Wu Chong-shi a0a1a2a3 Γ a4 a5 a61a7 a8a9a10a11 Γ a12 a13 §12.1 Γ a14a15a16a17a18 a19a20 Γ a21a22a23a24a25a26a27a28a29 Γ(z) = integraldisplay ∞ 0 e?ttz?1dt, Rez > 0. a30a31a32a33a34a35a36a37a38 Eulera32a33a39a40a41 a23 a32a33a42a43 t a44a45a46a47 a35 argt = 0 a48 star a49a50a51a52a53a54a55a56a57a58a59a60a61a62a63a48 a64a35a30 a29a65 a31a66 a25 a32a33a39a67a68 a29a65 a31a69a32a33( a70t = 0a71)a39a72a29a65 a31a73a74a32a33a39a75a76a77a78a67a79 a80a81a82a33a83a33a84a85a86 a48 integraldisplay ∞ 0 e?ttz?1dt = integraldisplay 1 0 e?ttz?1dt + integraldisplay ∞ 1 e?ttz?1dt. a87a88a36a37a82a33 a48 a89a90a39a91t ≥ 1 a92 a39a93a32 a21a22e ?ttz?1 a29 ta23a94a95a21a22 a39a96a97a98a35z a23a21a22 a39 a70 a99a100a101 a47a102a48a103a27a46 4.2a104a105 a39a77a106a107a67a108a109 a65 a31 a47a102a21a22 a39a110a111a112a106a107a32a33 a65a113a114a115a48 a64a35 et = ∞summationdisplay n=0 tn n!, a75a76a116a117a118a119a120a121 a22 N a39 et > t N N!, e ?t < N! tN . a122a116a117z a100a101a123a118 a65a124a125a126 (a127a125a126a128a23 a118a119 a65a129 a39a130a131 Rez<x 0 a39( a132a133 12.1) vextendsinglevextendsinglee?ttz?1vextendsinglevextendsingle < N!·tx 0?N?1. a13412.1 a30a135a39a111a77a136a137a138a139a140 a23N ( a141a142N > x0) a39a32a33 integraldisplay ∞ 1 tx0?N?1dta110a114a115a39a122 integraldisplay ∞ 1 e?ttz?1dta70za100 a101 a23 a118 a65a124a125a126 a41 a65a113a114a115 a39a64 a127a70 a99a100a101 a47a102a48 a77a106a107a36 a65 a82a33 a23 a32a33 a70a143a144 a100a101 a47a102 a39a145a146a147 a29 a106a107a67 a23a65a113a114a115a148a48 a64a35 vextendsinglevextendsinglee?ttz?1vextendsinglevextendsingle = e?ttx?1, x = Rez. Wu Chong-shi §12.1 Γa4a5a149a150a151 a62a7 a64 a127 a39a116a117z a100a101a123 a143a144 a100a101 a23 a118 a65a125a126 a39a131 Rez = x ≥ δ > 0a39 vextendsinglevextendsinglee?ttz?1vextendsinglevextendsingle ≤ tδ?1, a152 integraldisplay 1 0 tδ?1dta114a115 a39a122a32a33 integraldisplay 1 0 e?ttz?1dta70z a100a101a123a143a144a100a101a23a118a65a124a125a126a41a65a113a114a115a39a64a127a70a143 a144 a100a101 a47a102a48 a78a81a82a33a153a154a83a39a110 a142a155 Γ(z) = integraldisplay ∞ 0 e?ttz?1dt a70z a23a143a144 a100a101 a47a102a48 square star a49a50a156a157a158a159a160 ? a123a101a23 a32a33 a27a28 a41a39a32a33a161a162a96a163a112a77a164 a27a70a165a166 a123a39a152 a104a167a168 a35 Γ(z) = integraldisplay L e?ttz?1dt, Rez > 0, a32a33a161a162L a29ta100a101a123a169t = 0a170a171a23a144a172a173 a39argt = αa35 a25a22 a39|α| <pi/2 a48 a174a175a176C a177a13312.2a39a44a26a178a22a27a46 a85a86a179a42a32a33 contintegraldisplay C e?ttz?1dt,a110a180a106a142 a30a31 a181a86 