a0 a1 star a2a3a4a5a6a7a8a9a10 6 star §5.8 a11a12a8a13a14a15a16a17a18a6a7 a0a1a2 a3a4a5a6a7a8a9a10a11a12 a131 a14 a15a16a17 a18a19a20a21a22a23a24a25a26a27 §5.1 a28a29a30a31a32 Taylor a33a34 a35a36a37a38a39a40a41a42a43a44a45a46a47a48a35a36a49a50a38a39a51 a52a53a54 a35a36a49a50a38a39a48a55a56a37a57a39a58 a59a60 5.1 (Taylor) a61a62a63f(z)a64a65aa66a67a68a69a67C a70a71C a72a73a74a75a76a77a78a67a70a69a79a80z a81 a75f(z)a82a83a84a85a63a86a87a66(a88a89a90a75f(z)a82a64a a81 a86a87a66a84a85a63) f(z) = ∞summationdisplay n=0 an(z ?a)n, a91a92 an = 12pii contintegraldisplay C f(ζ) (ζ ?a)n+1dζ = f(n)(a) n! , C a93a94a95a96a97a98a99a51 a100 a101a102Cauchy a103a104a105a106a75a77a78a67C a70a79a107a108 a81z a75a109 f(z) = 12pii contintegraldisplay C f(ζ) ζ ?zdζ. a110a111 a75 1 ζ ?z = 1 (ζ ?a)?(z ?a) = 1 ζ ?a ∞summationdisplay n=0 parenleftbiggz ?a ζ ?a parenrightbiggn . a112 a85a63a64 vextendsinglevextendsingle vextendsinglevextendsinglez ?a ζ ?a vextendsinglevextendsingle vextendsinglevextendsingle≤ r < 1 a69a113a114 a92 a108a115a116a117a75a118 a112 a82a65a119a120a103a104a75 f(z) = 12pii contintegraldisplay C bracketleftBigg ∞summationdisplay n=0 (z ?a)n (ζ ?a)n+1 bracketrightBigg f(ζ)dζ = ∞summationdisplay n=0 bracketleftbigg 1 2pii contintegraldisplay C f(ζ) (ζ ?a)n+1 dζ bracketrightbigg (z ?a)n = ∞summationdisplay n=0 an(z ?a)n, an = 12pii contintegraldisplay C f(ζ) (ζ ?a)n+1 dζ = f(n)(a) n! . square a90a121a122 1. a123a124a69a125a126a82a65a127a128a75a129a130f(z)a64C a70a73a74a131a82 a51 a99a132a133a134a135a136a137a138a139a140a141a142a143a144a134a132a145a139a146a147a148a149a150a151a152a153 §5.1 a3a4a5a6a7Taylora11a12 a132a14 a154a155a156a157a158a159 a42z a75 a160a161a162a162a a163 a45a164a165a35a45Cprime a75 a54z a166a167 a40a45a46a51 f(z)a40Cprime a46a168 Cprimea169a170a49a50a42a51 2. a171a172Taylora86a87a69a173a106a174a175a176a62a63 a92 a69 Taylora105a106a177a178a75 a110a111 a125a126a179a178 a51 star a64a175a176a62a63 a92 a75f(x)a69a79a80a180a181a63a182a64a75a183a179a184a65a185a186Taylora105a106a182a64(a88Taylora105a106a116 a117)a51 star a64a187a176a62a63 a92 a75a73a74a69a130a188 (a108a180a181a63a182a64)a189a184a65a185a186Taylora85a63a116a117 a51 3. a190a191a192a193 a62a63f(z)a69a194 a81a195a196a197 a123a198 Taylora85a63a69a116a117a199a200 a51 a61 ba111f(z)a69a201 aa81a202 a203 a69a194 a81 a75a76a108a204a90a205a75a116a117a199a200 R = |b?a|a51 f(z)a40a45|z ?a| < |b?a| a46a206a206a49a50a75f(z)a161a162a40a45a46a207a208a163Taylora57a39(a209a210a211a75 Taylora57a39a40a45|z ?a| < |b?a| a46a43a44)a51a154a212a170a211a75f(z)a42Taylora57a39a43a44a213a214a215a216 a157|b?a| a51 a43a44a213a214a35a217a218a215a219a220 a157|b?a| a51 a221a222 a75ba223 a212 a166a224 a40a43a44a45a46 a75a225a226 a37a57a39a40a43a44 