!" $ %!!"!!&' ()&(*+ -.- /01213124 563718191 :17; . . . ...
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!" $ %!!"!!&' ()&(*+ -.- /01213124 563718191 :17; . . . . . . . . . . . . -.= >? 263@6AB . . . . . . . . . . . . . . . . . . . . . . -.D E01048 8 F0G9GH I6JG . . . . . . . . . . . . . . . # L!*(M'"(NO ! MPQ =.- S8? TBFB56 . . . . . . . . . . . . . . . . . . . . . . =.= VF? 63@6II4? A10I6W . . . . . . . . . . . . . . . . =.D XB9BFAB 3622?@066 . . . . . . . . . . . . . . . . . , YMQ!Z [(\N (!+(+] D.- ^:?_6`6AB n 6T2?0?I6a . . . . . . . . . . . . . . D.= >? 5?0@18b6IB 76W . . . . . . . . . . . . . . . . . R Y d)Q( )(Q( OQ'e f(*&!" B8BxF?I6? 0B8I1313@B87?II13@6 . . . . . . . . . y z(\(*( &! "!N&Q!&] c.- {1:0?A6 1x6FBI6;2 . . . . . . . . . . . . . . . . c.= w3@15I6A6 TBh7GxF?I6a . . . . . . . . . . . . . . c.D E01A7;@6? 676 :1FB01AW . . . . . . . . . . . . . . } !"!Q' f' !M&[Z(+ p.- E013@B; 718GJAB . . . . . . . . . . . . . . . . . . p.= E06?2 86TGB76TB_66 . . . . . . . . . . . . . . . . p.D S12BJI6? TB91@18A6 m810_B . . . . . . . . . . . . v ~fd*(+ *(&(&!f' K.- E0656I4 2?@B210`1T . . . . . . . . . . . . . . . . K.= ^@0GA@G0IB; G3@1a56813@o . . . . . . . . . . . . . K.D E1F81FIB; 5B3@o Ba3h?09B . . . . . . . . . . . . . $ !Q M!'" " )*!QPQ!&O -|.- ?0@84 . . . . . . . . . . . . . . . . . . . . . . . . -|.= GIA_61IB7oI4a BIB76T . . . . . . . . . . . . . . -|.D h1hb?II4? `GIA_66 . . . . . . . . . . . . . . . $$ L
&! --.- E;@4a :13@G7B@ . . . . . . . . . . . . . . . . . . . --.= j7GxFBI6? 8:1@o2Bk . . . . . . . . . . . . . . . . --.D wTF?8B@?7o3A6? @?10?24 . . . . . . . . . . . . . . --.< 1AG3 7?aIB . . . . . . . . . . . . . . . . . . . . --.U S0G9B; 3@101IB 2?FB76 . . . . . . . . . . . . . . . --.C >?A1@104? :1F01hI13@6 . . . . . . . . . . . . . . $# (*+N *(* !&(Q d& -=.- nB3:0?F?7?I6? 3A1013@?a. . . . . . . . . . . . . . -=.= g10B7o . . . . . . . . . . . . . . . . . . . . . . . . -=.D 1I@0B3@ :0656I 6 37?F3@86a . . . . . . . . . . . -=.< wF?17196; 6I8B06BI@I13@6 . . . . . . . . . . . .
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Q]Q' N" QN $# -D.- i@@0BA@104 6 `0BA@B74 . . . . . . . . . . . . . . -|= -D.= ^B21:1F1h6? 6 {3?7?IIB; . . . . . . . . . . . . . -|< -D.D S?@?026I6018BII4a kB13 . . . . . . . . . . . . . -|U -D.< j19B@3@81 1FI12?0I191 37G5B; . . . . . . . . . . -|c $R %'P [!&O !*(Zd[!&O $$ -B:0BJ68B?@3; :01_?FG0B :137?F18B@?7oI191 845637?I6;f (n)_6`0 F?3;@65I191 0BT71x?I6; 5637B π . 376 IB ABA12@1 JB9? :1;87;?@3; n 313?FI6k 3?2? 01A :01_?33 13@BIB8768B?@3; f (n) = 1 . FIBA1 ?376 f (n) = 0 --=
F7; I?A1@10191 n @1 GABTBIIB; :01_?FG0B I?s``?A@68IB I6A19FB I? 13@BI186@3;. nBTG2??@3; I?0Bh1@13:131hI13@o A1IA0?@I1a 3k?24 84563 7?I6a I65?91 I? F1ABT48B?@.{1T21xI1 3Gb?3@8G?@ F0G9B; s` `?A@68IB; :01_?FG0B. mB x? :?3I; 801F? h4 81TI6AB?@ 8 36@GB_66 g(n) =
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% ' 3 4 % 5 0 ,' 2( ' # " 6 " 2 , 7 ;
>1 `GIA_6; g(n) 84563762B :13A17oAG g(n) ≡ 1 . S1ABTB@?7o3@81 s@191 `BA@B 7?9A1 8481F6@3; 6T 6T8?3@I1a @?10?24 01I?A?0BÜ 0,
& 0 2 , ) % & + , 6 # $ 3 * / ".