a48 a13412.2 ? a182a65a183a167a168a184 a32a33a161a162L a104 a76 a29 t a100a101a123a169 t = 0 a170a171a23 a118a119a33a185a186a187a188 a173 a39a111a77 a24a189 a76 Ret → +∞a23a190a191a192 a117a73a74a193 a129a194a104a48 star a60a61a195a196 a197a198a199a200a201Γ a202a203 a201a204a205a206a207a208a209Rez > 0 a48 a210a211a212a213a201a214 a215a216 a213a217a218a219a220a198a221a222a201a39 a223a224a39a225a226a227a228a229za201a219a220a198a39a206a230a208a207a231a201a232a233a234a212a213a214a235 a216 a213a227a228a229a219a220a198a236a237 a48 a238a239a240a241 a23a190a242a29a243a244a22a21a22 a98 Taylor a245a246 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. a30a31a181a247 a29a70Rez > 0a23a248a249a250a142a155a23a48a251a252a191a253a71a70a143a144 a100a101 a47a102 a39a152 a143a71a23a254a22 a89a90 a70 a99a100 a101a123(z negationslash= 0,?1, ?2,···) a65a113a114a115 a39a64a152 a70 a99a100a101 a47a102 (z negationslash= 0,?1,?2,···)a48 a30a255a107a39 a252a191a143a71a23a254 a22 a109a0 a191 a110 a29a253a71 a32a33a109a0 a191a70 a99a100a101a123 a23a47a102a1a2a48 Γ(z) = integraldisplay ∞ 1 e?ttz?1dt + ∞summationdisplay n=0 (?)n n! 1 n + z. Wu Chong-shi a0a1a2a3 Γ a4 a5 a63a7 §12.2 Γ a14a15a16a3a4a5a6 a7a8 1 Γ(1) = 1 a48 a240a241 a70 Γa21a22a23a27a28 a41a108a9 z = 1 a194a104a142a155 a30a31a181a247 a48 a7a8 2 Γ(z + 1) = zΓ(z) a48 a10 a11a12 Γ a21a22a23a27a28 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 a13a209a14a15a16a17a237a18a19a20a15a21a22a23a24a221 a48 ? a235a217a25a26a218a27a28a29a30a31a208a229a226a32a33 Rez > 0a48 a34a35a209 Γ(z + 1) a36zΓ(z)a37 a218a219a220a198a221 a222 (z = 0, ?1, ?2, ··· a38a39) a39a223a224a39a40a41a221a222a227a228a42a24a39a237a18a43a204a39a14a15a44a45a46a47a218 a219a220a198a48a49a50 a48 ? a51 a235a232a198a39a52a237a18a53a54a55a29a44a45a46a47a23a56a49 Γ a202a203 a201a221a222a227a228 a48 a14a57a39a237a234a44a45a46a47 a58a59a49 Γ(z) = 1zΓ(z + 1). a197a60a61a62a201 a202a203 a218a63a220a198Rez > 0a197a221a222a39a64a62a201 a202a203 a218a63a220a198 Rez > ?1a197a221a222a65 a20a66a218a67a68a69a70 Rez > 0 a197a71a72 a65a35a224 a237a73 a39 Γ(z + 1)/z a74 a217a64a62a201 Γ(z) a218a69a70 Rez > ?1a197a201a221a222a227a228a48a75a76 a39a77a17a78 a227a228a79a80a229a201a16a17a81a82a225 Γ(z) a39a14 a74 a217a83a39 a237a18a78 Γ(z) = 1zΓ(z + 1), z negationslash= 0 a84 a49a217Γ(z)a218a69a70Rez > 1a197a201a204a205 a39 a75z = 0a85 a217Γ a202a203 a201a235a86a87 a85 a39resΓ(0) = 1 a48 ? a88a89 a197a90a91a92 a39a93 a237a18a234 Γ a202a203 a227a228a229a69a70 Rez > ?2 a39 Γ(z) = 1z(z + 1)Γ(z + 2), z negationslash= 0,?1. z = ?1a52a217Γa202a203 