a45a46a206a206a49a50 a75a227ba223a163a228a223 a42a229a230a231a232 ( a233a234ba223 a170 a161a235 a228a223a75a2365.5 a237)a51 1 1 + z2 = ∞summationdisplay n=0 (?)nz2n, |z| < 1. a62a63a69a194 a81z = ±i a189 a197 a123a198Taylora85a63a69a116a117a199a200R = |±i| = 1a51 a238 a64a175a63a239a240a70a75Taylora85a63a69a116a117a199a200a241a62a63a242a243a244a245a69a246a247a189a248a65a249a250 a51 1 1 + x2 = ∞summationdisplay n=0 (?)nx2n, ?1 < x < 1, a189a248a65a124a73a116a117a199a200a66a80 a1111 a75a118a66a62a631/(1+x2)a64a251a252a175a253a72a254 a111a255a0 a82a181a1 a2a3 a79a80a180a181a63 a254 a111 a182a64a69a4 4. Taylor a5a6a7a8a9a10 a11a123a108a252a64a67C a70a73a74a69a62a63a75a76a12a69 Taylora86a87 a111a13 a108a69a75a131 a86a87a247a63an a111a195a196a14a123a69 a51 a100 a15 a123a109a16a252Taylora85a63a64a67C a70a254a116a117a17a178a108a252a73a74a62a63f(z)a75 f(z) = a0 + a1(z ?a) + a2(z ?a)2 +···+ an(z ?a)n +··· = aprime0 + aprime1(z ?a) + aprime2(z ?a)2 +···+ aprimen(z ?a)n +···. a93a18a19z→aa75a76a20a78a85a63a64C a70a69a79a108a21a113a114 a92 a108a115a116a117a75a22a109 a0 = aprime0. a119a120a23a24a75a25a93a18a19 z → aa75a26a27 a1 = aprime1. a0a1a2 a3a4a5a6a7a8a9a10a11a12 a133 a14 a28a112a29a0 a75a131a82a186a27 an = aprimen, n = 0,1,2,···. square Taylora86a87a69a13a108a242a30a31a32a33a122 star a179a250a83a34a35a97a36a75a27a17a69f(z)a64a178a108a252a67a70a69Taylora86a87 a111a13 a108a69 a51 a118 a112 a75a179a108a123a130a83a188 a181a63a69a37a36a123a86a87a247a63 a51 star a28a38a64a178a108 a81 a86a87a69a16a252Taylora85a63a177a39a75a76a82a65a119a120a40a41a247a63 a51 ? a42a43 a111 a64a178a108 a81 a86a87a69a16a252 Taylora85a63a177a39a75a44a82a65a119a120a40a41a247a63 a51 ? a178a108a252a62a63a64a179a178 a81 a86a87a27a17a69a16a252Taylora85a63a75a131a45a109a105a46a69a116a117a113a114a75 a47 a179a48a49 a50 a40a41a86a87a247a63 a51 §5.2 Taylora51 a6a52a53a54a55 a134 a14 §5.2 Taylor a56a31a57a58a59a60 a188Taylora85a63a69a97a36a61a248a108a108a62a63 a51 a171a172a129a64a65a108a66a67a68a69a70a69a97a36 a51 star a71a72a105a106a122 ez = 1 + z + z 2 2! +···+ zn n! +··· = ∞summationdisplay n=0 zn n!, |z| < ∞, sinz = e iz ?e?iz 2i = ∞summationdisplay n=0 (?)n (2n + 1)!z 2n+1, |z| < ∞, cosz = e iz + e?iz 2 = ∞summationdisplay n=0 (?)n (2n)!z 2n, |z| < ∞, 1 1?z = ∞summationdisplay n=0 zn, |z| < 1. star a77a78 a91a73 a62a63a75a74 a111a75a76a77 a83a171a66a71a72a105a106 a51 1 1 + z2 = ∞summationdisplay n=0 parenleftbig?z2parenrightbign = ∞summationdisplay n=0 (?)nz2n, |z| < 1. a109a124a62a63a74a82a65a83a78a104a104a106a69a97a36a79a66a80a81a82a69a173a106a75 1 1?3z + 2z2 = ? 