 for every unlimited n ::; w then an � 0 for all unlimited =
n. Proof. Suppose, on the contrary, that there is some m > w for which am is not infinitesimal. Then, there is a standard 6 such that Jam l > 6. Define the set of integers S by n E S if n E N and Jan ! > 6. This defines S uniquely and, since N , {a.,} and 6 are standard, makes it standard. S contains an unlimited element m and so, by virtue of Theorem A. l . 1 , must be infinite. Moreover, S must contain some standard n being standard and non-empty. The set n E S and n :S w contains all the standard elements of S and hence contains an unlimited integer v i.e. there is unlimited v ::; w such that Jav l > 6. But that contradicts
the hypothesis of the Corollary and so the opening assumption is false. The proof is finished. • Remark that, if the standard sequence {a.,} is such that an � 0 for all standard n, then an � 0 for all unlimited n. For, Robinson's Lemma extends the formula to some unlimited n and Corollary A.2.2a completes the range. The validity of Corollary A.2.2a cannot be guaranteed when {an } is not a standard sequence. A simple counterexample is an = n 1-1 0 where 1-1 0 is a fixed 1 infinitesimal. Here an � 0 for n up to 1/ 1-16 say but is certainly not infinitesimal for n > 1/ !-1 0. Indeed, the set S in the proof of Corollary A.2.2a is no longer standard because it has no standard elements but is not empty. The theorems on sequences can be extended to sequences of functions as will
Section A.3
Continuity
149
be shown now. It will be assumed that the functions are defined on a standard interval I of the real line. Theorem A.2.3 Let fn be a standard function on I for standard n and f
standard. Then, for eve:ry x E I, fn (x) -> f(x) pointwise if, and only if, fn (x) � f(x) for every standard x E I and all unlimited integers n. Proof By the Transfer Rule fn (x) -> f(x) for all standard x E I if, and only if, fn (x) -> f(x) for all x E I. For a given standard x {fn (x)} is a standard sequence and f(x) is standard. From Theorem A.2.2 it follows that fn (x) � f(x) for unlimited n provides the same information and the proof is complete. • Theorem A.2.4 Under the same conditions as Theorem A.2.3 fn -> f uni formly on I if, and only if, fn (x) � f(x) for all x E I and all unlimited integers n. Proof If g(x) � 0 for all x E I, ig(x) i < E for standard E > 0. Hence g is bounded and sup ig(x)i � E. Hence sup ig(x) i � 0. Consequently, the statement fn (x) � f(x) for all x E I is equivalent to sup ifn (x) - f(x)l � 0. The sequence { sup lfn (x) - f(x) l } is standard by (T ) and so Theorem A.2.2 applies. The proof is finished. • Robinson's Continuous Lemma If f(x) � 0 for all limited x there is an unlimited X such that f(x) � 0 for ixi � X. Proof Let m E N be such that if(x) i < 1/m for ixi � m. The set of m includes
all the standard m by hypothesis. Therefore, the set contains an unlimited integer v such that lf(x) i < 1/v for ixi � v. The result stated can be inferred immediately. •
A.3
Continuity
Definition A.3. 1 The function f is said to be S-continuous at x if, and only if, f(y) � f(x) for all y � x. Theorem A.3.1 If f is standard f is S-continuous at standard x if, and only if, f is continuous at x. Proof Given S-continuity it is valid to make the statement: there is a b > 0 such that if(x) - f(y) i < E for all y in iY - xi < b and for every standard E
>
0
by choosing b as infinitesimal. Keep b constant. Then the Transfer Rule can be applied legitimately and the statement holds for all E > 0. Thus f is continuous at x. Conversely, when f is continuous at x, choose any standard E > 0. Then the statement there is a b(E)
>
0 such that
lf(x) - f(y) j < E for all y in iY - xi < b(E)
Appendix
150
Nonstandard Analysis