l Sj R 3W] `3S hZ3 y 3 W QW3Z QS hZ3 y 3 kWTS Q W S i f[2 a X XX a x y `T1 y _Q m 3Y Z3 UQ] QV h 1S [ 1 W V QUW1a kV 3 \ (x < y) m n x < ma − n < y.
E1ABx?2 @?:?0o5@1 83?9FB 3Gb?3@8G?@ A8BF0B@ _?7191 563 7B F?3;@65IB; TB:63o A1@10191 IB56IB?@3; 3 7Hh1a IB:?0?F TBFBII1a :137?F18B@?7oI13@6 _6`0. EG3@o TBFBIIB; :137?F1 8B@?7oI13@o _6`0 1h1TIB5B?@ 8 F?3;@65I1a TB:636 56371 A . >B2 IBF1 :1ABTB@o 5@1 IBaFG@3; @BA6? _?74? k 6 p 5@1 A · 10p < k 2 < (A + 1) · 10p .
E137? 719B06`26018BI6; I?0B8?I3@81 :?0?k1F6@ 8 lg A < 2 lg k − p < lg(A + 1).
E17B9B; k = 2 p = 2q :17G5B?2 m
lg A < 2m lg 2 − 2q < lg(A + 1).
SB7?? 13@B?@3; 3137B@o3; IB @?10?2G 01I?A?0B. >B s@12 `1I? 21x?@ 81TI6AIG@o 8:?5B@7?I6? 5@1 6 3 36@GB_6; @BA 676 6IB5? 0BT0?JB?@3;. >1 s@1 s21_66. f (n) 102B7oI1? F1ABTB@?7o3@81 3Gb?3@818BI6; I?845637624k `GIA_6a F1817oI1 :013@1. w3k1FI42 :GIA@12 ;87;?@3; IG2? 0B_6; 8456376@?7oI4k :0190B22 5@1 7?9A1 3F?7B@o. ^IB5B7B 21xI1 :?0?356@B@o :0190B224 6T 1FI1a A12BIF4 :1@12 6T --D
F8Gk :1@12 6T @0?k 6 @BA FB7??. { 0?TG7o@B@? ABxFB; :01 90B22B :17G56@ 381a I12?0 k . E0190B22G 1h1TIB562 5?0?T B 311@8?@3@8GHbGH `GIA_6H 5?0?T ϕ (n) . P nB3321@062 @?:?0o `GIA_6H ?376 ϕ (n) 1:0?F?7?I1 ϕ (n) + 1, ?376 I? 1:0?F?7?I1. h(n) = Ú-1 s@191 I? 21x?@ h4@o @BA ABA h(p) 6= ϕ (p) ?376 TIB5?I6? ϕ (p) 1:0?F?7?I1 6 1:0?F?7?I1 ?376 ϕ (p) I? 1:0?F?7?I1. h(p) Ë ÌÞ ð Ô âÔ SBIIB; 1h7B3@o 6T1h67G?@ :B0BF1A3B26 I1 TF?3o 1I6 8 h17o J6I3@8? 381?2 6T 1h7B3@6 1J6h15I4k G21TBA7H5?I6a. m?2 I? 2?I?? s@6 1J6hA6 :17?TI1 :?0?x6@o 5@1h4 TIB@o AGFB k1 F6@o I? IBF1. {1@ 1F6I 6T :062?018. ï ìôéò ûçî + ó ô öóëõóòì í éëõó ùõð õ ýì ì ëû éëèôûéè øêéùøóëïçóéí øì îìèìêìòû ï îìçéöçìé öóëõì úï ìô ô öóëõðéèëð . êìçûòéêûéò èïîóé ûçî + óó ó ìøêéùéõóò k
k
n
n
n
p
p
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p
f (n)
n
f (n)
