Γ a0 a1 star a2a3a4a5a6a7a8a9a10 4 star §8.3(ψ(z)) a11a12a13a14a15a16a17 star §8.5(Γ a18a19a20a21a22a23a24a25) a26a6a7 a0a1a2 Γ a3 a4 a51a6 a7a8a9 Γ a10 a11 §8.1 Γ a12a13a14a15a16 a17a18 Γ a19a20a21a22a23a24a25a26a27 Γ(z) = integraldisplay ∞ 0 e?ttz?1dt, Rez > 0. a28a29a30a31a32a33a34a35a36 Eulera30a31a37a38a39 a21 a30a31a40a41 t a42a43a44a45 a33 argt = 0 a46 star a47a48a49a50a51a52a53a54a55a56a57a58a59a60a61a46 a62a33a28 a27a63 a29a64 a23 a30a31a37a65a66 a27a63 a29a67a30a31( a68t = 0a69)a37a70a27a63 a29a71a72a30a31a37a73a74a75a76a65a77 a78a79a80a31a81a31a82a83a84 a46 integraldisplay ∞ 0 e?ttz?1dt = integraldisplay 1 0 e?ttz?1dt + integraldisplay ∞ 1 e?ttz?1dt. a85a86a34a35a80a31 a46 a87a88a37a89t ≥ 1 a90 a37a91a30 a19a20e ?ttz?1 a27 ta21a92a93a19a20 a37a94a95a96a33z a21a19a20 a37 a68 a97a98a99 a45a100a46a101a25a44 4.2a102a103 a37a75a104a105a65a106a107 a63 a29 a45a100a19a20 a37a108a109a110a104a105a30a31 a63a111a112a113a46 a62a33 et = ∞summationdisplay n=0 tn n!, a73a74a114a115a116a117a118a119 a20 N a37 et > t N N!, e ?t < N! tN . a120a114a115z a98a99a121a116 a63a122a123a124 (a125a123a124a126a21 a116a117 a63a127 a37a128a129 Rez<x 0 a37( a130a131 8.1) vextendsinglevextendsinglee?ttz?1vextendsinglevextendsingle < N!·tx 0?N?1. a1328.1 a28a133a37a109a75a134a135a136a137a138 a21N ( a139a140N > x0) a37a30a31 integraldisplay ∞ 1 tx0?N?1dta108a112a113a37a120 integraldisplay ∞ 1 e?ttz?1dta68za98 a99 a21 a116 a63a122a123a124 a39 a63a111a112a113 a37a62 a125a68 a97a98a99 a45a100a46 a75a104a105a34 a63 a80a31 a21 a30a31 a68a141a142 a98a99 a45a100 a37a143a144a145 a27 a104a105a65 a21a63a111a112a113a146a46 a62a33 vextendsinglevextendsinglee?ttz?1vextendsinglevextendsingle = e?ttx?1, x = Rez. §8.1 Γa3a4a147a148a149 a52a6 a62 a125 a37a114a115z a98a99a121 a141a142 a98a99 a21 a116 a63a123a124 a37a129 Rez = x ≥ δ > 0a37 vextendsinglevextendsinglee?ttz?1vextendsinglevextendsingle ≤ tδ?1, a150 integraldisplay 1 0 tδ?1dta112a113 a37a120a30a31 integraldisplay 1 0 e?ttz?1dta68z a98a99a121a141a142 a98a99 a21 a116 a63a122a123a124 a39 a63a111a112a113 a37a62 a125a68a141 a142 a98a99 a45a100a46 a76a79a80a31a151a152a81a37a108 a140a153 Γ(z) = integraldisplay ∞ 0 e?ttz?1dt a68z a21a141a142 a98a99 a45a100a46 square star a47a48a154a155a156a157a158 ? a121a99a21 a30a31 a25a26 a39a37a30a31a159a160a94a161a110a75a162 a25a68a163a164 a121a37a150 a102a165a166 a33 Γ(z) = integraldisplay L e?ttz?1dt, Rez > 0, a30a31a159a160L a27ta98a99a121a167t = 0a168a169a21a142a170a171 a37arg t = αa33 a23a20 a37|α| <pi/2 a46 a172a173a174C a175a1318.2a37 a42a24a176a20a25a44 a83a84a177a40a30a31 contintegraldisplay C e?ttz?1dt,a108a178a104a140 a28a29a179a84 a46 a1328.2 ? a180a63a181a165a166a182a30a31a159a160La102a74a27 t a98a99a121a167 t = 0 a168a169a21a116a117a31a183a184a185a186a171a37a109a75a22a187a74 Ret → +∞a21a188a189a190 a115a71a72a191 a127a192a102a46 star a58a59a193a194 a195a196a197a198a199Γ a200a201 a199a202a203a204a205a206a207Rez > 0 a46 a208a209a210a211a199a212 a213a214 a211a215a216a217a218a196a219a220a199a37 a221a222a37a223a224a225a226a227za199a217a218a196a37a204a228a206a205a229a199a230a231a232a210a211a212a233 a214 a211a225a226a227a217a218a196a234a235 a46 a236a237a238a239 a21a188a240a27a241a242a20a19a20 a96 Taylor a243a244 integraldisplay 1 0 e?ttz?1dt = ∞summationdisplay n=0 (?)n n! integraldisplay 1 0 tn+z?1dt = ∞summationdisplay n=0 (?)n n! 1 n + z. a28a29a179a245 a27a68Rez > 0a21a246a247a248a140a153a21a46a249a250a189a251a69a68a141a142 a98a99 a45a100 a37a150 a141a69a21a252a20 a87a88 a68 a97a98 a99a121(z negationslash= 0,?1, ?2,···) a63a111a112a113 a37a62a150 a68 a97a98a99 a45a100 (z negationslash= 0,?1,?2,···)a46 a28a253a105a37 a250a189a141a69a21a252 a20 a107a254 a189 a108 a27a251a69 a30a31a107a254 a189a68 a97a98a99a121 a21a45a100a255a0a46 a0a1a2 Γ a3 a4 a53a6 Γ(z) = integraldisplay ∞ 1 e?ttz?1dt + ∞summationdisplay n=0 (?)n n! 1 n + z. §8.2 Γa3a4a147a1a2a3a4 a54a6 §8.2 Γ a12a13a14a5a6a7a8 a9a10 1 Γ(1) = 1 a46 a238a239 a68 Γa19a20a21a25a26 a39a106a11 z = 1 a192a102a140a153 a28a29a179a245 a46 a9a10 2 Γ(z + 1) = zΓ(z) a46 a12 a13a14Γ a19a20a21a25a26 Γ(z + 1) = integraldisplay ∞ 0 e?ttzdt = ?e?ttz vextendsinglevextendsingle vextendsingle ∞ 0 + integraldisplay ∞ 0 e?tztz?1dt = z integraldisplay ∞ 0 e?ttz?1dt = zΓ(z). square a15a207a16a17a18a19a235a20a21a22a17a23a24a25a26a219 a46 ? a233a215a27a28a216a29a30a31a32a33a206a227a224a34a35 Rez > 0a46 a36a37a207 Γ(z + 1) a38zΓ(z)a39 a216a217a218a196a219 a220 (z = 0, ?1, ?2, ··· a40a41) a37a221a222a37a42a43a219a220a225a226a44a26a37a235a20a45a202a37a16a17a46a47a48a49a216 a217a218a196a50a51a52 a46 ? a53 a233a230a196a37a54a235a20a55a56a57a31a46a47a48a49a25a58a51 Γ a200a201 a199a219a220a225a226 a46 a16a59a37a235a232a46a47a48a49 a60a61a51 Γ(z) = 1zΓ(z + 1). a195a62a63a64a199 a200a201 a216a65a218a196Rez > 0a195a219a220a37a66a64a199 a200a201 a216a65a218a196 Rez > ?1a195a219a220a67 a22a68a216a69a70a71a72 