is true. Since the parameters f and € are standard (T) confirms that the state ment is true for some standard positive 6(t:). Now, if y ':::'. x, iY - xi < 6(t:) for standard positive 6(t:) and so if(x) - f (y) i < €. Since € is arbitrary, f(x) ':::'. f (y) and S-continuity has been affirmed. • On account of Theorem A.3.1, the set of x of a standard interval I at which the standard f is continuous is the same as the set of the Standardisation Rule with the property that f is S-continuous at x. Both sets are standard with the same standard elements. As a consequence the statement that the standard f is continuous on the standard I offers the same information as saying that f is S-continuous at every standard x of I. To put it another way, if the standard set of functions which are S-continuous at x is formed by the Standardisation Rule it contains precisely the functions which are continuous at x. In this case there is no ambiguity about the elements of the standard set provided by the Standardisation Rule although generally there is information about the standard elements solely. When x is limited st(x) exists and st(x) ':::'. x by Theorem A.l.4. Therefore, if f is standard and S-continuous at x, f(x) ':::'. f(st(x)). The right-hand side is standard and, therefore, limited. Hence f(x) is limited and so st(f(x)) exists. Applying Theorem A.l .4 we conclude that st(f(x))
=
f(st(x))
(A.3.1 )
when f i s standard and S-continuous at limited x. Violation of the conditions of Theorem A.3. 1 can destroy any connection between continuity and S-continuity. We illustrate with a few examples. The function f(x) x2 is standard. It is continuous at every x because if(y) - f(x) i < € for iY - xi < 6 so long as 6 < (x2 + t:)t - x. However, f is not S-continuous at unlimited x. For, with y = x + 1/x so that y ':::'. x, =
f(y) - f (x)
=
2 + 1/x2
which is not infinitesimal. As an example of a nonstandard f take f(x) = 1-1 /( 1-1 2 +x2) with 1-1 a positive infinitesimal. At the standard x = 0, f is continuous but not S-continuous because f (O) - !( 1-1 ) = 1/21-1 which is unlimited. In contrast, the selection f(x) = 1-1 sgn x makes f discontinuous at x = 0 but it is S-continuous at every x because if(x) - f(y) i � 2 1-1 . A standard interval I of the real line is said to be compact when every x E I is limited and st(x) E I.
Section A.4
151
The derivative
Theorem A.3.2 If f is continuous on the compact standard I [a, b] with f(a) > 0 and f(b) < 0 then there is c E I where f(c) 0. Proof. Assume that f is standard. Let J be a finite subset of I which contains all standard x E I. Let x0 be the largest element of J for which f (xo ) ;:::: 0. Since b is standard it is in J and so x0 < b. Also x0 is limited by assumption and =
=
f(st (xo )) ;:::: 0 by Eq. (A.3.1 ) . Let x1 be the next element of J above XI Xo + 6 and 6 is not infinitesimal =
Xoj
then
f (xi ) < 0.
If
2x0 < st (2x0 + 6) � st ( 2x1 - 6) < 2x1. This means that st (x0 + xi )/2 is a point of J between x0 and xi contrary to the definitions of x0 and x1 . Hence 6 must be infinitesimal. By Theorem A. l .4
which forces f(st(x0)) = 0. Thus the c of the theorem is standard when f is standard. The Transfer Rule extends the theorem to more general f. • Of course, the theorem does not assert that there is only one zero with a < c < b. There can be many others and they may be nonstandard even when f is standard. A.4
The derivative
The function f is said to be standard d such that
S-differentiable at the point a when there exists a
f(x) - f (a) =d x-a for all x � a, and d is called the S-derivative of f. Theorem A.4.1 If f is standard, f is S-differentiable at standard a if, and only if, f is differentiable at a. Proof. When f is differentiable define g(x)
=
f(x) - f(a) . x-a
Then, with g(a) = lim.,_,a g (x) , g is a continuous standard function and also g(a) = f'(a) with f'(a) standard. By Theorem A.3. 1 , when x � a, g(x) � g(a) and f is S-differentiable; the S-derivative coincides with the usual derivative.