èï )ûçîìè+ óð óçéèïîô ùïõéé öóëõóòïí èïî îïî øêó ìèõóöïéèëð ìè í øêó ( ùêû ì ëèìêìç í ìçï ô öóëõóòïí øìëîìõüîû î ô öóëõðéòìòû çïöéçó ôëéùï òì çì øêóýïôóèü ì çóîïéè ôìøêìëí ïöéò ý õì òûùêóèü ë ûçî + óé # & . êóöó "# ## ûçî + ó çï ôéëüòï ëéêüé ìçï çïð ý Dõï ýïðô öóëõóòï øìø èîï øêó ìøêéùéõóèü ô öóëõóòû èïîí öèìý õ ýìò í ) øêóç + óøóïõüçì ìý êéöéçï çï øêìôïõ æõó øìõûöï èëð ëõóúîìò û îóé îõïëë ûçî + ó ùóë " îêéùóèóêû óé øìçðèóé ô öóëõóòìëèó & í óõó îïîóé èì ìýð ëóõìô ïèéõüçì óòé è ìýõïëèü çéìøêéùéõéççìëèó . ìø èîï ùììøêéùéõóèü ò òéèìùìò èì
é çé øêì ìùóè çéêï êéúóòìëèü øêìýõéò ìëèïçìôï & ìêéçü çéøêóðèçìëèé "ùìôìõüçì ìöéôóùéç - øêéùéõéçóé ûçî + ó ë øìòì ü ïôéùìòì ô øìõçóò øêðò ùé ëèôó èóøï ëõì óèüí ûòçì "
óèü & øêóôìùóè ô óèì é î ûêïôçéçóðò g(n) = fn (n) + 1.
n = 1
n=2
f1
f2
fn (n)
f (n)
n
f (n) f (n)
, ,& "# +
îìèìê éëèéëèôéççì ìèçìëóèü î ô öóëõóò ò ûçî ó ðòêéúéçóé øêìèóôçìò ëõûöïé èéìêóð øêéôêï ïéèëð ô òïõìøìõé çû ó êûúîû èì éé ôêéòð , ,& øêóç + óøóïõüçì çé ôëé ùï êï êéúóòì "# øêóòéê ìùóç øìøûõðêç u(a, n) = 0,
n = f (a)
--
?A1@104? 6T I6k 3@1;@ :11FB7o 6 8 I6k 6T1h0Bx?I6? @191 x? ;87?I6; F1:17I;?@3; :13@101II626 F?@B7;26 Ú@?10?24 V?F?7; IB:062?0Û. wT8?3@I4 2I1915637?II4? :1:4@A6 1:0?F?7?I6; 8456376 21a `GIA_66. { -KDC 91FG moH06I9 88?7 `GIA_66 8456376 24? A1I?5I1a 2BJ6I1a 6T8?3@I1a I4I? ABA m Q WZ Q hiX WZ^ y Qb { @12 x? 91FG V?F?7o t0h0BI 76I6 88?76 X 1U \X TW`Z[ 1 º ZUWW b XB@?2 ?05 E13@ gB0A18 84F?7;76 6I4? A7B334 \ `GIA_6a. 3676;26 2I196k 2B@?2B@6A18 h471 G3@BI187?I1 5@1 83? :G@6 ABA 918106@3; 8?FG@ 8 n62 :0681F;@ A 1FI12G 6 @12G x? A7B33G 845637624k `GIA_6a. Ë Ì N ÔÏ ã áÏÖÒ Ð ÏÍ àÑáÖã 0B8I?I6? - êì õü øêóøóëïõ öïëèó + ïò ùõóç ôìõç øì ïçïõì óó ë
ìèìçïòóíèì ô îïöéëèôé ôìõçìôì ûçî + óó ý õì éëèéëèôéççì óëøêìýìôïèü îìøó & ë ïòéçì êï ôé öèì ýûîô çï öèì çóýûùü ùêû ìé * ïî "# ôìõçìôïð ## ôì çóîõï
ûçî + óð E = E0 ei(ωt−kr) ,
ω r = {x, y, z}
(ω = 2πν) k
E
,&