a201a235a86a87 a85 a39 resΓ(?1) = ?1 a48 ? a77a224a94a95a39a74 a237a18a234 Γ a202a203 a221a222 a227a228a229a219a220a198 a39 a75 z = 0, ?1, ?2, ···a37 a217Γ a202a203 a201a235 a86a87 a85 a39 resΓ(?n) = (?1) n n! . Wu Chong-shi §12.2 Γa4a5a149a96a97a98a99 a64a7 a100a101 1 a116a117a120a121 a22 na39 Γ(n) = (n?1)!. a120 a29 a64a35a30a31a102a64a39 Γ a21a22 a72a34a35a103a104 a21a22a48 a7a8 3 a105a106a107 a43 a27a46 Γ(z)Γ(1?z) = pisinpiz. a30a31a108 a191a23 a106a107 a132a189 a101 a23 a36 12.4 a109a48 a100a101 2 Γ(1/2) = √pi a48 a111a77 a70 a123a101 a23a148a110 3 a41a108a9 z = 1/2 a39a96a97a111a119 Γ(1/2) >0(a64a35a93a32a21a22a112a113 a35a120) a194a104a142a155a127 a181a247 a48 a100a101 3 Γ a21a22a70 a99a100a101a73a114 a129a48 a10 a64a35pi/sinpiz negationslash= 0a39a75a76Γ(z)Γ(1?z) negationslash= 0 a48 a30 a135a39 a177 a247 a70 z = z0 a129 a131 Γ(z 0) = 0 a39a115a116a131 Γ(1?z 0) = ∞ a48a30a111a180a171a117a70 1 ? z0 = ?n(a118a194 z0 = n + 1) a39 n = 0,1,2,···a92a48a251a127a92 Γ(z0) = Γ(n + 1) = n! a39a119a75 a120a121a122 a48 a64 a127 Γa21a22a70 a99a100a101a73a114 a129a48 square a13312.3 a41a123a170a124Γ(x)(xa35a165a22)a23a133a125a48 a67a169 a165a22 a126a175a240a127a128a109a129 a170 a30a31a130a86a76a131 Γ a21a22a23a132a129 a33a133 a48 a13412.3 a134 a135a136a137a138a139a140a141 Γ a142 a139a143 a7a8 4 a144 a104a108 a191 Γ(2z) = 22z?1pi?1/2Γ(z)Γ parenleftbigg z + 12 parenrightbigg . a30a31a108 a191a23 a106a107a147 a132 12.4a109a48 a7a8 5 Γ a21a22a23a145a146a245a246 a39 a194 Stirlinga108a191a184 a91|z|→∞a39|argz| <pi a92 a39a131 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 +···. a70a147a46 a41a148 a25a26a23 a181a247 a29 lnn! ～ nlnn?n. Wu Chong-shi a0a1a2a3 Γ a4 a5 a65a7 §12.3 ψ a14 a15 ψa21a22a29 Γa21a22a23 a116 a22a149a150 ψ(z) = dlnΓ(z)dz = Γ prime(z) Γ(z). a11a12Γ a21a22a23a148a110 a39 a104 a76 a142a170ψ(z)a23a250a151a148a110a184 1. z = 0,?1,?2,···a152a29ψ(z)a23a65 a103a153 a129 a39 a178a22 a130a35?1a65a154 a124 a30a155 a129 a76a156a39ψ(z) a70 a99a100a101 a47a102a48 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. ψa21a22a23a157a158a112a131 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. a159a31γ = ?ψ(1)a217 a203a160 a31a201a235a15a161a162a163 a203 a39a164a225 Eulera163 a203 γ = 0.5772 1566 4901 5328 6060 6512 0900 8240 ···. a165a201a204a205a217 γ = limn→∞ bracketleftBigg nsummationdisplay k=1 1 k ?lnn bracketrightBigg . star a166a26ψa21a22 a39 a104 a76 a190a167 a128a168 a170a169a170 a35a131 a46a191a23 a73a74 a254a22 ∞summationdisplay n=0 un = ∞summationdisplay n=0 p(n) d(n) Wu Chong-shi §12.3 ψ a4 a5 a66a7 