152
Nonstandard Analysis
Appendix
When f is S-differentiable g(x) � d for x � a. Define g(a) d so that g is standard and S-continuous at a. Theorem A.3.1 shows that g is continuous so that f is differentiable at a with f' (a) d. • Although the requirement that d be standard in S-differentiability looks rather restrictive it is not as confining as appears at first sight. Suppose that f is S-differentiable at every standard x E I with I standard. Then there is a standard function which reproduces the values of d at the standard x and this function may be taken as the derivative for nonstandard x. From another point of view, the standard set of the Standardisation Rule with the property f is S differentiable is the same as the standard set with f differentiable. In particular, if f is standard, the set of points at which f is differentiable is standard. For example, let f(x) x2 • If 1-1. is any infinitesimal =
=
=
which shows that the S-derivative is 2x at standard x. The domain x 2: 0 contains all standard x 2: 0 and so 2x can be taken as the derivative throughout X 2: 0. These remarks indicate that there is little point in distinguishing between the notation for the S-derivative and the conventional derivative. Theorem A.4.2 If f is standard on the standard I and has derivative f'(x) at x then there is o > 0 such that
f(y) - f (x) y-x
�
f' (x)
for 0 < I Y - xi < 0.
Note that there is no necessity for x to be standard in this theorem. Proof. We know that the set of points at which f is differentiable is standard call it J. J is not empty because it contains x. If there were no standard point in
J then J would be empty, the empty set being standard and having no standard points. Therefore there is a standard point, say a, in J. Pick any infinitesimal 1-1. > 0. Then, on account of the differentiability at a, for 0 < IY - ai < 1-1.
f (y) - f (a) y-a whence
l
�
f ' (a)
\
f(y) - f (a) J'(a) <E y-a for all standard E > 0. This is a statement in which f, f', E and a are standard.
Therefore, by the Transfer Rule, it holds for all points of J. The proof is terminated. •
Section A.4
The derivative
153
It is fairly obvious now that all the usual rules for the derivative are valid e.g. (!g)' = f'g + fg', Rolle's theorem, f' = 0 on I entails f = constant, f' > 0 means f is increasing and we shall not bother to prove them in detail. For later purposes it is helpful to have a stronger version of differentiability, namely The standard f is said to be strongly if there is a standard d such that
differentiable at the standard a
f(x) - f(y) �d x-y for all x � a and all y � a with x i= y. Observe that , by selecting y = a, strong differentiability enforces differentia bility with d = J'(a) . Theorem A.4.3 The standard f is strongly differentiable at all interior points of the standard I if, and only if, f is differentiable and f' is continuous on I. Proof. Let a E I be a standard interior point and x � a. Since strong differen tiability entails differentiability Theorem A.4.2 gives
f(y) - f (x) J' � (x) y-x y � x and y i= x. By virtue of strong differentiability at a f(y) - f (x) � J' (a). y-x Hence f'(x) � f'(a) for x � a and continuity follows from Theorem A.3. 1 since f' is standard. The only if part of the theorem has been proved. To show the converse take x � a, y � a with x i= y. Differentiability provides, via Rolle's theorem, a c between x and y such that f(y) - f(x) = f' (c). y-x From the continuity of f' and c � a, Theorem A.3.1 reveals that J'(c) � f' (a)
for
whence strong differentiability can be affirmed. The proof is complete. • Higher derivatives ( when they exist ) can be calculated in a natural way since f' is standard when f is. Thus Taylor's expansion
f(x) = f (a) + f' (a)(x - a) + . . . + f (n) (a)(x - at jn! + R, can be available. When f is infinitely differentiable at a f is said to be analytic at a when the remainder R,. satisfies R,. � 0 with lx - al < o for some standard o and n unlimited.
The extension to functions of a complex variable z is immediate. Analytic functions can be defined in the usual way and have the usual properties.
A.5
Nonstandard Analysis
Appendix
154 Integration
As in conventional theory we start with integrals on the real line. Let f be bounded on the interval [a, bJ with a < b. Given any infinitesimal 1-1. let N be the largest integer such that a + N 1-1. � b. Definition A.5.1 J! f(x)dx st I:f=o f (tj)l-1 for any infinitesimal 1-1. and every tj E [a + j 1-1. , a + (j + 1 ) 1-1. ] It is necessary to check that the right-hand side has a meaning. Since f is bounded there is M such that \f\ S M throughout the interval. Hence =
•
{
}
lt 1
f(tj) l-1 � M(N + 1) 1-1 � M(b - a + 1-1 ) .