"# # ìèõóöï ïðëð ìè & õóúü øéêé ïøóëü øïêïòéèêìô ë ûöéèìò ìêìúì ó ôéëèç ëôð é "# # # $& "# # ùé ) øìëèìðççïð . õïçîïí ùéõéççïð çï ó éêéç + óïõüçìé ûêïôçéçóéí îìèìêìòû ûùìôõéèôìêðéè ,& í õé "# # îì øìùýóêïéèëð ó éêéç + óêìôïçóé ,& ùïéè "# # i
ψ(r, t) = ψ0 e ~ (Et−pr) ,
E = ~ω,
~
p = k~, 2π
ï èïî é *éøéêü ó ëõéùûéè & "# # öèì çóð ó øêéùëèïôõðéè êéùóç éêï ëìýì ìùóç ó ôì òì ç ôïêóïçèìô ïøóëó ûêïôçé D ýì úï ô çìôìò çïøêïôõéçóóí îïî øêïôóõìí ìîï ôïéèëð øéêô òí èêéýûð ùïõüçé úé ì ùôó éçóð èìò ëò ëõé ðëçìíöèì çï ïøóëó "# # & îôïçèìôïð òé ïçóîï çé ïîïçöóôïéèëð Ë Ë ÌÞ ¯ ãÑØÎÏÑáÎã à Ôà ÖÏÕ ^:G3@623; 3 I?h?3 IB X?27H. :0?F?762 3A1013@o 0B3:01 3@0BI?I6; 62:G7o3B 3xB@6; 8F17o 3@?0xI; 3 3?5?I6?2c S Ú063. -B :0BA@6A? @BA191 310@B A17?hBI6; F?a3@86@?7oI1 Ú62?H@ 2?3@1. SB7??. E0?F3@B862 5@1 8 @15A? A :016T81F6@3; 1@3@0?7 817 A18 6 363@?2B :?0?:04968B?@ 8 @15AG B .SB7oI?aJ?? F86x?I6? hGF?@ :0163k1F6@o :1 s776:3G 2?IoJ?91 0BT2?0B 5@1 6I@G6 @68I1 1x6FB?21. m?:?0o G2?IoJ62 5637?II13@o 817A18 8 @15 A? C . ^63@?2B :?0?aF?@ 8 D 6 hGF?@ F869B@o3; :1 h17oJ?2G s776:3G. i2:76@GFB A17?hBI6a G8?7656@3;. t@1 Gx? :01@681 0?56@ 6I@G6_66 I1 :0BA@6AB :1F@8?0xFB?@ @BA191 01FB 2?@B 210`1T4. {4h10 212?I@B 1@3@0?7B 1ABT48B?@3; 3Gb?3@8?II1 876;?@ IB :18?F?I6? 363@?24. - ìõéé èì ìí ëïòï ó óîï ôëé öï é ëèïõîóôï éèëð ë øêìýõéòïòó çìôì ì èóøïí ùé èìöç é øêéùëîï ïçóð øêóç + óøóïõüçì çéôì òì ç ( . )
( . +
Ë Ò Ì ßàÖàÎÏÙá Ï Ï ãâá g1xI1 76 F8B @?7B 0BTI4k @?2:?0B@G0 :068?3@6 8 A1I@BA@ @BA62 1h0BT12 5@1h4 1I6 1h2?I;763o @?2:?0B@G0B26W gI1 962 81:013 :1ABx?@3; I?7?:42 6h1 k101J1 6T8?3@I1 5@1 :06 A1I@BA@? @?:71 :?0?k1F6@ 1@ 910;5?91 @?7B A k171FI12G. 19FB @?2:?0B@G04 30B8I68BH@3; :01_?33 :0?A0BbB?@3; 6 IB3@G:B ?@ 0B8I18?36?. m?2 I? 2?I?? :0B867oI4a 1@8?@ R3S 3Y W^ V 1S hZ[ 4 . XBFB5G 0?JB?@ F6IB265?3A6a A1I@BA@ª. g4 I? hGF?2 FB 8B@o @15I191 0?J?I6;.nB3321@062 690GJ?5IGH 21F?7o.EG3@o 62??@3; F8B @?7B ABxF1? 6T A1@104k 313@16@ 6T F8Gk :1 7186I1A A = {A A, AB} B = {B , B } . m?:71:?0?FB5B 2?xFG :17186IAB26 1FI191 @?7B 15?Io 2?F7?IIB; 2?xFG A B h43@0B;. P