Rez > 0 a195a73a74 a67a37a222 a235a75 a37 Γ(z + 1)/z a76 a215a66a64a199 Γ(z) a216a71a72 Rez > ?1a195a199a219a220a225a226a46a77a78 a37a79a19a80 a225a226a81a82a227a199a18a19a83a84a223 Γ(z) a37a16 a76 a215a85a37 a235a20a80 Γ(z) = 1zΓ(z + 1), z negationslash= 0 a86 a51a215Γ(z)a216a71a72Rez > 1a195a199a202a203 a37 a77z = 0a87 a215Γ a200a201 a199a233a88a89 a87 a37resΓ(0) = 1 a46 ? a90a91 a195a92a93a94 a37a95 a235a20a232 Γ a200a201 a225a226a227a71a72 Rez > ?2 a37 Γ(z) = 1z(z + 1)Γ(z + 2), z negationslash= 0,?1. z = ?1a54a215Γa200a201 a199a233a88a89 a87 a37 resΓ(?1) = ?1 a46 ? a79a222a96a97a37a76 a235a20a232 Γ a200a201 a219a220 a225a226a227a217a218a196 a37 a77 z = 0, ?1, ?2, ···a39 a215Γ a200a201 a199a233 a88a89 a87 a37 resΓ(?n) = (?1) n n! . a0a1a2 Γ a3 a4 a55a6 a98a99 1 a114a115a118a119 a20na37 Γ(n) = (n?1)!. a118 a27 a62a33a28a29a100a62a37 Γ a19a20 a70a32a33a101a102 a19a20a46 a9a10 3 a103a104a105 a41 a25a44 Γ(z)Γ(1?z) = pisinpiz. a28a29a106 a189a21 a104a105 a130a187 a99 a21 a34 8.4 a107a46 a98a99 2 Γ(1/2) = √pi a46 a109a75 a68 a121a99 a21a146a1083 a39a106a11z = 1/2a37a94a95a109a117Γ(1/2) >0(a62a33a91a30 a19a20a110a111 a33a118) a192a102a140a153 a125 a179a245 a46 a98a99 3 Γ a19a20a68 a97a98a99a71a112 a127a46 a12 a62a33pi/sinpiz negationslash= 0 a37a73a74 Γ(z)Γ(1?z) negationslash= 0 a46 a28a133a37 a175 a245 a68 z = z0 a127 a129 Γ(z 0) = 0 a37a113 a114a129 Γ(1?z 0) = ∞ a46 a28a109a178 a169a115a68 1 ? z0 = ?n(a116a192 z0 = n + 1) a37 n = 0,1,2,···a90a46a249a125a90 Γ(z0) = Γ(n + 1) = n!a37a117a73a118a119a120a46a62a125 Γa19a20a68 a97a98a99a71a112 a127a46 square a131 8.3 a39a121a168a122 Γ(x)(x a33a163a20) a21a131a123a46 a65a167 a163a20a124 a173a238a125a126a107a127 a168 a28a29a128a84a74a129 Γ a19a20a21 a130 a127 a31a131 a46 a1328.3 a132 a133a134a135a136a137a138a139 Γ a140 a137a141 a9a10 4 a142 a102a106 a189 Γ(2z) = 22z?1pi?1/2Γ(z)Γ parenleftbigg z + 12 parenrightbigg . a28a29a106 a189a21 a104a105a145 a130 8.4a107a46 §8.2 Γa3a4a147a1a2a3a4 a56a6 a9a10 5 Γ a19a20a21a143a144a243a244 a37 a192 Stirlinga106a189a182 a89|z|→∞a37|argz| <pi a90 a37a129 Γ(z) ～ zz?1/2e?z√2pi braceleftBig 1 + 112z + 1288z2 ? 13951840z3 ? 5712488320z4 +··· bracerightBig , lnΓ(z) ～ parenleftbigg z ? 12 parenrightbigg lnz ?z + 12 ln(2pi) + 112z ? 1360z3 + 11260z5 ? 