3 a0 a1 F a2a4a3a4a22a8a23a24a5a4a6a16a7a11a9a11a25 7 F a26 10{11 a27 (a28a30a29 J (z) a31 N (z) a32a30a33a30a34 ) a35a24a36 a7a24a37a8a38a8a39a41a40a16a42a44a43a13a5a11a6 ? E tc`?`f` ??1: ? E tc`?`f` ? |HelmholtzZ??US"/s ?M  H;ü¤?è±sZ? 1 r d dr ? rdR(r)dr ? + h k2 ??? ?r2 i R(r) = 0: ?Tk2 ?? 6= 0TMDx = pk2 ??r; y(x) = R(r)5Z?M1(”¨)BesselZ? 1 x d dx ? xdy(x)dx ? + ? 1? ” 2 x2 ? y(x) = 0; ?? = ”2 BesselZ?μ ? ?x = 0?x = 1 x = 0 ^?5 ?x = 1 ^d?5 ? ?5 ?x = 0)·S‰ = §” ? Bc?Xü p BesselZ?x = 0?￥?53 / ?211 ? B/Xü¤?￥2T ?” 6=? ? HBesselZ?￥ ?(L?í1)?53 ^ J§”(x) = 1X k=0 (?)k k!Γ(k§” +1) ?x 2 ·2k§” : ?T” =? ?n5Jn(x)?J?n(x)L?M1 J?n(x) = (?)nJn(x); ? HBesselZ?￥?B3 ˉ ^Jn(x)?=35 V |1 Nn(x) = 2…Jn(x)ln x2 ? 1… n?1X k=0 (n?k?1)! k! ?x 2 ·2k?n ?1… 1X k=0 (?)k k!(k +n)! ￡?(n+k +1)+?(k +1)??x 2 ·2k+n ; i O???n = 0 H3 ??Vr T??=[￥μK? x17.1 Besself ?￥'?é?2: x17.1 Besself ?￥'?é BesselZ??￥”2'?Yè ^?'?′ù5 '00 +?' = 0; '(0) = '(2…); '0(0) = '0(2…) %?￥? = m2; m = 0;1;2;￠￠￠ yN'??|"×o ? ?¨Besself ?￥?é/ ?5 [ -Xü¤?V￥Bt2 T(m17.1?ó  -+?Besself ?￥m?) m17.1`Besself ? 1. J?n(x)?Jn(x)L?M1 J?n(x) = (?)nJn(x): ￡ ün6.4? 2. Jn(x)￥  }? Jn(?x) = (?)nJn(x): V[V” = n H￥Vr T°¤ A 3. Jn(x)￥ 3?f ? exp ?x 2 t? 1t ?? = 1X n=?1 Jn(x)tn; 0 < jtj < 1: ￡ ün5.4? è7 ? ?¨Besself ?￥  eBt?é x17.1 Besself ?￥'?é?3: 4. Jn(x)￥sV U Jn(x) = 1… Z … 0 cos(xsin ?n )d : ￡￡￡ 3?f ?Vr T exp ?x 2 t? 1t ?? = 1X n=?1 Jn(x)tn ? 7t = ei  1 2 t? 1t ? = 12 ?ei ?e?i ￠ = isin eixsin = 1X n=?1 Jn(x)ein : ?ü ^f ?eixsin ￥FourierZ 7 T(ˉ ?? T)?FourierZ 7￥" ? Tü ?￡¤ Jn(x) = 12… Z … ?… eixsin ? ein ·? d = 12… Z … ?… [cos(xsin ?n )+isin(xsin ?n )]d : · s￥$f ??′? ^ f ? ?[s10 L? ^ }f ? ?[ü ?°¤?1  ?￥sV U ?T|$f ??￥? ?n?1 ?iˉ ?”?"¤?￥i? ^f ?J”(x)￥sV U 5. ?T 3?f ?Vr T? 7t = iei ? V[¤? eixcos = 1X n=?1 Jn(x)inein =J0(x)+ 1X n=1 h inJn(x)ein +J?n(x)i?ne?in i =J0(x)+ 1X n=1 h inJn(x)ein +(?)ni?nJn(x)e?in i =J0(x)+2 1X n=1 inJn(x)cosn : +Y ^ ?T 7x = kr? ^üμ eikrcos = J0(kr)+2 1X n=1 inJn(kr)cosn : ü  T?￥r? ?31?US"?￥USM i Oük ?31o ?] H |Mê￥ HWy 01e?i!t5  T  ?sY??o?V?Mêy0￥ bW?sP ^?xàZ_. x17.1 Besself ?￥'?é?4: l?￥ ü ?oy1 ?￥?Mê ? ^ krcos ?!t =è ?; 7· ò[?￥J0(kr)?Jn(kr) í ?￥ ^? ?o( ??n/ ?￥?é8)yN??Z 7 T￥i lü ^ ü ?o?? ?oZ 7 [ o ￥? ^? ?¨Besself ?￥?é/ ?o +??é ?i¨Besself ?? ? ? 6. Besself ?J”(x)?J?”(x)￥Wronski?  T W [J”(x); J?”(x)] · flfl flfl fl J”(x) J?”(x) J0”(x) J0?”(x) flfl flfl fl = ? 2 …x sin…”: ￡￡￡? BesselZ? 1 x d dx ? xdJ”(x)dx ? + ? 1? ” 2 x2 ? J”(x) = 0; 1 x d dx ? xdJ?”(x)dx ? + ? 1? ” 2 x2 ? J?”(x) = 0: [xJ?”(x); xJ”(x)sYe? ?Z?Mh'¤ J?”(x) ddx ? xdJ”(x)dx ? ?J”(x) ddx ? xdJ?”(x)dx ? = ddx n x￡J?”(x)J0”(x)?J”(x)J0?”(x)? o = 0: ?[ J”(x)J0?”(x)?J?”(x)J0”(x) · W [J”(x); J?”(x)] = Cx: sè ?Cü ^J”(x)J0?”(x)?J?”(x)J0”(x)?x?1[￥" ? J”(x) = 1X k=0 (?)k k!Γ(k +” +1) ?x 2 ·2k+” = 1Γ(” +1) ?x 2 ·” ? 1Γ(” +2) ?x 2 ·”+2 +￠￠￠ J0”(x) = 1Γ(” +1) ”2 ?x 2 ·”?1 ? 1Γ(” +2) ” +22 ?x 2 ·”+1 +￠￠￠ J?”(x) = 1X k=0 (?)k k!Γ(k?” +1) ?x 2 ·2k?” = 1Γ(?” +1) ?x 2 ·?” ? 1Γ(?” +2) ?x 2 ·?”+2 +￠￠￠ J0?”(x) = 1Γ(?” +1)?”2 ?x 2 ·?”?1 ? 1Γ(?” +2)?” +22 ?x 2 ·?”+1 +￠￠￠ ?.lZ_? ?DMê￥ HWy0￥??μ1 ?T | HWy01ei!t * 1?? ü ?oü ^_μxàZ_.l ￥ x17.1 Besself ?￥'?é?5: yN C = 1Γ(” +1) 12” ￠ 1Γ(?” +1) ?”2?” ? 1Γ(?” +1) 12?” ￠ 1Γ(” +1) ”2” = ? 2”Γ(” +1) Γ(?” +1) = ? 2Γ(”) Γ(1?”) = ? 2… sin…”: ?[ü￡¤ W [J”(x); J?”(x)] = ? 