If f i s standard, M can b e taken as standard. Therefore, the right-hand side of the inequality is limited for b - a limited. Thereby, the left-hand side is limited and it is permissible to take the standard part of the definition. It has been demonstrated that the integral certainly exists when a, b and f are standard. Consequently, it can be extended to other (a, b , f) by identifying it with the standard function which takes the standard value of the Definition for the standard triples (a, b, f) with f bounded on [a, bJ. Remark The confirmation that a meaning can be attached to the sum in Defi nition A.5.1 is unaffected by changing f to \fJ . So J! \f(x)\ dx exists and
If c satisfies
can
ll
l l
f(x)dx �
\f(x)\ dx.
a < c < b the additive property of the integral
l
f(x)dx
=
[ f (x)dx + l f(x)dx
be deduced in a trivial manner. It is possible that, for some functions, the value of the sum depends upon the choice of the infinitesimal 1-1. . The next theorem reveals that this surmise is false for continuous functions. Theorem A.5 .1 If f is continuous on the compact interval [a.b] the value of
the integral is independent of 1-1. . Proof. It is sufficient to prove the result when f takes real values. Let x, y be standard points of the interval with x < y. Assume to begin with that f is standard. The continuity of f on [x , yJ ensures that min f � f � max f there. Accordingly, as in the justification of the definition,
(y - x) mi n f S
[ f(t)dt � (y - x) max f.
155
Integration
Section A.5
Since the integral has been identified with a standard function the inequality can be extended by the Transfer Rule from the standard triples (y, x, f) to any triples (y, x , f) with f continuous on [x, y]. Then, if F(x) = J: f(t)dt,
F(y) - F(x) :::; max f y-x by the additive property. Pick a standard c and choose y � c, x � c wi th y i= x. Because f is continuous max f and min f differ infinitesimally from f (c) on [x, y]. Hence F(x) is strongly differentiable at any standard c, and has derivative f(c). Since the integral is a standard function the Transfer Rule extends this min f :::;
result to every point of the interval. Suppose now that a different infinitesimal is used in the definition of the inte gral, leading to F1 (x) say. Then F1 (x) is strongly differentiable with derivative f(c) at c. Hence F'(x) F;(x) whence F(x) - F1 (x) = constant = 0 because both integrals vanish at x = a. The Transfer Rule extends the theorem to any continuous f. The proof is terminated. • Piecewise continuous functions can be treated by first splitting the integral into portions where there is continuity by the additive property and then ap plying Theorem A.5.1 to each of the separate portions. Since the conventional indefinite integral enjoys the property that its deriva tive reproduces f where f is continuous our i ntegral agrees with the conventional whenever the two exist. The additive property extends the equality to piecewise continuous functions. Another useful result is Theorem A.5.2 If f(x) � 0 for a :::; x .:::; b with b - a limited =
Proof. Obviously
ll
l f (x)dx
�
0.
i
f (x)dx :::; (b - a) sup I fl .
As in Theorem A. 2 .4 f (x) � 0 implies sup I f I � 0. The stated value of the inte gral can be inferred at once from b- a being limited and the proof is complete. • Corollary A.5.2 If f is bounded and integrable over [a - 11 o , b + 11 0] with 11 0
infinitesimal, then
l lf(x) - f(x + 11 o) l dx � 0.
Proof. Since both J: f (x)dx and J: f(x + 11 0)dx exist by hypothesis so does the given integral by the Remark after Definition A.5. 1. We can assume 11 0 > 0 without loss. Choose any infinitesimal 11 > 11 0. Then
1 1 f(x) - f(x + 11 o) l dx � � lf(a + j 11 ) - f(a + j 11 + 11o)l 11 . b
N
156 Also
N
l: f(tj) V! �
whence
Nonstandard Analysis
Appendix
j =O
N
1b f(x)dx a
N
�
l:U(tj) - f(tj)} vt
j=O
L f(tj) vt
j=O �
o.