a171a172a39a40a41p(n)a172d(n) a152a29na23a173a170a191a48 a35 a124a174 a106 a254a22a114a115 a39 p(n) a23a175a22a176a177 a77a238d(n) a23a175a22 a1782a39 a194 limn→∞un = limn→∞n·un = 0. a177 a247d(n) a29na23ma175a173a170a191 a39a96a97a99a82a114 a129a152a29a65 a103a114 a129 a39 d(n) = (n + α1)(n + α2)···(n + αm), a194un a111a131a65 a103a153 a129 a39a115 a104 a82a33a33 a191 a35 un = p(n)d(n) = msummationdisplay k=1 ak n + αk. a166a26ψa21a22a23a179 a130a145a180a39 a194a104 a168 a142 Nsummationdisplay n=0 un = msummationdisplay k=1 ak [ψ(αk + N)?ψ(αk)] = msummationdisplay k=1 ak [ψ(αk + N)?lnN ?ψ(αk)], a40a41 a166a26a124 msummationtext k=1 ak = 0a48a174a153a164N →∞a39a194a142 ∞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). a181 12.1 a168a73a74 a254a22 ∞summationtext n=0 1 (3n + 1)(3n+ 2)(3n + 3) a171a172 a48 a60 a64a35 1 (3n + 1)(3n + 2)(3n + 3) = 1 6 1 n + 1/3 ? 1 3 1 n + 2/3 + 1 6 1 n + 1, a75a76a39a11a12a123a101a123 a170a23 a168a172a108 a191 a39a131 ∞summationdisplay n=0 1 (3n + 1)(3n + 2)(3n + 3) = ? 1 6 bracketleftbigg ψ parenleftbigg1 3 parenrightbigg ?2ψ parenleftbigg2 3 parenrightbigg +ψ(1) bracketrightbigg . a108a9ψ a21a22a23a157a158a112 a39 a194a142 ∞summationdisplay n=0 1 (3n + 1)(3n+ 2)(3n + 3) = 1 4 bracketleftbigg pi √3 ?ln3 bracketrightbigg . a181 12.2 a168a73a74 a254a22 ∞summationtext n=0 1 n2 + a2 a171a172a39a40a41a > 0 a48 Wu Chong-shi a0a1a2a3 Γ a4 a5 a67a7 a60 a64a35 1 n2 + a2 = i 2a parenleftbigg 1 n + ia ? 1 n?ia parenrightbigg , a75a76 ∞summationdisplay n=0 1 n2 + a2 = ? i 2a [ψ(ia)?ψ(?ia)]. a166a26 a123a101 a151a170a23ψa21a22a23a148a110 4a39 ψ(ia)?ψ(?ia) = ? 1ia ?picotipia = i bracketleftbigg1 a +pi coth pia bracketrightbigg , a110 a104 a76a168 a142 ∞summationdisplay n=0 1 n2 + a2 = 1 2a2 bracketleftbig1 +piacothpiabracketrightbig. star a177a247un a182 a131a37a103a153 a129 a39a183 a177 a39 d(n) = (n + α1)(n + α2)···(n + αm)(n + β1)2(n + β2)2···(n + βl)2, a115 un = p(n)d(n) = msummationdisplay k=1 ak n + αk + lsummationdisplay k=1 bracketleftbigg b 1k n + βk + b2k (n + βk)2 bracketrightbigg . a184 a44 a128a39 a254a22a114a115a23a248a249a29 msummationdisplay k=1 ak + lsummationdisplay k=1 b1k = 0. a11a12ψ a21a22a23a179 a130a145a180a39 a194a142 ∞summationdisplay n=0 un = ? msummationdisplay k=1 akψ(αk)? lsummationdisplay k=1 bracketleftbigb 1kψ(βk)?b2kψprime(βk) bracketrightbig. a181 12.3 a168a73a74 a254a22 ∞summationtext n=0 1 (n + 1)2(2n + 1)2 a171a172 a48 a60 a64a35 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 , a75a76 ∞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. Wu Chong-shi §12.4 B a4 a5 