2…x sin…”: F?” 6=? ? HW [J”(x); J?”(x)] 6= 0J”(x)?J?”(x)L?í1 F?” =? ?n HW [Jn(x); J?n(x)] = 0Jn(x)?J?n(x)L?M1 7. Besself ?J”(x)?J?”(x)￥?w1" d dx [x ”J”(x)] = x”J”?1(x); d dx ￡x?”J ”(x) ? = ?x?”J ”+1(x): ￡￡￡°¤VBesself ?￥) ?Vr T???) ? ? ü ? l ? ?[ V[?[±  d dx [x ”J”(x)] = d dx 1X k=0 (?)k k!Γ(k +” +1) x2k+2” 22k+” = 1X k=0 (?)k k!Γ(k +”) x2k+2”?1 22k+”?1 =x”J”?1(x): ?ü ^?B T]" d dx h x?”J”(x) i = ddx 1X k=0 (?)k k!Γ(k +” +1) x2k 22k+” = 1X k=1 (?)k (k?1)!Γ(k +” +1) x2k?1 22k+”?1 = 1X k=0 (?)k+1 k!Γ(k +” +2) x2k+1 22k+”+1 = ?x?”J”+1(x): ?"ü?￡ ü ?= T V? ??w1"?h ?J”(x)J0”(x)? V[¤? ??￥?w1" J”?1(x)?J”+1(x) = 2J0”(x); J”?1(x)+J”+1(x) = 2”x J”(x): x17.1 Besself ?￥'?é?6: +Y ^ 7” = 0 J00(x) = ?J1(x): 8. Besself ?￥víZ 7Besself ?￥víZ 7μ ?'￥ ??B? a¨?x ! 0 J”(x) = 1Γ(” +1) ?x 2 ·” +O ? x”+2 · : ? V[°¤?Besself ?￥) ?Vr T¤? 6B?víZ 7 a¨?x !1 J”(x) ? r 2 …x cos ? x? ”…2 ? …4 · ; jargxj < …: 1 I 1J”(x) í ?￥ ^? ?o$ ? ??é5? ?T￥ *" 7x = kri Oür ?31?US"?￥USM ük ?31 o ? | HWy01e?i!t5?r@v HJ”(kr) ? í ?￥o?V?￥Mêü ^ cos ? kr? ”…2 ? …4 · e?i!t = 12 ‰ exp h i ? kr? ”…2 ? …4 ?!t ·i +exp h ?i ? kr? ”…2 ? …4 +!t ·i ; ?Mê ? ^? ? kr? ”…2 ? …4 currency1!t =è ?; sY í ?￥ ^?Mê ? ?" HW? ?v l ê￥? ?? ￥? ?o7 O?? J”(kr) ? r 2 …kr cos ? kr? ”…2 ? …4 · ; jarg(kr)j < … ??cμDpr?Q1￥??y0o?V?￥ ? @ áDr?Q1????￥§ ?Dr?? 1 ?[?ê HW =YV ???? ? @V￥9 ? ?M?ü ^ a í ?￥? ^B?? ?h￥ ? ?o 9. L ?¨Besself ?￥ ,??” > ?11? ? HJ”(x)μí k? ,? ? ì ??? ^ L ??1s? Là  1?J”(x) ,?￥i?? ú?￡ o￡ üa ?2 ?7 O?A) ?μ? ?¨￥Besself ? ?[o3) ?” > ?1￥ f? F J”(x)￥ ,?? V ? ^B′ ?y1?x ^B′ ? HJ”(x)￥í k) ?V U ^B??[) ?) ??? V ?10 x17.1 Besself ?￥'?é?7: F !fi ^J”(x)￥B? ,?'J”(fi) = 05fi￥ˉ afi?9B? ^J”(x)￥ ,? J”(fi?) = [J”(fi)]? = 0: 'J”(fix)?J”(fi?x) ([x = 11 ,? ? ìsY ?@Z? 1 x d dx ? xdJ”(fix)dx ? + ? fi2 ? ” 2 r2 ? J”(fix) = 0; 1 x d dx ? xdJ”(fi ?x) dx ? + ? fi?2 ? ” 2 r2 ? J”(fi?x) = 0: | ?Z?sYe[xJ”(fi?x)?xJ”(fix)Mh uW[0; 1] s'¤ ? fi2 ?fi?2 ·Z 1 0 xJ”(fix)J”(fi?x)dx = ?x ? J”(fi?x)dJ”(fix)dx ?J”(fix)dJ”(fi ?x) dx ?flfl flfl 1 0 = 0: ?? xJ”(fix)J”(fi?x) = xjJ”(fix)j2 ? 0; O??10 ?[?” > ?1 Hs Z 1 0 xJ”(fix)J”(fi?x)dx 6= 0; ?"ü￡¤ fi2 = fi?2; 'fi2 ^ L ? F? Hμ ? V ? fi2 ? 0'fi1 L ? ? fi2 < 0'fi1B′ ? ???fi? V ?1B′ ? ?[fiB? ^ L ? FB?J”(fi) = 05?J”(x)￥) ?Vr T V[ A9B?μJ”(?fi) = 0 ?[J”(x)￥ ,??1s? Là  ÷éB?? ?w1"?Rolle? ?ü V[??J”(x)￥M #￥ ? ,?-WA? μJ”§1(x)￥B? ,? x17.2 Neumannf ??8: x17.2 Neumannf ? F BesselZ?￥ ?3J§”(x)?” 6=? ? H ^L?í1￥ W [J”(x); J?”(x)] = ? 2…x sin…”; Z?￥Y3 V[V U1J§”(x)￥L?F? F?” =? ?n HJ§n(x) ^L?M1￥yN?31×? pZ?￥?=3 FVe5  ? aK'￥÷E ^ |?=31c ?[￥?53} ?Z??" ? F1? F ó￥÷E ^?” 6=? ? Hü?=39? ^e?1 |1J?”(x)7 ^ ˉ ? | 1J§”(x)￥L?F?? ? V[ a?1ê4F?" ? è ? | y2(x) = cJ”(x)?J?”(x)sin”… ; üB?μ W [J”(x); y2(x)] = 2…x: ?"' P” !? ?ny2(x) ˉ ?DJn(x)L?í1 F?” !? ?n H3 Ty2(x)￥s sin”… ! 0yN?A? a?ê4 6B?F?" ?c P¤y2(x)￥s09M103 T? V ?μil F I n? J?n(x) = (?)nJn(x) = cosn…Jn(x); # V |c = cos”… ?"ü?l Neumannf ?? N”(x) = cos”…J”(x)?J?”(x)sin”… ; ? ?” ^?1? ? ?9 V[ |1BesselZ?￥?=3 ? ?¨￥Neumannf ?Nn(x)?? ?31” ! n HN”(x)￥K Nn(x) = lim”!n cos”…J”(x)?J?”(x)sin”… = 1… ?@J ”(x) @” ?(?) n@J?”(x) @” ? ”=n = 2…Jn(x)ln x2 ? 1… n?1X k=0 (n?k?1)! k! ?x 2 ·2k?n ? 1… 1X k=0 (?)k k!(n+k)! ￡?(n+k +1) +?(k +1)? ?x 2 ·2k+n ; jargxj < …: ?μ￥óD?9TY”(x) x17.2 Neumannf ??9: i O???n = 0 H1 ??· ?=[￥μK? ?x ! 0; Re” > 0 HN”(x)￥ví?1? ??J?”