If f(a + j vt ) 2: f (a + j vt + vto) choose ti = a + j vt , tj = a + j vt + vt o; otherwise reverse the roles of tj and tj. The stated result follows immediately. • Singular integrals occur whenever f is unbounded in the range of integration or the range is infinite. It is customary to define such integrals (when they exist) by means of limits e.g.
100 f(x)dx = lim 1 f(x)dx. Theorem A.5.3 Let ln = J: f(x)dx with n E N. If {ln} is a standard sequence the infinite integml exists if, and only if, ln l with l standard for all unlimited n. U!hen the integml exists J:' f (x)dx = l and J:" f (x)dx 0 for any unlimited a
t-----) oo
t
a
�
�
integer w . Proof. Application of Theorem A.2.2 covers all the assertions except the last
which is a consequence of the first part and the additive property. There is nothing more to prove. • There are similar theorems for integrals to oo and from oo to oo but details will be omitted. For f (x) unbounded at a point , which may be taken as the origin without loss of generality, one is concerned with limits such as -
lim
or lim
€-++0
{J.b €
J.b
€--+ + 0 €
-
f(x)dx
f (x)dx +
�-e- b f(x)dx } .
{J:;n f (x)dx}
Here the theorem analogous to Theorem A.5.3 depends on sequences like but details are left to the reader. For many applications it is useful to have Theorem A .5.4 If f(x) � 0 for all limited x 2: 0 and lf(x) l :S: h(x) x E R with x 2: 0 where standard h is integmble from 0 to oo then
fo
oo
f (x)dx � 0.
Proof. If n is a positive limited integer
ion f (x)dx
�
0
for all
Section A.5
Integration
157
by virtue of Theorem A.5.2. Robinson's Lemma assures us that there is an unlimited integer w such that
low f(x)dx 0. I f" f(x)dxl Loo h(x)dx �
Also
�
:::;
0
from Theorem A.5.2. Addition completes the proof. • Actually f(x) � 0 for more x than specified in Theorem A.5.4 by Robinson's Continuous Lemma but that fact is not needed in the proof. Corollary A.5 .4 If f�V! o lf(x) l dx exists then
fooo lf(x) - f(x + V! o) l dx
�
0
for infinitesimal V! 0 . Proof. The proof is the same as for Theorem A.5.4 but calling on Corollary A.5.2. • By replacing f by Theorem A.5.5 If
f - g in Theorem A.5.4 we obtain f(x) � g(x) for all limited x 2 0 and l f (x) l :::; h(x) , lg(x) l :::; h(x) for all x E R with x 2 0 where standard h is integrable from 0 to oo then f(x)dx � g(x)dx.
fooo
fooo
A variant of this theorem is Theorem A.5.6 Let f(x) g(x)h(x) with g(x) � 0 for all limited integrable from 0 to oo If lf(x) l :::; 2h(x) for all x E R =
.
x and h > 0
fooo f(x)dx V! fooo h(x)dx =
for some infinitesimal V! . Proof. For limited positive n E N
so that
n If f(x)dx l :::; sup 19 1 Jo h(x)dx [ f(x)dx/ion h(x)dx 0. �
By Robinson's Lemma this is true for an unlimited w i.e.
low f(x)dx V! low h(x)dx. =
Since f:" f(x)dx � 0 and J:" h(x)dx � 0 as in theorem A.5.4 the stated formula follows. •
158
Appendix
Nonstandard Analysis
Corresponding theorems can be developed for infinite series by employing the sequence { Sn} where Sn L�=O am . They will be stated without proof. Theorem A .5. 7 If { sn} is a standard sequence 2.:::=0 an exists if, and only =
if, sn
c:=
l with l standard for all unlimited n. When the infinite series exists c:= 0 for any unlimited w . Theorem A .5.8 If an c:= 0 for all limited integers n and lanl :S hn for every n E N where 2.:::=0 hn is converyent and hn is standard, then L::=0 an c:= 0. Theorem A.5.9 If an c:= bn for all limited integers n and lan l :S hn , Ibn I :'S: hn where L:=0 hn is convergent and hn standard, then 2.:::=0 an c:= L::=o bn . Theorem A.5.10 If an = bn hn with bn c:= 0 for all limited integers n, standard hn > 0 with L::=0 hn convergent and I an i :'S: 2hn for all n E N then I::=0 an = l and L::=w an
00
00
L an = V1 L hn n=O n=O for some infinitesimal
vt .