1
2
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A2
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¸º®Æ ÆA Æ o¯½ º ì èéøõìôìò ìýòéçé ëôð ïç ëì ëèïèóöéëîóòó « - ý öç çé òóøêéùëèïôõéçóð çéøìùôó & îìçèïîèïòó " Q
-B5B7oIB; @?2:?0B@G0B A 100 B 0 . 1I@BA@ :01 81F62 8 < s@B:B. ^IB5B7B :0681F62 8 A1I@BA@ 6 B . { 0?TG7o@B@? @?2:?0B@G04 A 6 B G0B8I68BH@3; 6A3@BI18;@3; 0B8I426 50 Ú063. -.B 37?FGHb?2 s@B:? A1I@BA@ 6T1h0Bx?I IB 063.-.U. ?9 A1 86F?@o 5@1 :137? 840B8I68BI6; @?2:?0B@G04 3@BI18;@3; @BA626 A1 B1
A2 B2
¸º®Æ Æ Æ µ´¸´ 9 ȵ³ A1 , B1 − 75◦ ,
A2 , B2 − 25◦ .
^7?FGHb6a s@B: FB?@ @?2:?0B@G04
A1 − 50◦ , A2 − 25◦
B1 − 75◦ , B2 − 50◦ .
E137?FI6a s@B:Ü @?7B 0BT81F;@3; 6 :137? 840B8I68BI6; @?2:?0B@G0 2?xFG :17186IAB26 62??2 8IB5B7? h471 100 ), A − 37,5 ( 8IB5B7? h471 0 ), B − 62,5 ( - ìõéé èìöçìí èéò øéêïèûêï øêìøìê + óìçïõüçï í ùé ) êïëëèìðçóé ùì + éçèêï ìè êïë ëòïèêóôïéòì èìöîó . êéùøìõì óò ùïõééí öèì ôëé èéõï óòé è ìùóçïîìô îì ó + óéçè êïëúóêéçóðí øêóöéò õóçé ç é êï òéê õ ýì ì èéõï ëèêì ì øêìøìê + óì çïõüç ïýëìõ èçì èéòøéêïèûêé . ûëèüí çïîìçé + í ëôì ëèôï òìùéõó èïîì ôèéòøéêïèûêì í öèì õ ýì øêéùòéèí øéêéçéëéçç ó ìùçì èìöîó ô ùêû û í ë óçì í èìèöïë øêó ìùóè ô ëìëèìðçóé èéøõìôì ì êïôçìôéëóð ë çìôì ìîêûD ïýìø é ëêéùì èçìí öèì èï ëéõéççïð ýûùéè îï ïèüëð é ì ìýóèïèéõðò ýéë îìçéöçì ëïòìò ùéõéí éëõó ìçó øì éõï è øêóýõó óèüëð î êïçóöçì ëòéêï
éêéí èì ýûùûè ì õï ùïèüëð øì òéêé øêóýõó éçóð ó ûòéçüúïèüëð ô êï ï ó ó í ëììèôéèëèôéççìí ýûùûè ûîìêïöóôïèüëðí ó ìçó çóîì ùï çé ëòì ûè ùìëèóöü êïçó + ëôìé ì òóêï = ùéëü éëèü çïù öéò øìêï ò ëõóèü ïøêóòéêí çïù èéòí òì ûè õó ìýóèïèéõó èì ì òóêï ìýçïêû óèü ëû é ëèôìôïçóé ìùéõüíèéòøéêïèûê îìçéöçìí çéìý ; ðëçðéèí îïî ûëèêìéç çïú òóê = ïèì ) ùïéè
R
R2 − r 2
H
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-