(x) %? N”(x) ??Γ(”)… ?x 2 ·?” : 7?N0(x) N0(x) ? 2… ln x2: ?[? ?” ^?1? ?N”(x)x = 0?? ^? ?￥ ?x !1 HNeumannf ?￥víVr T ^ N”(x) ? r 2 …x sin ? x? ”…2 ? …4 · ; jargxj < …: yNN”(x)9 V[¨ ? í? ?o]"9 ^? ?￥? ?o?? ￥? ?o￥?F N”(x)￥?w1"￥? T?Besself ?? ?M] d dx [x ”N”(x)] = x”N”?1(x); d dx ￡x?”N ”(x) ? = ?x?”N ”+1(x): Besself ???1?B ??f ?Neumannf ???1?= ??f ? N0(x); N1(x)?N2(x)￥m?nm17.2 m17.2`Neumannf ?  ? N Hankel9;üBesselZ?￥?=3 |1 J”(z)?(?)nJ?”(z) ” ?n : ?” ! n H lim”!n J”(z)?(?) nJ?”(z) ” ?n = lim”!n ?J ”(z)?Jn(z) ” ?n ?(?) nJ?”(z)?J?n(z) ” ?n ? = ?@J ”(z) @” ?(?) n@J?”(z) @” ? ”=n =…Nn(z): (Watson, x 3.5) x17.2 Neumannf ??10: Zeros of the functions J”(z) & N”(z) 1. Real zeros When ” is real, the functions J”(z) & N”(z) each have an infinite number of zeros, all of which are simple with the possible exception of z = 0. For non-negative ” the sth positive zeros of these functions are denoted by j”;s and n”;s respectively. s j0;s j1;s n0;s n1;s 1 2.40483 3.83171 0.89358 2.19714 2 5.52008 7.01559 3.95768 5.42968 3 8.65373 10.17347 7.06805 8.59601 4 11.79153 13.32369 10.22235 11.74915 5 14.93092 16.47063 13.36110 14.89744 6 18.07106 19.61586 16.50092 18.04340 7 21.21164 22.76008 19.64131 21.18807 8 24.35247 25.90367 22.78203 24.33194 9 27.49348 29.04683 25.92296 27.47529 10 30.63461 32.18968 29.06403 30.61829 2. McMahon’s expansions for large zeros j”;s;n”;s ? fl ? ??18fl ? 4(??1)(7??31)3(8fl)3 ?32(??1)(83? 2 ?982?+3779) 15(8fl)5 ?64(??1)(6949? 3 ?153855?2 +1585743??6277237) 105(8fl)7 ?￠￠￠￠￠￠ ; s ”; ? = 4”2; fl = s+ ”2 ? 14 ? …; for j”;s fl = s+ ”2 ? 34 ? …; for n”;s 3. Complex zeros of J”(z) When ” ? ?1 the zeros of J”(z) are all real. If ” < ?1 and ” is not an integer the number of complex zeros of J”(z) is twice the integer part of (?”); if the integer part of (?”) is odd two of these zeros lie on the imaginary axis. 4. Complex zeros of N”(z) When ” is real the pattern of the complex zeros of N”(z) depends on the non-integer part of ”. Attention is confined here to the case ” = n, a positive integer or zero. x17.2 Neumannf ??11: Zeros of Nn(z) The figure above shows the approximate distribution of the complex zeros of Nn(z) in the region jargzj? …. The figure is symmetrical about the real axis. The two curves on the left extend to infinity, having the asymptotes Imz = §12 ln3 = §0:54931:::::: There are an infinite number of zeros near each of these curves. The two curves extending from z = ?n to z = n and bounding an eye-shaped domain intersect the imaginary axis at the points §i(na+b), where a = q t20 ?1 = 0:66274:::::: b = 12 q 1?t?20 ln2 = 0:19146:::::: and t0 = 1:19968:::::: is the positive root of cotht = t. There are n zeros near each of these curves. Complex zeros of N0(z) Complex zeros of N1(z) Real part Imaginary part Real part Imaginary part ?2:40302 0:53988 ?0:50274 0:78624 ?5:51988 0:54718 ?3:83353 0:56236 ?8:65367 0:54841 ?7:01590 0:55339 x17.3?f ??12: x17.3?f ? O ^ ?@?w1" d dx [x ”C”(x)] =x”C”?1(x); d dx h x?”C”(x) i = ?x?”C”+1(x) ￥f ?fC”(x)gd?1?f ? - ?o ￥Besself ??Neumannf ?? ^?f ? ???fff ? ? ?BBB??? ^ ^ ^BesselZZZ???￥￥￥333 n5ü?w1"?? C0”(x)+ ”xC”(x) = C”?1(x) (z) C0”(x)? ”xC”(x) = ?C”+1(x): (#) |(z) T± ¤ C00”(x)+ ”xC0”(x)? ”x2C”(x) = C0”?1(x): (##) |(#) T?