The Maclaurin-Cauchy test for convergence offers a connection between se ries and integrals. Theorem A.5.11 If f(x) > 0 and f decreases steadily then
t f(m) 2:
m=O
J:+I O
for any n E N . If, in particular, f (x)
t f(m)
m=O
1mm+I
f (x)dx 2: c:=
c:=
'f f (m)
m=l
0
ion f(x)dx. 0
Proof. For any integer m the decrease in f enforces f(m) 2:
f (x)dx 2: f(m + 1).
Addition over m supplies the first part of the theorem. When f(x) c:= 0 the two extremes of the inequality differ only by an infinitesimal. The second part is an instant consequence. •
References
Abramowitz, M. and Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover, New York. Bernstein, A. R. and Robinson, A. 1966 Pacific Journal of Math. 16(3) , 421431. Berry, M. V. and Howls, C. J. 1990 Proc. Roy. Soc. Lond. A430, 653-668. Berry, M. V. and Howls, C. J. 1991 Proc. Roy. Soc. Lond. A434, 657-675. Berry, M. V. 1991 Proc. Roy. Soc. Lond. A435, 437-444. Bleistein, N. and Handelsman, R. A. 1975 A symptotic Expansions of Integrals. Holt , Rinehart and Winston, New York. Boyd, W. G. C . 1 990 Proc. Roy. Soc. Lond. A429, 227-246. Boyd, W. G. C. 1 993 Proc. Roy. Soc. Lond. A440, 493-518. Boyd, W. G. C. 1994 Proc. Roy. Soc. Lond. A447, 609-630. Chester, C., Friedman, B. and Ursell, F. 1957 Proc. Cambridge Philos. Soc. 53, 599-611. Dingle, R. B. 1973 Asymptotic Expansions: Their Derivation and Interpreta tion. Academic Press, London. Howls, C. J. 1992 Proc. Roy. Soc. Lond. A439, 373-396. Izumi , S. 1927 Japan J. Math. 4, 29-32. Jones, D. S. 1982 The Theory of Generalised Functions. Cambridge University Press, Cambridge. Lighthill, M. J. 1958 Fourier Analysis and Generalised Functions. Cambridge University Press, Cambridge. McLeod, J. B. 1992 Proc. Roy. Soc. Lond. A437, 343-354. Nelson, E. 1977 Bull. Amer. Math. Soc. 83, 1 165-1198. Olde Daalhuis, A. B. 1992 !.M.A. Jour. Appl. Math. 49, 203-216. Olde Daalhuis, A. B. 1993 Proc. Roy. Soc. Edin. A123 , 731-743. Olde Daalhuis, A. B. and Olver, F. W. 1 994 Proc. Roy. Soc. Lond. A445, 39-56. Olde Daalhuis, A. B. and Olver, F. W. J. 1995 Methods and Applications of Analysis. 2, 348-367. Olver, F. W. J. 1974 Asymptotics and Special Functions. Academic Press, New York. 1 59
160 Olver, F. W. J. 1991a SIAM J. Math. Anal. 22, 1460-1474. Olver, F. W. J. 1991b SIAM J. Math. Anal. 22, 1475-1489. Olver, F. W. J. 1994 Methods Applic. Analysis. 1, 1-13. Paris, R. B. 1992a J. Comp. appl. Math. 41, 117-133. Paris, R. B. 1992b Proc. Roy. Soc. Lond. A436, 1 65-186. Paris, R. B. and Wood, A. D. 1992 J. Comp. appl. Math. 41, 135-143. Paris, R. B. and Wood, A. D. 1995 Bull. IMA. 31. 21-28. Robert, A. 1 988 Nonstandard Analysis. Wiley, New York. Robinson, A. 1966 Non Standard Analysis. North-Holland, Amsterdam. Stieltjes, T. J. 1886 Ann. Sci. Ecole Norm. Sup. [3] , 3, 201-258. van den Berg, I. 1987 Nonstandard Asymptotic Analysis. Springer-Verlag, Berlin. Wong, R. 1989 Asymptotic Approximations of Integrals. Academic Press, Boston.