11680z7 +···. a68a145a44 a39a146 a23a24a21 a179a245 a27 lnn! ～ nlnn?n. a0a1a2 Γ a3 a4 a57a6 §8.3 ψ a12 a13 ψa19a20a27 Γa19a20a21 a114 a20a147a148 ψ(z) = dlnΓ(z)dz = Γ prime(z) Γ(z). a13a14Γ a19a20a21a146a108 a37 a102 a74 a140a168ψ(z)a21a248a149a146a108a182 1. z = 0,?1,?2,···a150a27ψ(z)a21a63 a101a151 a127 a37 a176a20 a128a33?1a67a152 a122 a28a153 a127 a74a154a37ψ(z) a68 a97a98a99 a45a100a46 2. ψ(z + 1) =ψ(z) + 1z. ψ(z + n) =ψ(z) + 1z + 1z + 1 +···+ 1z + n?1, n = 2,3,···. 3. ψ(1?z) =ψ(z) +picotpiz. 4. ψ(z)?ψ(?z) = ?1z ?picotpiz. 5. ψ(2z) = 12ψ(z) + 12ψ parenleftbigg z + 12 parenrightbigg + ln2. 6. ψ(z) ～ lnz ? 12z ? 112z2 + 1120z4 ? 1252z6 +···, z →∞, |argz| <pi. 7. limn→∞bracketleftbigψ(z + n)?lnnbracketrightbig = 0. ψa19a20a21a155a156a110a129 ψ(1) = ?γ, ψprime(1) = pi 2 6 , ψ parenleftbigg1 2 parenrightbigg = ?γ ?2ln2, ψprime parenleftbigg1 2 parenrightbigg = pi 2 2 , ψ parenleftbigg ?12 parenrightbigg = ?γ ?2ln2 + 2, ψprime parenleftbigg ?12 parenrightbigg = pi 2 2 + 4, ψ parenleftbigg1 4 parenrightbigg = ?γ ? pi2 ?3ln2, ψ parenleftbigg3 4 parenrightbigg = ?γ + pi2 ?3ln2, ψ parenleftbigg1 3 parenrightbigg = ?γ ? pi2√3 ? 32 ln3, ψ parenleftbigg2 3 parenrightbigg = ?γ + pi2√3 ? 32 ln3. a157a33γ = ?ψ(1)a215 a201a158 a33a199a233a17a159a160a161 a201 a37a162a223 Eulera161 a201 γ = 0.5772 1566 4901 5328 6060 6512 0900 8240 ···. a163a199a202a203a215 γ = limn→∞ bracketleftBigg nsummationdisplay k=1 1 k ?lnn bracketrightBigg . star a164a24ψa19a20 a37 a102 a74 a188a165 a126a166 a168a167a168 a33a129 a44a189a21 a71a72 a252a20 ∞summationdisplay n=0 un = ∞summationdisplay n=0 p(n) d(n) §8.3 ψ a3 a4 a58a6 a169a170a37a38a39p(n)a170d(n) a150a27na21a171a168a189a46 a33 a122a172 a104 a252a20a112a113 a37 p(n) a21a173a20a174a175 a75a236d(n) a21a173a20 a1762a37 a192 limn→∞un = limn→∞n·un = 0. a175 a245d(n) a27na21ma173a171a168a189 a37a94a95a97a80a112 a127a150a27a63 a101a112 a127 a37 d(n) = (n + α1)(n + α2)···(n + αm), a192un a109a129a63 a101a151 a127 a37a113 a102 a80a31a31 a189 a33 un = p(n)d(n) = msummationdisplay k=1 ak n + αk. a164a24ψa19a20a21a177 a128a143a178a37 a192a102 a166 a140 Nsummationdisplay n=0 un = msummationdisplay k=1 ak [ψ(αk + N)?ψ(αk)] = msummationdisplay k=1 ak [ψ(αk + N)?lnN ?ψ(αk)], a38a39 a164a24a122 msummationtext k=1 ak = 0a46a172a151a162N →∞a37a192a140 ∞summationdisplay n=0 un = lim N→∞ msummationdisplay k=1 ak [ψ(αk + N)?lnN ?ψ(αk)] = lim N→∞ msummationdisplay k=1 ak [ψ(αk + N)?lnN]? msummationdisplay k=1 akψ(αk) = ? msummationdisplay k=1 akψ(αk). a179 8.1 a166a71a72 a252a20 ∞summationtext n=0 1 (3n + 1)(3n + 2)(3n + 3) a169a170 a46 a58 a62a33 1 (3n + 1)(3n + 2)(3n + 3) = 1 6 1 n + 1/3 ? 