￥”?1” ?1i|(z) T} ? C0”?1(x) = ” ?1x C”?1(x)?C”(x) = ” ?1x h C0”(x)+ ”xC”(x) i ?C”(x): } ?(##) T'¤ C00”(x)+ ”xC0”(x)? ”x2C”(x) = ” ?1x C0”(x)+ ”(” ?1)x2 C”(x)?C”(x); F? ?ü¤? C00”(x)+ 1xC0”(x)+ 1? ” 2 x2 ? C”(x) = 0: ?ü￡ ü ?f ?fC”(x)gB? ^BesselZ?￥3 x17.4 BesselZ?￥'?′ù5?13: x17.4 BesselZ?￥'?′ù5 p 1?%?￥?? ￥%μ ? q ??ù5?]?V ?) ?V￥ ê±sZ??3ù5Ci àμó SHq ? 1 p￥9? ^ í?? ??￥êM ?? ? HW? bW7M? C1 p￥ ^%μ ? q' pó? ê±sZ??H?Hq/￥ ?μò???  T ￥? ? q | ü ?US"i|USe?b??? ￥???" ê±sZ??H?Hqü ^ @2u @t2 ?c 2 ?1 r @ @r r@u@r ? + 1r2 @ 2u @`2 ? = 0; uflflr=0μ?, uflflr=a = 0; uflfl`=0 = uflfl`=2…, @u@` flfl flfl `=0 = @u@` flfl flfl `=2… : ù5C￥3u μ??M?(D`í1) ?$ C1 p￥ü ^H?Hq￥K?/??? V *t!′ P¤Z?μd ,3 u(r;`;t) = v(r;`)ei!t: |N3 T} ?Z?#H?Hqü V[¤? 1 r @ @r r@v@r ? + 1r2 @ 2v @`2 + ?! c ·2 v = 0; vflflr=0μ?vflflr=a = 0; vflfl`=0 = vflfl`=2…, @v@` flfl flfl `=0 = @v@` flfl flfl `=2… :  7v(r;`) = R(r)'(`)s ?M ü¤? ?'?′ù5 '00(`)+m2'(`) = 0; '(0) = '(2…), '0(0) = '0(2…) ? 1 r d dr ? rdR(r)dr ? + k2 ? m 2 r2 ? R(r) = 0; R(0)μ?, R(a) = 0; ?k = !=c ?B?'?′ù5XüQn?V?? ?￥'?′ m2; m = 0;1;2;3;￠￠￠ ; x17.4 BesselZ?￥'?′ù5?14: '?f ?1 'm(`) = ‰cosm`; sinm`: ?[?=?'?′ù5?? ?m2 ^X?￥7k2 ^'?′? p p3?=?'?′ù5 ?k = 0 Hè±sZ?￥Y3 ^ R(r) = C0rm +D0r?m: } ?H?Hq V[? C0 = D0 = 0: yNk = 0? ^'?′ ?k 6= 0YVMDx = kr; y(x) = R(r) V[|±sZ??1BesselZ?V7 p¤ ?￥ Y3 R(r) = CJm(kr)+DNm(kr): I n?H?Hq￥1 p R(0)μ?) D = 0; R(a) = 0; ) Jm(ka) = 0: !?(m)i ^m¨Besself ?Jm(x)￥?i?? ,?(?l?v ? )i = 1;2;3;￠￠￠5'?′ù5 ￥3 ^ '?′k2mi = ? ?(m)i a !2 ; i = 1;2;3;￠￠￠ ; '?f ?Rmi(r) = Jm(kmir): ? ^ü p¤ ?? ￥%μ??￥? ? q !mi = ? (m) i a c; ??(m)i ^m¨Besself ?Jm(x)￥?i?? ,? 1 s ?M E?￥?¨?1 ?11) ?  ?¤?￥'?f ?￥??BB1" o B? {1?]￥SE V[] H¤?'?f ?￥??BB1" n5'?f ?Rmi(r) = Jm(kmir) ? ?@￥±sZ??H?Hq 1 r d dr ? rdJm(kmir)dr ? + k2mi ? m 2 r2 ? Jm(kmir) = 0; Jm(0)μ?, Jm(kmia) = 0: x17.4 BesselZ?￥'?′ù5?15: ] Hf ?R(r) = Jm(kr) ? ?@￥±sZ??H?Hq 1 r d dr ? rdJm(kr)dr ? + k2 ? m 2 r2 ? Jm(kr) = 0; Jm(0)μ?: ?￥k1 ?i L ? ?[B? a ???μJm(ka) = 0 C¨rJm(kr)?rJm(kmir)sYe ?Z?Mh uW[0; a] sü¤? ? k2mi ?k2 ·Z a 0 Jm(kmir)Jm(kr)rdr = r ? Jm(kmir)dJm(kr)dr ?Jm(kr)dJm(kmir)dr ?flfl flfl r=a r=0 : } ?H?Hq V[|  ?￥2T?1 ? k2mi ?k2 ·Z a 0 Jm(kmir)Jm(kr)rdr = ?aJm(ka)dJm(kmir)dr flfl flfl r=a = ?kmiaJm(ka)J0m(kmia):(#)” (”) F?B? f? ^k = kmj 6= kmi,'k9 ^B?'?′,????kmi.? HüμJm(kmja) = 0,yN(#) T￥· 10.???kmj 6= kmi, ?[ Z a 0 Jm(kmir)Jm(kmjr)rdr = 0; kmi 6= kmj; '???]'?′￥'?f ? uW[0; a] [ ?×r?? F 6B? f? ^k = kmi? H(#) T￥  (10 V[5|(#) T￥  ]"[k2mi ? k2 ?a |Kk ! kmiü¤? Z a 0 J2m(kmir)rdr = ? lim k!kmi kmia k2mi ?k2Jm(ka)J 0 m(kmia) = a 2 2 ￡J0 m (kmia) ?2 : ?? ^'?f ?Jm(kmir)￥ Z ?T|  ?'?′ù5(17.46b)?r = a ￥ QH?Hq?1?= ?? ? ?H?Hq9 V [ ? ?1) ? 1?Besself ?B￥?!?oó2 ? ?Tf ?f(r) uW[0; a]  ?? Ooμμ K?v?l5 V?'?f ?Jm(kir)Z 7 f(r) = 1X i=1 biJm(kir); x17.4 BesselZ?￥'?′ù5?16: ?Jm(kir) ^'?′ù5￥37Z 7" ?1 bi = Z a 0 f(r)Jm(kir)rdr Z a 0 J2m(kir)rdr : ?"¤?￥) ? uW[–; a?–] (– > 0)  ^Bá l ?￥ è è è1??8￥ ? ? !μB?í ké￥??8??1a?1 ?1??ê¨?US"zà'1??8￥à ?T?8￥V ????10?1u0f(r) k p?8 =?￥s??M? 333A ?? H?uD`; zí1'u = u(r;t) ???3ù5 @u @t ? ? r @ @r r@u@r ? = 0; uflflr=0μ?, uflflr=a = 0; uflflt=0 = u0f(r) %? ? - ?￥B?) ? ?^N?3ù5￥B?3 u(r;t) = 1X i=1 ci J0 ? ?ira · exp ? ?? ??i a ·2 t ? ; ??i ^J0(x)￥?i?? ,?} ?Hqμ u(r;t)flflt=0 = 1X i=1 ci J0 ? ?ira · = u0f(r): ?