Index
B
A
90, 91 , 159 Airy function, 45, 63 analytic continuation, 47 asymptotic behaviour, 46, 63 differential equation, 47 integral representation, 45
Abramowitz, M. ,
algorithm for nonstandard statement ,
R. , 141 , 159 V., 99, 159
Bernstein, A. Berry, M.
Bessel function,
61 , 62, 69, 1 1 2 115 Bleistein, N., 2, 159 Boyd, W. G. C . , 99, 159 modified,
Beta function,
3 74 24, 47, 51, 66,
almost minimal term, analytic continuation,
c C, 142 Chester, C.,
76 singularities,
asymptotic expansion, coefficient in, definition,
phase determination,
5
compact interval,
6
complex contours ,
5
10, 1 5 123 function of, 9 generalised , 12 integral of, 7 inversion of, 10 recessive, 1 23 uniform in angle,
92 93
150 27
confluent hypergeometric function,
division by,
21,
61
dominant,
uniqueness,
92, 159
coalescing saddle points,
51
connection formulae,
132 , 133
continuity, and S-continuity,
149
and strong differentiability, on compact interval ,
12
6
153
150
conventional quantities standard,
143
convergence, Maclaurin-Cauchy test ,
asymptotic expansions, addition of, 7
158
148 39
of functions,
multiplication of,
8
asymptotic sequence,
critical point ,
3
4 derivative, 5 example, 4
curve of steepest descent ,
28
definition,
subsequence,
D derivative,
4
asymptotic series ,
strong,
43
general theorems,
43
119
analytic continuation o f series,
132, 133 series, 129, 132
connection formulae,
asymptotics, uniform,
153, 155
differential equation,
79
error bound for
161
132
162
Index
Asymptotics
strongly differentiable, 153 error bound variation with path, 125 hyperasymptotics, 135 V"(x) , 90 solution in series, 126 various cases, 1 26 G generalised function, 2 with parameter, 123 with pole, 1 38 H with zero, 136, 137 Handelsman, R. A., 2, 159 WKB approximation, 119 Howls, C. J., 99 , 159 Dingle, R. B., 99 , 159 hyperasymptotics, 99 divergence, 1 accuracy, 106 dominant asymptotic expansion, 1 23 analytic continuation, 109 differential equation, 135 E numerical examples, 107, 109, 113, endpoint, 117, 118 near saddle point, 89 Stieltjes transform, 110 entire function, 45 technique, 106 asymptotic behaviour, 59 exponential type, 50 integral of, 51 I idealisation rule, 142 expansion, incomplete gamma function, 61 asymptotic, 5 infinite integral, 156 F of infinitesimal, 156 Fourier integral, 32 infinite series, 158 sum of, 158 as limit, 34 integrand of bounded variation, 34 infinitesimal, 3, 20 Friedman, B., 92, 159 standard, 145 function, infinitesimal number, 145 Airy, 45, 63 integer, nonstandard, 142 Beta, 1 1 5 bounded variation, 34 integral, and series, 158 confluent hypergeometric, 21, 6 1 as standard part, 154 entire, 45, 59 asymptotic behaviour, 17 generalised, 2 contour, 27 incomplete gamma , 61 Fourier, 32 J (p,, z) , 66, 67, 79 infinite, 156 parabolic cylinder, 90 regular, 1 1 Laplace, 23, 44, 100 Laplace-type, 26 S-continuous, 149 S-differentiable, 151 of continuous function, 154 of entire function, 51 standard, 144
of infinitesimal, partial sum of, Pearcey, ray,
163
Index
Asymptotics
155, 156 65
nonstandard i nteger, nonstandard set ,
142
141
number,
95
145 145 unlimited, 19, 142
infinitesimal,
25
156 swallowtail, 95
limited,
singular,
121 121
i ntegral equation, iteration in,
0
99, 134, 159 2, 99, 134, 159, 160 optimal remainder , 100 Olde Daalhuis, A. B.,
interval,
150 standard, 143 iteration, 121 Izumi , S., 56, 159
Olver,
compact ,
F. W.
J. ,
p
parabolic cylinder function, Paris, R. B . ,
J
J(p,, z), 66, 67, 79
part ,
analytic continuation, Jones , D.