a68a7 §12.4 B a14 a15 Ba21a22a29a103 a36 a65 a38 Eulera32a33 a27a28a23a184 B(p,q) = integraldisplay 1 0 tp?1(1?t)q?1dt, Rep > 0, Req > 0. a185t = sin2θ a39 a182 a104 a76 a142a155 Ba21a22a23a186a65 a31a109a0 a191 B(p,q) = 2 integraldisplay pi/2 0 sin2p?1θcos2q?1θdθ. a70Ba21a22a23a27a28 a41a98a42a187 s = 1?ta39a110 a104 a76 a142a155 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, a194B(p,q)a116a117pa172q a29 a116a34 a23a184 B(p,q) = B(q,p). Ba21a22a104 a76 a26 Γa21a22 a109a188 a170 a83a39 B(p,q) = Γ(p)Γ(q)Γ(p + q) . a10 a70Rep > 0a39Req > 0a23a248a249a250 a39a89a90a131 Γ(p) = integraldisplay ∞ 0 e?ttp?1dt = 2 integraldisplay ∞ 0 e?x2x2p?1dx, Γ(q) =2 integraldisplay ∞ 0 e?y2y2q?1dy. a117 a29 Γ(p)Γ(q) = 4 integraldisplay ∞ 0 integraldisplay ∞ 0 e?(x2+y2)x2p?1y2q?1dxdy. a185x = rsinθ a39y = rcosθ a39 a142 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 a189a208a14a15a46a47a60a39a237a78 B a202a203 a221a222 a227a228a229 p a36q a201a219a220a198a48 a19a14a15a46a47a60 a39 a52a237a18a190a191a192 a84a193B(p,q)a13a209p a36q a217a13a164a201a48 Wu Chong-shi a0a1a2a3 Γ a4 a5 a69a7 a129 a70 a11a12 B a21a22 a172Γ a21a22a23 a145a180 a191 B(p,q) = Γ(p)Γ(q)Γ(p + q) . (star) a194a106Γ a21a22a23 a81a31 a148a110 a39 a194a105a106a107 a43 a27a46 Γ(z)Γ(1?z) = pisinpiz a172 a144 a104a108 a191 Γ(2z) = 22z?1pi?1/2Γ(z)Γ parenleftbigg z + 12 parenrightbigg . a87a106a107 a105a106a107 a43 a27a46 a39 a70 (star)a191 a41a185p = z, q = 1?z a39 B(z, 1?z) = Γ(z)Γ(1?z)Γ(1) = Γ(z)Γ(1?z). a186a65a190 a101a39 B(z, 1?z) = integraldisplay 1 0 tz?1(1?t)?zdt. a185x = t/(1?t)a39a123 a191a194a104a195 a35 B(z, 1?z) = integraldisplay ∞ 0 xz?1 1 + xdx. a30a31a32a33 a70 a36 7.6 a109 a41a196a197a198a199a200a39a30a135a110a106 a142 Γ(z)Γ(1?z) = B(z, 1?z) = pisinpiz. a30a31a106a107a91a90 a29a700 < Rez < 1a23a248a249a250a142a155a23a48a251a29 a39 a103 a117 a252a191a23 a81 a71a70 a99a100a101 a152a47a102 a39a64 a127 a39 a30a31 a252a191a70 a99a100a101a130a80a201 a48 square a202a106a107 a144 a104a108 a191a48 a30 a104 a76 a169 a200a32a33 integraldisplay 1 ?1 (1?x2)z?1dx, Rez > 0 a23 a198a199 a142a155a48 a185 x2 = ta39a115 a142integraldisplay 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) . a203a98a42a187 1 + x = 2ta391?x = 2(1?t)a39a115a131 a186a65a204a125a191a23 a181a247 a184integraldisplay 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) . a117 a29 Γ(z)Γ(1/2) Γ(z + 1/2) = 2 2z?1Γ(z)Γ(z) Γ(2z) a194a144 a104a108 a191 Γ(2z) = 22z?1pi?1/2Γ(z)Γ parenleftbigg z + 12 parenrightbigg . a30a205 a23 a106a107a206a90 a29a70Rez > 0 a23a248a249a250a182a207a23a48a251a29 a39a120 a177a208 a101 a173a175 a86a106a200 a23 a39a30a31a181a247 a70 a99a100a101 a152 a80a201 a48 square Wu Chong-shi ?