[ ci = 2u0a2J2 1(?i) Z a 0 f(r)J0 ? ?ira · rdr: o1?? f(r)￥ 8? Tü V[ ?  ?￥s è ? ! f(r) = 1? ?r a ·2 ; Lμ ci = 2u0a2J2 1(?i) Z a 0 ? 1? ?r a ·2? J0 ? ?ira · rdr: 7x = r=a5 V?1 ci = 2u0J2 1(?i) Z 1 0 ? 1?x2 · J0(?ix)xdx: ?¨?w1"ü V[ ?  ?￥s Z 1 0 ? 1?x2 · J0(?ix)xdx = Z 1 0 ? 1?x2 · 1 ?i d dx [xJ1(?ix)]dx = ? 1?x2 · 1 ?ixJ1(?ix) flfl fl 1 0 + 2? i Z 1 0 x2J1(?ix)dx = 2?2 i x2J2(?ix) flfl fl 1 0 = 2?2 i J2(?i): x17.4 BesselZ?￥'?′ù5?17: y1 J0(x)+J2(x) = 2xJ1(x); i I n?J0(?i) = 0üμ J2(?i) = 2? i J1(?i): } ?'¤Z 1 0 ? 1?x2 · J0(?ix)xdx = 4?3 i J1(?i): ? ^KaL V p?F" ? ci = 8u0?3 iJ1(?i) : V  ?￥2T 1?x2 = 1X i=1 8 ?3iJ1(?i)J0(?ix) ? V[w?Bt? T è ? 7x = 0'¤ 1X i=1 8 ?3iJ1(?i) = 1: ?  ±  7x = 1? V¤? 2 = 1X i=1 8 ?2i '1 X i=1 1 ?2i = 1 4: è è è2?ìT ü ??_??￥%μ ? q !?ì￥ =???sY1a?b ? =H?( = ?)%??H?(??)1? p?ìT ü ??_??￥%μ ? q 333A ???ê¨ ü ?US"C￥ù5 ^1 u(r;t) ? ?@￥±sZ??H?H q? H? ? "B??￥ ? 4 ?? ? 4 ?1?¨ w ?US") ? ?Dù5?(?)??ù5 ? a í?V?￥ t ?  ^êM ?Yè ^B? O ?^¨ O AV U ? ^ bWUS? HW￥f ?°? US"/ V[°? ?19 ? O A? ?￥ ?iB?s  bWUS￥ ê± +Y ^ ?T¨Laplace ??? O f ?A￥T¨ ? T??S f ?￥T¨ àμ I 1?] r2A· @ 2A @x2 + @2A @y2 : 7 ?TA￥s Vr T A= Axex +Ayey; ¨Laplace ??T¨?A￥ Bs  è ?Ax9μ ? ?￥2T r2Ax · @ 2Ax @x2 + @2Ax @y2 : x17.4 BesselZ?￥'?′ù5?18: ey? ? ^°?US"/US O ex?ey? ^è O ? ? bWê?M?  e?? w ?US"￥ f ?ü?M] è ? ü ?US"/ O A￥s Vr T ^ A= Arer +A e ; US O er?e ? ^è O 7 ^ ￥f ?yN ?^ ?3 y z x ez er eθr z θ m17.3` ü ?US"?￥US O  r2A = r2 (Arer)+r2 (A e ) 6= h r2Ar i er + h r2A i e : y1?A? I nUS O ?USM ￥ ê±  Y L  ? ? ?D?Xü?? der d =e ; de d = ?er: ?[ r2A = ? r2Ar? 1r2Ar? 2r2 @A @ ? er + ? r2A ? 1r2A + 2r2 @Ar@ ? e : ?"' PA = 0￥Hq/o I nA￥?_s Ar9??μ r2A= r2 [Arer] = ? r2Ar ? 1r2Ar ? er + 2r2 @Ar@ e : Cí?e ?￥ù5 è5 ? ?￥Hq/?_êMu?? ?@±sZ?F @2u @t2 ?c 2 r2 ? 1r2 ? u = 0; @u@ = 0 ?[?_êMD í1u = u(r;t) ?@￥±sZ?F?H?Hq?? ^ @2u @t2 ?c 2 ?1 r @ @r r@u@r ? ? ur2 ? = 0; u flfl fl r=a = 0; @u@r flfl fl r=b = 0: x17.4 BesselZ?￥'?′ù5?19: 7 u(r;t) = R(r)e?i!t; } ?Z??H?HqL¤? 1 r d dr ? rdR(r)dr ? + ? k2 ? 1r2 ? R(r) = 0; R(a) = 0; R0(b) = 0; ?k = !=c ?^ A?k = 0 H'?′ù5í3? ^ VTMDx = kr?y(x) = R(r)7|Z??1BesselZ??N' V¤? R(r) = CJ1(kr)+DN1(kr): } ?H?Hq'¤ CJ1(ka)+DN1(ka) = 0; CJ01(kb)+DN01(kb) = 0: ? V[ A? ^1?C?D￥L?} ?Z?Fμd ,3￥ sA1Hq ^ flfl flflJ1(ka) N1(ka) J01(kb) N01(kb) flfl flfl = 0: ?"ü p¤ b???8?_??￥%μ ? q!i = kic ?ki ^ J1(ka)N01(kb)?N1(ka)J01(kb) = 0 ￥?i???(?l?v ? )M?￥%μ??  T ^ ui(r;t) = [N1(kia)J1(kir)?J1(kia)N1(kir)]e?ikict: x17.5cBesself ?￥s?20: x17.5cBesself ?￥s ?¨Besself ? p3 ê±sZ??3ù5 H1 ?? #?9 ?cBesself ?￥ sKe?￥μ/ +? ??￥s 1.$f ?1 af ?DBesself ?￥e ?? ??￥s?¨Besself ?￥?w1" ?9 ?17.4?￥ è5?ü??V?? ?? ￥s/ ?TB?÷ ?R￥) ? n5 I nsR x?J”(x)dx?¨?w1" d dx [x ”J”(x)] = x”J”?1(x); Qˉs?süμZ x?J”(x)dx = Z x??”?1x”+1J”(x)dx =x??”?1x”+1J”+1(x) ?(??” ?1) Z x??”?2x”+1J”+1(x)dx =x?J”+1(x)?(??” ?1) Z x??”?3x”+2J”+1(x)dx = h x?J”+1(x)?(??” ?1)x??1J”+2(x) i +(??” ?1)(??” ?3) Z x??”?5x”+3J”+2(x)dx = ￠￠￠ : ?s?sBQ" s?s￥[?9FB[??C￥s?$f ?￥ af ?￥Q ?ü??BQ7Besself ?￥¨ ?5 6ú1?"s?snQaL???sZ x??nJ”+n(x)dx: ?(??n)§(” +n) = 1' ?+” = 1??” = 2n+1; 5??s V[V U1μK￥? Ti O ?¨?ZE ?T?¨?w1" d dx h x?”J”(x) i = ?x?”J”+1(x); ×ˉ  ?￥) ?? V[¤?Z x?J”(x)dx = Z x?+”?1x?”+1J”(x)dx = ?x?+”?1x?”+1J”?1(x) ?(?+” ?1) Z x?+”?2x?”+1J”?1(x)dx = ?x?J”?1(x)+(?+” ?1) Z x??1J”?1(x)dx: x17.5cBesself ?￥s?21: ?" ?s?sBQ?C￥s?$f ?￥ af ?