90
99, 160
80
S., 2, 1 59
145, 150 53, 56, 65 o f integral , 65 Pearcey integral , 95 standard ,
partial sum,
L
23, 44 100 Laplace-type integral , 26 Lighthill, M. J. , 2, 159 limited number, 145
Laplace integral,
hyperasymptotics ,
pole near saddle point,
87
10, 49 coefficients, 49 derivative of, 1 1 , 1 2
power series,
growth of coefficients,
51
11, 15 normalised, 49 integral of,
M Maclaurin series ,
6, 70, 100 158
Maclaurin-Cauchy test, MAPLE,
R R, 142
83, 107
MATHEMATICA ,
ray i ntegral,
McLeod, J. B . ,
recessive asymptotic expansion,
83, 107 99, 159
39 method of steepest descents , 28 modified Bessel function, 6 1 , 62, 69, 112 method of stationary phase ,
25, 48, 67
regular function, remainder,
11
1, 2
expansion of,
99
73 135 minimising, 102, 103 numerical examples, 69 optimal, 73, 100 Riemann surface, 4 exponentially small,
hyperasymptotics,
N
N , 142 Nelson, E.,
3, 141 , 1 59
nonstandard analysis,
141
123
164
Asymptotics
Index
standard infinitesimal, 145 Riemann-Lebesgue Lemma, 32 standard interval, 143 Robert, A., 2, 160 Robinson, A. , 141 , 159, 160 standard part, 145, 150 Robinson's Continuous Lemma, 13, 149 standard sequence, 146, 148 standard set, 141 Robinson's Lemma, 147 Rolle's theorem, 152 standard sets, equality of, 143 standardisation rule, 142, 150 s stationary phase, 38 S-continuity, 149, 150 and continuity, 149 method of, 39 examples, 1 50 steepest descent, S-continuous, 17 curve of, 28 method of, 28 S-continuous function, 149 Stegun, I. A., 90, 91, 159 S-derivative, 1 51 higher, 153 Stieltjes, T. J. , 99, 160 Stieltjes transform, 110, 135 saddle point, 27 analytic continuation, 1 13 near endpoint, 89 hyperasymptotics, 116 near pole, 87 Stirling's formula, 17, 19 saddle points, Stokes line, 79 coalescing, 92 sequence, 146 Stokes' phenomenon, 79, 113, 135 strong derivative, 155 asymptotic, 3 strong differentiability, 153 of functions, 148 and continuity, 153 standard, 146, 148 series, 43 sum, of infinite series, 158 and integral, 1 58 of infinitesimals, 158 divergent, 1 partial, 53 infinite, 158 Maclaurin, 6, 70, 100 asymptotics, 53, 56, 65 of integral, 65 power, 10, 49 set, 141 surface, nonstandard, 141 Riemann, 4 of all subsets, 143 swallowtail integral, 95 of standard elements, 144 standard, 141 T set theory, 141 (T), 143 Taylor's theorem, 153 Zermelo-Fraenkel system, 141 transfer rule, 3, 143 singular integral, 156 truncation, 1 standard function, 144, 147 best, 74 standard functions, numerical examples, 69 equality of, 144 ·
Asymptotics u
Index
uniform asymptotic expansion, 12 uniform asymptotic representation, 79 uniform asymptotics, 79 unlimited number, 19, 142 Ursell, F., 92, 159 v
V"(x), 90 as parabolic cylinder function, 90 asymptotic behaviour, 91 van den Berg, I., 2, 160 w
Watson's Lemma, 44, 45 WKB approximation, 119, 120, 126 error bound, 1 20 Wong, R., 2, 160 Wood, A. D., 99, 160 z
Zermelo-Fraenkel system, 141
165