§12.5 Γ a4a5a149a209a210a211a212a213 a610a7 ?§12.5 Γ a14a15a16a214a215a216a217a218 Γa21a22a23a27a28 (a36a37a38a219a220a32a33) Γ(z) = integraldisplay ∞ 0 e?ttz?1dt, Rez > 0. a111a221 a26 a117 a143a144 a100a101 a48 a35 a124a222 a194a30 a65a223a224 a39a225 a109 a163a226a106a107a128a227a228 Γ a21a22a23a186 a156a229 a204 a109a0 a191 a39 a190a191a230 a231a175a176a32a33a109a188a172a73a74a104a32a109a188a39a67a232 a152a70 a99a100a101a80a201 ( a233 a155 a129a104 a180a183a156) a48 a131a145a106a107 a132a234a235a236a237 [12]a39a363a238a48 1. Γa62a63a158a239a240a49a50a57a241 Γ(z) = ? 12isinpiz integraldisplay (0+) ∞ e?t(?t)z?1dt, |arg(?t)| <pi, a40a41 a23 a32a33a175a176a35 a184 a169a123 a144 a100a101a242 a146 a120 a165a166 a73a74a193a243 a170a171 a39 a253a207a244 a102 a129 a120a245 a65a246 a39a202 a143a207a155a250a144 a100a101a242 a146 a120 a165a166 a73a74a193a243 ( a132a133 12.4)a48a127a191a70 a99z a100a101a80a201a39 a251 z =a121a22 a154a156 a48 a134 12.4 a134 12.5 Γa21a22a23a186a65 a31a175a176a32a33a109a188 a29 Γ(z) = 12pii integraldisplay (0+) ?∞ ett?zdt, |argt| <pi, a32a33a175a176a169 a250a144 a100a101a242 a146a247a165a166 a73a74a193a243 a170a171 a39 a143a207a244 a102 a129 a120a245 a65a246 a39a202 a253a207a155 a123 a144 a100a101a242 a146a247 a165a166 a73a74a193a243 ( a132a133 12.5)a48a127a191a70 a99z a100a101a80a201a39 a230 a231 z =a121 a22a48 2. Γa62a63a158a248a249a250a251a252a49a57a241 Γ(z) = 1z ∞productdisplay n=1 braceleftbiggparenleftBig 1 + zn parenrightBig?1 parenleftbigg 1 + 1n parenrightbiggzbracerightbigg , a127a191 a116a118a253 z a130a80a201a39 a251 a153 a129 z =a247 a121 a22 a154a156 a48 3. Γa62a63a158a254a255a0a1a249a0a250a251a252a49a57a241 1 Γ(z) = ze γz ∞productdisplay n=1 bracketleftBigparenleftBig 1 + zn parenrightBig e?z/n bracketrightBig , a40a41 γ = limn→∞ bracketleftBigg nsummationdisplay k=1 1 k ?lnn bracketrightBigg = 0.5772156649··· a110 a29 a219a220 a25a22 (12.3a109)a48 a30a31a73a74a104a32a123 a170a124 a118a253 z a23 Γ(z)a39a2a92a244 a107 a124 Γ(z)a23a132a129 a35 a65 a103a153 a129z = 0,?1,?2,··· a152a73a114 a129a48 Wu Chong-shi a0a1a2a3 Γ a4 a5 a611a7 a169Γ a21a22a23 a73a74a104a32a109a188 a104a142a155a65 a180 a151 a131a119 a28a23 a181a247 a48 a183 a177 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 . a243 a30a81 a191 a168a116 a22a149a150 a39a72 a104 a76 a142a155 pitanpiz = ?8z ∞summationdisplay m=1 1 4z2 ?