￥Q ???BQ7Besself ?￥¨ ?9??1? ^s?snQaL???s Z x??nJ”?n(x)dx: ?(??n)§(” ?n) = 1' ??” = 1?+” = 2n+1; 5??s9 V[V U1μK￥? T9 ?¨?ZE ??§” = } ? O?§” 6=  ? Hs Z x?J”(x)dx B?? ?¨?ZE è ? Z z 0 z”J”(z)dz = 2”?1p…Γ(” +1=2) ￡z￡J”(z)H”?1(z)?J”?1(z)H”(z)?: H”(z)?1(”¨)Struvef ? H”(z) = 1X k=0 (?)k Γ(k +3=2)Γ(” +k +3=2) ?z 2 ·2k+”+1 : 2.$f ?1· ?f ?DBesself ?￥e è ? I nsZ 1 0 e?axJ0(bx)dx; Rea > 0: 9 ??? ??￥s1 s?è· ?f ?e?ax ￡s l ??￥T¨yN ?^ n5 I n?¨Besself ?￥) ?V Ui?[s? V[? ?"?¤? í k) ?7a?31 p?f ?/ ?ü¨??÷E9 ?  ?￥s Z 1 0 e?axJ0(bx)dx = Z 1 0 e?ax 1X k=0 (?)k (k!)2 bx 2 ?2k dx = 1X k=0 (?)k (k!)2 b 2 ?2kZ 1 0 e?axx2kdx = 1X k=0 (?)k (k!)2 b 2 ?2k (2k)! a2k+1 = 1a 1X k=0 1 k! ?12 ? ?32 ? ?52 ? ￠￠￠ ?2k?12 ? b a ?2k = 1a " 1+ b a ?2#?1=2 = 1pa2 +b2: x17.5cBesself ?￥s?22: ??SE￥ ? 4-)ü ^) ? p?7 O) ? p? H?aa1μB?￥K? Hq è ?  ? p? Hü1μjb=aj < 1￥K?? ? ?^￡ üe ?￥s Rea > 0￥ ?i> u×?Bá l ?y7Rea > 0￥ ?i u× =3 7s ￥2T9]B u× =3? 3ü?￥e ?ü V[ ??  ?￥K?H q 6B?SE ^|$f ??￥Besself ?¨ ?￥sV U T} ?i?DsQ?ü' 57y??SE11} ?Besself ?￥) ?Vr T÷ ?^ty1C￥9 ??? ü ?? ^) ? p?÷ μ/ F? Z 1 0 e?axJ0(bx)dx = Z 1 0 e?ax ? 1 2… Z … ?… eibxsin d ? dx = 12… Z … ?… d Z 1 0 e?(a?ibsin )xdx = 12… Z … ?… d a?ibsin V[¨ = ?? ?9 ???s Z 1 0 e?axJ0(bx)dx = 12… I jzj=1 1 a?bz ?z ?1 2 dz iz = 12…i I jzj=1 2dz ?bz2 +2az +b = 1?bz +a flfl fl z=(a?pa2+b2)=b = 1pa2 +b2: ü'57y??SE￥ 6B?z) ^?3T3ü?? ???ZE9μ K ? è ?' ?oo  ? ?¨Besself ?￥sV U?d? ?¨￥Besself ?D59μsV U?? T1?ˉ?¨  ?1 ?PBt V??s? V[wB?μi ±￥2T. ?T|??s ?31Besself ?J0(bt) ￥LaplaceMDZ 1 0 J0(bt)e?ptdt = 1pp2 +b2; * 1? LaplaceMD￥ ? ?(n10.3?)ü??μ Z t 0 J0(b?)J0(b(t??))d?; 1p2 +b2: 6BZ ?á ì??? 1 p2 +b2 : 1 b sinbt: ?[ü ?¤?sZ t 0 J0(b?)J0(b(t??))d? = 1b sinbt: x17.5cBesself ?￥s?23: 3.$f ?1 ??f ?DBesself ?￥e ?? ??￥sB? a ?1?ˉ? f?/2T ^Wf ?] H I n? s￥ l ??sZE￥ê4 31+Yl? x17.6 Hankelf ??24: x17.6 Hankelf ? - ?o ￥J”(x)?N”(x)? V[¨ ? í? ?o ? ì￥víZ 7sY ^ J”(x) ? r 2 …x cos ? x? ”…2 ? …4 · ; N”(x) ? r 2 …x sin ? x? ”…2 ? …4 · : ?^ A ? ì í￥? ?o?;μ? ?o?μ? o? ^ ?Tá ì) ?￥ù5? o #? ?o? o ^1 p ü ? uY? ?o? o? ?f ?A ?ü?ZL  ? HA ???TL?F? H(1)” (x) ·J”(x)+iN”(x) ? r 2 …x exp h i ? x? ”…2 ? …4 ·i ; H(2)” (x) ·J”(x)?iN”(x) ? r 2 …x exp h ?i ? x? ”…2 ? …4 ·i : ?T ￥? M?￥ HWy0e?i!t5H(1)” (x)}V? ?oH(2)” (x)}V? o H(1)” (x)?H(2)” (x)sY?1?B ???= ?Hankelf ? ? ìA ?? ^BesselZ?￥3 ? ^?f ?d?1? ? ??f ? è è è3èHoá ???V ? ￥ ?  ! ü ?èHo<° ? ?Bá ??? ?  ?èHo￥ ? qB? è O D?à ü? p$? ? ? ￥èHo 333 |?US"zà'D?à×??? ? èHo￥è O  ü???àyN??8 V ? ￥? 3è @9 ^à_￥ ? o￥è O 9 ^à_￥ ?T¨u}VèHo￥è O  5?ù5￥ í ? V?u = u(r;`;t)Dzí1 ? ?@Z? @2u @t2 ?c 2 ?1 r @ @r r@u@r ? + 1r2 @ 2u @`2 ? = 0: y1 ? qB?# V ! u(r;`;t) = v(r;`)e?i!t: v(r;`)ü ?@HelmholtzZ? 1 r @ @r r@v@r ? + 1r2 @ 2v @`2 +k 2v = 0; ?o ?k = !=cc ^; ?éB?|u(r;`;t)[#M?￥v(r;`)?s1 ?s ? o?s ? ? o?s v(r;`) = v1(r;`)+v2(r;`): x17.6 Hankelf ??25: F ? ￥èHo v1(r;`) = E0eikrcos`; ? ￥Z_ |1xà('` = 0)￥Z_v1(r;`) ?@HelmholtzZ? 1 r @ @r r@v1@r ? + 1r2 @ 2v1 @`2 +k 2v1 = 0; F ? ov2(r;`)9 ?@HelmholtzZ? 1 r @ @r r@v2@r ? + 1r2 @ 2v2 @`2 +k 2v2 = 0: 1  p3v2(r;`)?A? v2(r;`) ? ?@￥H?Hq n5 ??? ?@? ùHq v2(r;`)flfl`=0 = v2(r;`)flfl`=2…; @v2(r;`) @` flfl fl `=0 = @v2(r;`)@` flfl fl `=2… : QL !?8V ? ^ ?X?8è? <? ?r = a ￥ M_s 10 ?[ u(r;`;t)flflr=a = 0; ?N V[? v2(r;`)flflr=a = ?E0eikacos`: Kaí kùHq5 ^v2(r;`) ^_í kù ? ￥? ?o' v2(r;`)flflr!1?ocμ ? o￥?s:  ?