1 3 1 n + 2/3 + 1 6 1 n + 1, a73a74a37a13a14a121a99a121 a168a21 a166a170a106 a189 a37a129 ∞summationdisplay n=0 1 (3n + 1)(3n + 2)(3n + 3) = ? 1 6 bracketleftbigg ψ parenleftbigg1 3 parenrightbigg ?2ψ parenleftbigg2 3 parenrightbigg +ψ(1) bracketrightbigg . a106a11ψ a19a20a21a155a156a110 a37 a192a140 ∞summationdisplay n=0 1 (3n + 1)(3n+ 2)(3n + 3) = 1 4 bracketleftbigg pi √3 ?ln3 bracketrightbigg . a179 8.2 a166a71a72 a252a20 ∞summationtext n=0 1 n2 + a2 a169a170a37a38a39 a > 0 a46 a0a1a2 Γ a3 a4 a59a6 a58 a62a33 1 n2 + a2 = i 2a parenleftbigg 1 n + ia ? 1 n?ia parenrightbigg , a73a74 ∞summationdisplay n=0 1 n2 + a2 = ? i 2a [ψ(ia)?ψ(?ia)]. a164a24 a121a99 a149a168a21ψa19a20a21a146a108 4a37 ψ(ia)?ψ(?ia) = ? 1ia ?picotipia = i bracketleftbigg1 a +pi coth pia bracketrightbigg , a108 a102 a74a166 a140 ∞summationdisplay n=0 1 n2 + a2 = 1 2a2 bracketleftbig1 +piacothpiabracketrightbig. star a175a245un a180 a129a35a101a151 a127 a37a181 a175 a37 d(n) = (n + α1)(n + α2)···(n + αm)(n + β1)2(n + β2)2···(n + βl)2, a113 un = p(n)d(n) = msummationdisplay k=1 ak n + αk + lsummationdisplay k=1 bracketleftbigg b 1k n + βk + b2k (n + βk)2 bracketrightbigg . a182 a42 a126a37 a252a20a112a113a21a246a247a27 msummationdisplay k=1 ak + lsummationdisplay k=1 b1k = 0. a13a14ψ a19a20a21a177 a128a143a178a37 a192a140 ∞summationdisplay n=0 un = ? msummationdisplay k=1 akψ(αk)? lsummationdisplay k=1 bracketleftbigb 1kψ(βk)?b2kψprime(βk) bracketrightbig. a179 8.3 a166a71a72 a252a20 ∞summationtext n=0 1 (n + 1)2(2n + 1)2 a169a170 a46 a58 a62a33 1 (n + 1)2(2n + 1)2 = bracketleftbigg 4 n + 1 + 1 (n + 1)2 bracketrightbigg ? bracketleftbigg 4 n + 1/2 ? 1 (n + 1/2)2 bracketrightbigg , a73a74 ∞summationdisplay n=0 1 (n + 1)2(2n + 1)2 = ? bracketleftbig4ψ(1)?ψprime(1)bracketrightbig+bracketleftbigg4ψparenleftbigg1 2 parenrightbigg +ψprime parenleftbigg1 2 parenrightbiggbracketrightbigg = 2pi 2 3 ?8ln2. §8.4 B a3 a4 a510a6 §8.4 B a12 a13 Ba19a20a27a101 a34 a63 a36 Eulera30a31 a25a26a21a182 B(p,q) = integraldisplay 1 0 tp?1(1?t)q?1dt, Rep > 0, Req > 0. a183t = sin2θ a37 a180 a102 a74 a140a153 Ba19a20a21a184a63 a29a107a254 a189 B(p,q) = 2 integraldisplay pi/2 0 sin2p?1θcos2q?1θdθ. a68Ba19a20a21a25a26 a39a96a40a185 s = 1?ta37a108 a102 a74 a140a153 B(p,q) = integraldisplay 1 0 tp?1(1?t)q?1dt = integraldisplay 0 1 (1?s)p?1sq?1 (?ds) = integraldisplay 1 0 sq?1(1?s)p?1 ds, a192B(p,q)a114a115pa170q a27 a114a32 a21a182 B(p,q) = B(q,p). Ba19a20a102 a74 a24 Γa19a20 a107a186 a168 a81a37 B(p,q) = Γ(p)Γ(q)Γ(p + q) . a12 a68Rep > 0a37Req > 0a21a246a247a248 a37a87a88a129 Γ(p) = integraldisplay ∞ 0 e?ttp?1dt = 2 integraldisplay ∞ 0 e?x2x2p?1dx, Γ(q) =2 integraldisplay ∞ 0 e?y2y2q?1dy. a115 a27 Γ(p)Γ(q) = 4 integraldisplay ∞ 0 integraldisplay ∞ 0 e?