(2m?1)2, z negationslash= ± 1 2,± 3 2,···, (App1) picotpiz = 1z + 2z ∞summationdisplay m=1 1 z2 ?m2, z negationslash= 0,±1,±2,···, (App2) picscpiz =pi2 bracketleftBig tanpiz2 + cotpiz2 bracketrightBig = 1z + 2z ∞summationdisplay m=1 (?1)m z2 ?m2, z negationslash= 0,±1,±2,···, (App3) pisecpiz =picscpi parenleftbigg1 2 ?z parenrightbigg = 4 ∞summationdisplay m=1 (?1)m(2m?1) 4z2 ?(2m?1)2, z negationslash= ± 1 2,± 3 2,···. (App4) a30a155 a245a246a191 a34a35a131 a46 a33 a191a245a246a48 a30 a204a245a246a23a125a191 a163a2a117a3a232a200a4 a132 a200 a23 Taylora245a246 a172 Laurent a245 a246 a65 a114a115 a126a175 a152a29 a99a100a101 ( a132a129 a243a154a156) a39a147a163 a29a233a65a5a126a6a7a126a48a70 a30 a204a245a246 a41a39a180a78 a21a22a70 a67 a23 a99a82a153 a129a23a132a8a148 a2 a92 a109a129a73a9 a48 a180a139a98a131 a46 a33 a191a245a246a23 a39a10a164a117 a70 a131a164 a125a126a128 a111a131a153 a129a23a11 a112a12a13a14a15a16a17a12a13a18 a19(8.1) a20(8.2)a21a22a23a24a25a26a14a27a28a29a30a31a32a33a34 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,···. a19(App1)～(App4) a35a22a36a37a38z = 0a39a40Taylora41a42a14a43a44a45a24a20a46a47a14a48a28a29a33a34 piz tanpiz = 2 ∞summationdisplay n=1 bracketleftbigg ∞summationdisplay m=1 1 (2m?1)2n bracketrightbigg (2z)2n, (App5) piz cotpiz = 1?2 ∞summationdisplay n=1 bracketleftbigg ∞summationdisplay m=1 1 m2n bracketrightbigg z2n, (App6) pisecpiz = 4 ∞summationdisplay n=0 bracketleftbigg ∞summationdisplay m=1 (?)m?1 (2m?1)2n+1 bracketrightbigg (2z)2n, (App7) pizcscpiz = 1 + 2 ∞summationdisplay n=1 bracketleftbigg ∞summationdisplay m=1 (?)m?1 m2n bracketrightbigg z2n. (App8) a49 a31a50a51a14 z 2 cot z 2 = iz 2 eiz/2 + e?iz/2 eiz/2 ?e?iz/2 = ∞summationdisplay n=0 (?)n B2n(2n)!z2n, |z| < 2pi; Wu Chong-shi ?§12.5 Γ a52a53a54a55a56a57a58a59 a6012a61 z 2 tan z 2 = z 2 cot z 2 ?z cotz = ∞summationdisplay n=1 (?)n?12 2n ?1 (2n)! B2nz 2n, |z| <pi; z cscz = z2 cot z2 + z2 tan z2 = ∞summationdisplay n=0 (?)n?12 parenleftbig22n?1 ?1parenrightbig (2n)! B2nz 2n, |z| <pi; 2ez/2 ez + 1 = sech z 2 = ∞summationdisplay n=0 En n! parenleftBigz 2 parenrightBign , |z| <pi, a62a63 B n a20 En a64a65a66a67 Bernoulli a13a20 Euler a13 (a68a69 5.6 a70a14a71a72a73a74 a67a75a76 ) a14a20a77a51a78 (App5)～(App8)a22a79a80a81a14a48a28a29a33a82a83a84a85a86a87a13a20a88 ∞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, ∞summationdisplay m=1 (?1)m?1 (2m?1)2n+1 = (?)n 22n+2 pi2n+1 (2n)! E2n. a19 a77a51a78a89a31a22a90a91a92 B2n = (?)n?1 2(2n)!(2pi)2n ∞summationdisplay m=1 1 m2n, a93 a28a29a94a82a95 n →∞a96B2n →∞a18a97a98a14 B20 = ?5.291×102, B30 = 6.016×108, B40 = ?1.930×1016, B50 = 7.501×1024, ··· B200 = ?3.647×10215.