￥ ê±sZ?(HelmholtzZ?)?H?Hqü? B???￥?3ù5?vs ? M E￥S?? V[ ?@Z??? ùHq#í kùHq￥B?3 v2(r;`) = 1X m=0 (Am cosm`+Bm sinm`)H(1)m (kr): } ? :/￥H?Hqü¤? v2(r;`)flflr=a = 1X m=0 (Am cosm`+Bm sinm`)H(1)m (ka) = ?E0eikacos` = ?E0J0(ka)?2E0 1X m=1 imJm(ka)cosm`:  ?￥KaB?¨?  eikrcos = J0(kr)+2 1X n=1 inJn(kr)cosn : x17.6 Hankelf ??26: 1?" ?ü¤? A0 = ?E0 J0(ka) H(1)0 (ka) ; Am = ?2E0im Jm(ka) H(1)m (ka) , m = 1;2;3;￠￠￠ ; Bm = 0, m = 1;2;3;￠￠￠ : ? ^ ? oè O ￥ bW?sü ^ v2(r;`) = ?E0J0(ka) H(1)0 (ka) H(1)0 (kr)?2E0 1X m=1 im Jm(ka) H(1)m (ka) H(1)m (kr)cosm`: Kaü p¤ u(r;`;t) = E0ei(krcos`?!t) ? E0J0(ka) H(1)0 (ka) H(1)0 (kr)e?i!t ?2E0e?i!t 1X m=1 im Jm(ka) H(1)m (ka) H(1)m (kr)cosm`: x17.7′7 Besself ??27: x17.7′7 Besself ? C) ?+ ?+ y￥Besself ?, n5 ^+ y7 ￥Besself ?,7 1B′ ?￥Bessel f ? ?^ ˉV ê±sZ?￥?3ù5? ?? ?′7 ￥Besself ? !μ??8 =￥LaplaceZ??3ù5 1 r @ @r r@u@r ? + 1r2 @ 2u @`2 + @2u @z2 = 0; uflfl`=0 = uflfl`=2…, @u@` flfl fl `=0 = @u@` flfl fl `=2… ; uflflz=0 = 0, uflflz=h = 0; uflflr=0μ?, uflflr=a = f(`;z): ?vs ?M E￥SSE 7 u(r;`;z) = R(r)'(`)Z(z); } ?Z?#H?Hqs ?M ü?¤?'?′ù5 '00(`)+?'(`) = 0; '(0) = '(2…), '0(0) = '0(2…) ? Z00(z)+?Z(z) = 0; Z(0) = 0, Z(h) = 0 [#è±sZ? 1 r d dr rdRdr ? + ? ??? ?r2 · R = 0: ??B?'?′ù5 V[¤? '?′?m = m2, m = 0;1;2;3;￠￠￠ ; '?f ?'m(`) = Am cosm`+Bm sinm`; ?Am?Bm ^ ?iè ???=?'?′ù5? V[ p¤ '?′?n = ?n… h ·2 , n = 1;2;3;￠￠￠ ; '?f ?Zn(z) = sin n…h z: ?" :/￥è±sZ?üM? 1 r d dr rdRdr ? + ? ? ?n… h ·2 ? m 2 r2 ? R = 0: x17.7′7 Besself ??28: TMDx = (n…=h)r?y(x) = R(r)ü V[|NZ??1 1 x d dx xdydx ? + ? ?1? m 2 x2 · y = 0: ??Z??1′7 BesselZ?TMDt = ixü V[| ??1BesselZ?? ^ R(r) = CJm in… h r ? +DNm in… h r ? : ? úüC 7 1B′ ?￥Besself ??Neumannf ? B? a ??Besself ?￥7 1B′ ?xei…=2( ?x1 L ?) Hf ?′9 ^ˉ ? J”(xei…=2) = 1X k=0 (?)k k!Γ(k +” +1) ?x 2e i…=2·2k+” = ei”…=2 1X k=0 1 k!Γ(k +” +1) ?x 2 ·2k+” : ?"L?^?l?B ?′7 Besself ? I”(x) · e?i”…=2J”(xei…=2) = 1X k=0 1 k!Γ(k +” +1) ?x 2 ·2k+” : +Y ^?? ?¨￥?B ?′7 Besself ?e?1μ In(x) = i?nJn(ix): ?"S￥ "￥ Es üA P¤x?”? ^ L ? HI”(x)￥f ?′? ^ L ? ]"??I”(x)?I?”(x)? ^]B?′7 BesselZ?￥37 O I n? I?n(x) = inJ?n(ix) = (?)ninJn(ix) = i?nJn(ix) = In(x) V[?l?= ?′7 Besself ?1 K”(x) = …2sin”… h I?”(x)?I”(x) i : ???l￥z) ^?”1? ?n HKn(x) ˉ ?μil ODIn(x)L?í1 Kn(x) = lim”!nK”(x) = 12 n?1X k=0 (?)k(n?k?1)!k! ?x 2 ·2k?n +(?)n+1 1X k=0 ‰ 1 k!(n+k)! h ln x2 ? 12?(n+k +1) ?12?(k +1) i?x 2 ·2k+n : x17.7′7 Besself ??29: ? ú ˉ???n = 0 H? ??· ?B[￥μK? m17.4`′7 Besself ? ?  ?ó￥I”(x)?K”(x)￥?l ?^ ? ìx ! 0 H￥ví?1 F ?T” ? 05 ? I”(x) ^μ?￥ ? K”(x) ^í?￥ F?x !1 H ? ì￥ví?1? ^ I”(x) ? r 1 2…xe x; K”(x) ? r … 2xe ?x:  L¨?èè? ?tví?1Gê ?31￥3 è ?  ?￥?3ù5 1 r @ @r r@u@r ? + 1r2 @ 2u @`2 + @2u @z2 = 0; uflfl`=0 = uflfl`=2…, @u@` flfl fl `=0 = @u@` flfl fl `=2… ; uflflz=0 = 0, uflflz=h = 0; uflflr=0μ?, uflflr=a = f(`;z): ?üB?μ+3 umn(r;`;z) = (Am cosm`+Bm sinm`) ￡ h CmnIm ?n… h r · +DmnKm ?n… h r ·i sin n…h z: ??μ?Hq uflflr=0μ? ￥K? Dmn = 0: x17.7′7 Besself ??30: ? ^H?Hq?μ?Hq￥K?/Z?￥+3ü ^ umn(r;`;z) = (Amn cosm`+Bmn sinm`) Im ?n… h r · sin n…h z: |?í k?+3?F  ?¤?B?3 ?a ?¨d QH?Hq' V??F" ? x17.8?  ?¨Besself ??31: x17.8?  ?¨Besself ? ) ? 6B ?+ y￥Besself ? μ+ y¨ ?￥Besself ?¨1?  ?￥Besself ? 5) ?J1=2(x) J1=2(x) = 1X k=0 (?)k k!Γ(k +3=2) ?x 2 ·2k+1=2 = r 2 …x 1X k=0 (?)k (2k +1)!x 2k+1: ?[ J1=2(x) = r 2 …x sinx ^?f ? L=  ?iB??  ?¨￥Besself ?? ^?f ?? ^ af ?? ??f ?￥ˉ?f ?? V[V?w1" ?￡ ü n5ü?w1" d dx [x ”J”(x)] = x”J”?1(x) ?? 1 x d dx ? x”J”(x) = x”?1J”?1(x); ?[ 1 x d dx ?n x1=2J1=2(x) = 1 x d dx ?nr2 … sinx = x?n+1=2J?n+1=2(x): yNJ?n+1=2(x)? ^?f ?+Y ^ J?1=2(x) = r 2 …x cosx: ]"ü?w1" d dx h x?”J”(x) i = ?x?”J”+1(x) ?? 1 x d dx ? x?”J”(x) = ?x?(”+1)J”+1(x); ?[ 1 x d dx ?n x?1=2J1=2(x) = 1 x d dx ?nr2 … sinx x = (?)nx?n?1=2Jn+1=2(x): x17.8?  ?¨Besself ??32: yNJn+1=2(x)9? ^?f ? A ?Jn+1=2(x)DJ?(n+1=2)(x) ^L?í1￥ W[Jn+1=2(x); J?(n+1=2)(x)] = ? 2…x sin ? n+ 12 · … = (?)n+1 2…x: 7Nn+1=2(x)DJ?(n+1=2)(x)L?M1 Nn+1=2(x) = cos(n+1=2)… ￠Jn+1=2(x)?J?(n+1=2)(x)sin(n+1=2)… =(?)n+1J?(n+1=2)(x): x17.9 oBesself ??33: x17.9 oBesself ? á ì oUS"/|HelmholtzZ?r2u+k2u = 0s ?M  u(r; ;`) = R(r)￡( )'(`); ;ü¤? ??è±sZ? '00 +?' = 0; 1 sin d d ? sin d￡d ? + h ?? ?sin2 i ￡ = 0; 1 r2 d dr r2dRdr ? + h k2 ? l(l +1)r2 i R = 0: B? f ?/?B?Z??M?￥? ùHq?'?′ù5 V[?'?′ ?m = m2; m = 0;1;2;￠￠￠ : ?=?Z??M?￥μ?Hq?'?′ù5?'?′ ?l = l(l +1); l = m;m+1;m+2;￠￠￠ : ?B?) ?? ??Z?￥ p3ù5 ?A) ?k = 0￥ f?y1? H￥ p3ù5 BcXü) ?V ?￥ ?L? í13ü ^rl?r?l?1 y1k 6= 0# VTMDx = kr?y(x) = R(r)|Z?M1 1 x2 d dx x2dydx ? + h 1? l(l +1)x2 i y(x) = 0: ??Z??1 oBesselZ? oBesselZ?￥? T?BesselZ?dèM ?éB?sZ?￥ ? V[ A? oBesselZ?9μ ? ? FB? ^x = 0?5 ? FB? ^x = 1d?5 ? 9?BesselZ?M]yN V[ km| ??1BesselZ? I n???Z?x = 0?￥·SZ? ‰(‰?1)+2‰?l(l +1) = 0; y7·S1‰1 = l?‰2 = ?(l + 1)?BesselZ?￥·S‰ = §”￥+?‰1 + ‰2 = 0?]#? ?TMD y(x) = v(x)px ; x17.9 oBesself ??34: ?" V[? v(x)￥±sZ?x = 0?￥·Sü?M1 ‰ = § l + 12 ? ; ?BesselZ?￥+?? ?B"   ?MD/ dy dx = 1p x ?dv dx ? 1 2 v x ? ; d dx x2dydx ? = 1px ? x ddx xdvdx ? ? v4 ? ; ?[v(x) ? ?@￥±sZ?ü ^ 1 x d dx xdvdx ? + ? 1? (l +1=2) 2 x2 ? v = 0: ?? ^l+1=2¨￥BesselZ? ?￥ ?L?í13ü ^Jl+1=2(x)?Nl+1=2(x)N$  ü V[| oBesselZ?￥L?í13 |1 jl(x) = r … 2xJl+1=2(x) = r … 2x 1X n=0 (?)n n!Γ(n+l +3=2) ?x 2 ·2n+l+1=2 = p… 2 1X n=0 (?)n n!Γ(n+l +3=2) ?x 2 ·2n+l ? nl(x) = r … 2xNl+1=2(x) = (?) l+1 r … 2xJ?(l+1=2)(x) = (?)l+1 r … 2x 1X n=0 (?)n n!Γ(n?l +1=2) ?x 2 ·2n?l?1=2 = (?)l+1 p… 2 1X n=0 (?)n n!Γ(n?l +1=2) ?x 2 ·2n?l?1 ; sY?1l¨ oBesself ?? oNeumannf ? -+? oBesself ?? oNeumannf ?￥Vr T ^ j0(x) = sinxx ; j1(x) = 1x2?sinx?xcosx￠; j2(x) = 1x3 h? 3?x2￠sinx?3xcosx i ; n0(x) = ?cosxx ; n1(x) = ? 1x2?cosx+xsinx￠; n2(x) = ? 1x3 h? 3?x2￠cosx+3xsinx i : x17.9 oBesself ??35: m17.5` oBesself ? ? ?19? V[?l oHankelf ? h(1)l (x) = jl(x)+inl(x); h(2)l (x) = jl(x)?inl(x): ?w1" 1 x d dx ?nh xl+1jl(x) i = xl?n+1jl?n(x); 1 x d dx ?nh x?ljl(x) i = (?)nx?l?njl+n(x); jl?1(x)+jl+1(x) = 2l +1x jl(x); ljl?1(x)?(l +1)jl+1(x) = (2l +1)j0l(x); l +1 x + d dx ? jl(x) = jl?1(x); l x ? d dx ? jl(x) = jl+1(x): è è è4|f ?eikrcos ?Legendre[ TZ 7 333 ! eikrcos = 1X l=0 cl(kr)Pl(cos ); 5Z 7" ? cl(kr) = 2l +12 Z 1 ?1 eikrxPl(x)dx = 2l +12 1X n=0 (ikr)n n! Z 1 ?1 xnPl(x)dx: x17.9 oBesself ??36: ?¨16.4?￥2Tüμ cl(kr) = 2l +12 1X n=0 (ikr)l+2n (l +2n)! Z 1 ?1 xl+2nPl(x)dx = 2l +12 il 1X n=0 (?)n (l +2n)!(kr) l+2n ￠ (l +2n)! 2l+2nn! p… Γ(n+l +3=2) = 2l +12 ilp… 1X n=0 (?)n n!Γ(n+l +3=2) kr 2 ?l+2n = (2l +1)il jl(kr): ?[KaüμZ 7 T eikrcos = 1X l=0 (2l +1)il jl(kr)Pl(cos ): 9 V[?í??Z 7 TB?t ?3 d ü ?o? o ?oZ 7? ^y1 ?T? ?Mê￥ HWy01e?i!t7 Oür? ?31 oUS5  TP ^_ = 0(' ?zà)Z_.l￥ ü ?oo ?1k7· ?B[?￥jl(kr)5 μ o ?o￥Mê y0 jl(kr) ? 1kr sin kr? l…2 ? :