(x2+y2)x2p?1y2q?1dxdy. a183x = rsinθ a37y = rcosθ a37 a140 Γ(p)Γ(q) =4 integraldisplay ∞ 0 integraldisplay pi/2 0 e?r2(rsinθ)2p?1(rcosθ)2q?1rdrdθ =4 integraldisplay ∞ 0 e?r2r2(p+q)?1dr integraldisplay pi/2 0 sin2p?1θcos2q?1θdθ =Γ(p + q)B(p,q). square a187a206a16a17a48a49a62a37a235a80 B a200a201 a219a220 a225a226a227 p a38q a199a217a218a196a46 a21a16a17a48a49a62 a37 a54a235a20a188a189a190 a86a191B(p,q)a15a207p a38q a215a15a162a199a46 a0a1a2 Γ a3 a4 a511a6 a127 a68 a13a14 B a19a20 a170Γ a19a20a21 a143a178 a189 B(p,q) = Γ(p)Γ(q)Γ(p + q) . (star) a192a104Γ a19a20a21 a79a29 a146a108 a37 a192a103a104a105 a41 a25a44 Γ(z)Γ(1?z) = pisinpiz a170 a142 a102a106 a189 Γ(2z) = 22z?1pi?1/2Γ(z)Γ parenleftbigg z + 12 parenrightbigg . a85a104a105 a103a104a105 a41 a25a44 a37 a68 (star)a189 a39a183p = z, q = 1?z a37 B(z, 1?z) = Γ(z)Γ(1?z)Γ(1) = Γ(z)Γ(1?z). a184a63a188 a99a37 B(z, 1?z) = integraldisplay 1 0 tz?1(1?t)?zdt. a183x = t/(1?t)a37a121 a189a192a102a193 a33 B(z, 1?z) = integraldisplay ∞ 0 xz?1 1 + xdx. a28a29a30a31 a68 a34 7.6 a107 a39a194a195a196a197a198a37a28a133a108a104 a140 Γ(z)Γ(1?z) = B(z, 1?z) = pisinpiz. a28a29a104a105a89a88 a27a680 < Rez < 1a21a246a247a248a140a153a21a46a249a27 a37 a101 a115 a250a189a21 a79 a69a68 a97a98a99 a150a45a100 a37a62 a125 a37 a28a29 a250a189a68 a97a98a99a128a78a199 a46 square a200a104a105 a142 a102a106 a189a46 a28 a102 a74 a167 a198a30a31 integraldisplay 1 ?1 (1?x2)z?1dx, Rez > 0 a21 a196a197 a140a153a46 a183 x2 = ta37a113 a140 integraldisplay 1 ?1 (1?x2)z?1dx = 2 integraldisplay 1 0 (1?x2)z?1dx = integraldisplay 1 0 (1?t)z?1t?1/2dt = B parenleftbigg z, 12 parenrightbigg = Γ(z)Γ(1/2)Γ(z + 1/2) . a201a96a40a185 1 + x = 2ta371?x = 2(1?t)a37a113a129 a184a63a202a123a189a21 a179a245 a182 integraldisplay 1 ?1 (1?x2)z?1dx =22z?1 integraldisplay 1 0 tz?1(1?t)z?1dt = 22z?1B(z, z) =22z?1Γ(z)Γ(z)Γ(2z) . §8.4 B a3 a4 a512a6 a115 a27 Γ(z)Γ(1/2) Γ(z + 1/2) = 2 2z?1Γ(z)Γ(z) Γ(2z) , a192a142 a102a106 a189 Γ(2z) = 22z?1pi?1/2Γ(z)Γ parenleftbigg z + 12 parenrightbigg . a28a203 a21 a104a105a204a88 a27a68Rez > 0 a21a246a247a248a180a205a21a46a249a27 a37a118 a175a206 a99 a171a173 a84a104a198 a21 a37a28a29a179a245 a68 a97a98a99 a150 a78a199 a46 square a0a1a2 Γ a3 a4 a513a6 ?8.5 Γ a12a13a14a207a208a209a210a211 Γ a19a20a21a25a26 (a34a35a36a212a213a30a31 (8.1)a189) a109a214a24 a115 a141a142 a98a99 a46 a33 a122a215 a192a28 a63a216a217 a37a218 a107 a161a219 a104a105a126a220a221 Γ a19a20a21a184 a154a222 a202 a107a254 a189 a37 a188a189a223a224 a173a174a30a31a107a186a170a71a72a102a30a107a186a37a65a225 a150a68 a97a98a99 a78a199( a226 a153 a127a102 a178a181a154) a46 a129a143a104a105 a130a227a228a229a230[12]a37a34 3a231a46 1. Γa60a61a156a232a233a47a48a55a234 Γ(z) = ? 12isinpiz integraldisplay (0+) ∞ e?t(?t)z?1dt, |arg(?t)| <pi, a38a39 a21 a30a31a173a174a33 a182 a167a121 a142 a98a99a235 a144 a118 a163a164 a71a72a191a236 a168a169 a37 a251a205a237 a100 a127 a118a238 a63a239 a37a200 a141a205a153a248a142 a98a99a235 a144 a118 a163a164 a71a72a191a236 ( a130a131 8.4)a46a125a189a68 a97z a98a99a78a199a37 a249 z =a119a20 a152a154 a46 a132 8.4 a132 8.5 Γa19a20a21a184a63a29a173a174a30a31a107a186a27 Γ(z) = 12pii integraldisplay (0+) ?∞ ett?zdt, |argt| <pi, a30a31a173a174a167 a248a142 a98a99a235 a144a240a163a164 a71a72a191a236 a168a169 a37 a141a205a237 a100 a127 a118a238 a63a239 a37a200 a251a205a153 a121 a142 a98a99a235 a144a240 a163a164 a71a72a191a236 ( a130a131 8.5)a46a125a189a68 a97z a98a99a78a199a37 a223a224 z =a119a20a46 2. Γa60a61a156a241a242a243a244a245a47a55a234 Γ(z) = 1z ∞productdisplay n=1 braceleftbiggparenleftBig 1 + zn parenrightBig?1parenleftbigg 1 + 1n parenrightbiggzbracerightbigg , a125a189 a114a116a246 z a128a78a199a37 a249 a151 a127 z =a240 a119 a20 a152a154 a46 3. Γa60a61a156a247a248a249a250a242a249a243a244a245a47a55a234 1 Γ(z) = ze γz ∞productdisplay n=1 bracketleftBigparenleftBig 1 + zn parenrightBig e?z/n bracketrightBig , a38a39 γ = limn→∞ bracketleftBigg nsummationdisplay k=1 1 k ?lnn bracketrightBigg = 0.5772156649··· a108 a27 a212a213 a23a20(8.3a107)a46 a28a29a71a72a102a30a121 a168a122 a116a246z a21Γ(z)a37a251a90a242a105a122Γ(z)a21 a130 a127 a33 a63 a101a151 a127 z = 0,?1,?2,···a150a71a112a127a46 ?8.5 Γ a3a4a147a252a253a254a255a0 a514a6 a167Γ a19a20a21 a71a72a102a30a107a186 a102a140a153a63 a178 a149 a129a117 a26a21 a179a245 a46 a181 a175 sinpiz = piΓ(z)Γ(1?z) =piz ∞productdisplay m=1 bracketleftbigg 1? z 2 m2 bracketrightbigg , cospiz = sin2piz2sinpiz = ∞productdisplay m=1 bracketleftbigg 1? 4z 2 (2m?1)2 bracketrightbigg . a241 a28a79 a189 a166a114 a20a147a148 a37a70 a102 a74 a140a153 pitanpiz = ?8z ∞summationdisplay m=1 1 4z2 ?(2m?1)2, z negationslash= ± 1 2,± 3 2,···, (8.1) picotpiz = 1z + 2z ∞summationdisplay m=1 1 z2 ?m2, z negationslash= 0,±1,±2,···, (8.2) picscpiz = pi2 bracketleftBig tanpiz2 + cotpiz2 bracketrightBig = 1z + 2z ∞summationdisplay m=1 (?1)m z2 ?m2, z negationslash= 0,±1,±2,···, (8.3) pisecpiz =picscpi parenleftbigg1 2 ?z parenrightbigg = 4 ∞summationdisplay m=1 (?1)m(2m?1) 4z2 ?(2m?1)2, z negationslash= ± 1 2,± 3 2,···. (8.4) a28a153 a243a244a189 a32a33a129 a44 a31 a189a243a244a46 a28 a202a243a244a21a123a189 a161a251a115a1a225a198a2 a130 a198 a21 Taylora243a244 a170 Laurent a243 a244 a67 a112a113a124 a173 a150a27 a97a98a99 (a130 a127 a236a152a154) a37a145a161 a27a226a63a3a124a4a5a124a46a68 a28 a202a243a244 a39a37a178a76 a19a20a68 a65 a21 a97a80a151 a127a21 a130a6 a146 a251 a90 a107a127a71a7 a46 a178a137a96a129 a44 a31 a189a243a244a21 a37a8a162a115 a68 a129a162 a123a124a126 a109a129a151 a127a21a9 a110a19a20 a37 a192a10a11a12a13a14 a15(8.1) a16(8.2)a17a18a19a20a21a22a23a24a25a26a27a28a29a30a31 pi2sec2piz =4 ∞summationdisplay m=?∞ 1 [2z ?(2m?1)]2, z negationslash= ± 1 2,± 3 2,···, pi2csc2piz = ∞summationdisplay m=?∞ 1 (z ?m)2, z negationslash= 0,±1,±2,···. a15(8.1) ～ (8.4) a32a18a33a34a35z = 0a36a37 Taylora38a39a23a40a41a42a20a16a43a44a23a45a25a26a30a31 piz tanpiz = 2 ∞summationdisplay n=1 bracketleftbigg ∞summationdisplay m=1 1 (2m?1)2n bracketrightbigg (2z)2n, piz cotpiz = 1?2 ∞summationdisplay n=1 bracketleftbigg ∞summationdisplay m=1 1 m2n bracketrightbigg z2n, pisecpiz = 4 ∞summationdisplay n=0 bracketleftbigg ∞summationdisplay m=1 (?)m?1 (2m?1)2n+1 bracketrightbigg (2z)2n, piz cscpiz = 1 + 2 ∞summationdisplay n=1 bracketleftbigg ∞summationdisplay m=1 (?)m?1 m2n bracketrightbigg z2n, a19a165.8a46a47a48a49a50a51a18a49a52a53a23a54a25a26a30a55a56a57a58a59a60a13a16a61 ∞summationdisplay m=1 1 m2n = (?)n?1 2 (2pi)2n (2n)! B2n, ∞summationdisplay m=1 (?1)m?1 m2n = (?) n?122n?1 ?1 (2n)! pi 2nB2n, ∞summationdisplay m=1 1 (2m?1)2n = (?)n?1 2 22n ?1 (2n)! pi 2nB2n, a62a63a64 Γ a65 a66 a6715a68 ∞summationdisplay m=1 (?1)m?1 (2m?1)2n+1 = (?)n 22n+2 pi2n+1 (2n)! E2n. a15a69a70 a48a71a28a18a72a73a74 B2n = (?)n?1 2(2n)!(2pi)2n ∞summationdisplay m=1 1 m2n, a54a25a26a75a55a76 n →∞a77B2n →∞a14a78a79a23 B20 = ?5.291×102, B30 = 6.016×108, B40 = ?1.930×1016, B50 = 7.501×1024, ··· B200 = ?3.647×10215.