1 1?z + 2 1?2z = ? ∞summationdisplay n=0 zn + 2 ∞summationdisplay n=0 (2z)n = ∞summationdisplay n=0 parenleftbig2n+1 ?1parenrightbigzn, |z| < 1 2. a109a66a62a63a82a65a83a84a85a80a81a82a69a62a63a69a181a63a88a103a104a75a86 a238 a82a65a87a88a89a188a90 a91 Taylor a85a63 a51 1 (1?z)2 = d dz 1 1?z = d dz ∞summationdisplay n=0 zn = ∞summationdisplay n=1 nzn?1 = ∞summationdisplay n=0 (n + 1)zn, |z| < 1. star a28a38a62a63a82a65a83a84a85a16a252(a88a91a252)a62a63a69a92a103a75 a238a93 a108a78a104a69Taylora86a87a40a41a87a88a188a90a95a75 a76a82a94a83a85a63a177a92a69a97a36 a51 1 1?3z + 2z2 = 1 1?z · 1 1?2z = ∞summationdisplay k=0 zk · ∞summationdisplay l=0 2lzl = ∞summationdisplay k=0 ∞summationdisplay l=0 2lzk+l = ∞summationdisplay n=0 parenleftBigg nsummationdisplay l=0 2l parenrightBigg zn = ∞summationdisplay n=0 parenleftbig2n+1 ?1parenrightbigzn, |z| < 1 2. a0a1a2 a3a4a5a6a7a8a9a10a11a12 a135 a14 a20a78a84a85a63a64a116a117a67a70a95a77a116a117a75a22a85a63a177a92 a111a96 a36a69a75a92a103a64a16a116a117a67a69a105a46a113a114a70a97a95a77a116 a117 a51 star a98a123a247a63a36 a51 a99 5.1 a188tanza64z = 0a69Taylora86a87 a51 a100 a20a78tanza111a194a62a63a75a22 a91 a64z = 0a69Taylora86a87a101a129a109a194a102a84a75 tanz = ∞summationdisplay k=0 a2k+1z2k+1 = sinzcosz, sinz = cosz · ∞summationdisplay k=0 a2k+1z2k+1, ∞summationdisplay n=0 (?)n (2n + 1)!z 2n+1 = ∞summationdisplay l=0 (?)l (2l)!z 2l · ∞summationdisplay k=0 a2k+1z2k+1 = ∞summationdisplay n=0 parenleftBigg nsummationdisplay k=0 (?)n?k (2n?2k)!a2k+1 parenrightBigg z2n+1. a40a41a247a63a75a131a27 nsummationdisplay k=0 (?)k (2n?2k)!a2k+1 = 1 (2k + 1)!. a103 a65 n = 0 : a1 = 1; n = 1 : 12a1 ?a3 = 16, a3 = 13; n = 2 : 124a1 ? 12a3 + a5 = 1120, a5 = 215; ... a118 a112 a75a109 tanz = z + 13z3 + 215z5 + 17315z7 +···. a86tanza69a194 a81 a82a65a104a105a75a85a63a69a116a117a199a200a101a66 pi/2a51 a106a107a108a159a109 a39a110 a75 a219a111a112 a109 a39a113a114a42a115a116a117 a109 a75 a118a222 a169 a161a162a119 a36a120a121a207a208 a109 a39 a75 a122 a35a217 a215a123a124a120a121a57a39a42a125a126a127a128 ( a129 a207a208 a109 a39an a42a49a50a48a130a128)a51 a52a131a132a133a134 a120a121a57a39a135a42a136a35a126 a209 a136a137a126 a109 a39 a75 a218 a161a162a138a107a108a159a109 a39a110a51 star a139a140a141a142a7Taylor a5a6 a77a78a143a144a62a63a75a64a145a146a147a123a198a82a144a104a148a149a75a131a82a150a82a144a62a63a151a152 a153Taylor a86a87 a51 a99 5.2 a188a143a144a62a63(1 + z)α a64z = 0a69Taylora86a87a75a147a123z = 0a95 (1 + z)α = 1a51 §5.2 Taylora51 a6a52a53a54a55 a136 a14 a100 a82a49 a50 a188a90a62a63(1 + z)α a64z = 0a81a69a154a180a181a63a144a75 f(0) =1, fprime(0) =α (1 + z)α?1vextendsinglevextendsinglez=0 = α, fprimeprime(0) =α(α?1) (1 + z)α?2vextendsinglevextendsinglez=0 = α(α?1), ... f(n)(0) =α(α?1)(α?2)···(α?n + 1) (1 + z)α?nvextendsinglevextendsinglez=0 =α(α?1)···(α?n + 1), ... a118 a112 (1 + z)α =1 + αz + α(α?1)2 z2 +··· + α(α?1)···(α?n + 1)n! zn +··· = ∞summationdisplay n=0 parenleftbiggα n parenrightbigg zn, a91a92 parenleftbiggα 0 parenrightbigg = 1 a174 parenleftbiggα n parenrightbigg = α(α?1)···(α?n + 1)n! a155 a66a67a156a69a157a120a106a86a87a247a63 a51 a85a63a69a116a117a113a114a75a183a130a158a159a160a69 a153 a36 a238 a123 a51 a116a117a199a200a39a78 z = 0a17a159a160a69 a202a161a162 a201a75 a103 a65a75 a202a163 a82a48a69a116a117a113a114 a111|z| < 1, R = 1 a51 a99 5.3 a188a143a144a62a63ln(1 + z)a64z = 0a69Taylora86a87a75a147a123 ln(1 + z)vextendsinglevextendsinglez=0 = 0a51 a100 a64a72a164a147a123a165a75a62a63ln(1 + z)a82a83a84a66a123a103a104a75a118 a112 ln(1 + z) = integraldisplay z 0 1 1 + zdz = integraldisplay z 0 ∞summationdisplay n=0 (?)nzndz = ∞summationdisplay n=0 (?)n integraldisplay z 0 zndz = ∞summationdisplay n=0 (?)n n + 1z n+1 = ∞summationdisplay n=1 (?)n?1 n z n. a116a117a113a114 a47 a130a166a159a160a167a35 a153 a51 a116a117a199a200a39a78 z = 0 a17a159a160a69 a202a161a162 a201a75 a202a163 a82a48a69a116a117a113a114 a111 |z| < 1a75R = 1a51 star a168a169a170a171a172a7 Taylor a5a6 a28a38 a62a63f(z)a64z = ∞a81a73a74a75a76 a47 a82a65a64z = ∞a81a86a87a85 Taylora85a63 a51 a0a1a2 a3a4a5a6a7a8a9a10a11a12 a137 a14 a173a174 f(z) a40∞ a223 a207a208a56Taylora57a39 a75 a212 a170a165a175a176 z = 1/t a75a226a177 f(1/t)a40t = 0a223 a207 a208a56Taylora57a39a51 a225a163f(1/t)a40t = 0a223 a49a50 a75a178 f parenleftbigg1 t parenrightbigg = a0 + a1t + a2t2 +···+ antn +···, |t| < r; f(z) = a0 + a1z + a2z2 +···+ anzn +···, |z| > 1r. a179a180 a122f(z)a40∞a223 a42Taylora57a39a135 a132a181a182 a39a126a168a183a37a126 a75 a184a181a185 a37a126 a75a226 a43a44a186 a167a163 |z| > 1/ra75 a218 a212 a170 a211a75 a57a39a40 a162∞ a163 a45a164a42a136a36a45a46a43a44a51 §5.3 a3a4a5a6a7a187a188a189a190a10a191a3a4a5a6a7a192a193a10 a138a14 §5.3 a28a29a30a31a32a194a195a196a197a198a199a28a29a30a31a32a200a201a198 star a64a65a73a74a62a63a69a16a252a202a130a242a243a75a12a33a203a109a204a69a202a130a69a124a250a205a144 a51 a59a206 a28a38f(z) a64aa81a71a91a207a114a70a73a74a75f(a) = 0a75a76 a155z = a a66 f(z)a69a208 a81 a51 a61f(z)a64z = a a81 a71 a91a207 a114a70a73a74a75a76a146|z ?a|a209a104a210a95a75 f(z) = ∞summationdisplay n=0 an(z ?a)n, a22a211z = aa66a208 a81 a75a76a42a109 a0 = a1 = ··· = am?1 = 0, am negationslash= 0. a112 a95a75 a155 z = aa81 a66f(z)a69ma180a208 a81 a75a177a101a89a75 f(a) = fprime(a) = ··· = f(m?1)(a) = 0, f(m)(a) negationslash= 0. a208 a81 a69a180a63a254 a111a14 a123a69a212a251a63 a64a62a63a69a73a74a113a114a70a75a179a82a48a109a104a63a102a69a208 a81 a51 a49a50a38a39a213 a223 a42a35a36a214 a134a215a216 a170a41a42a217a218 a215 a59a60 5.2 a211 f(z)a179a219a39a78a208a75 a3 a64a220a221 z = aa64a70a69a113a114a70a73a74a75a76a42a48a222a17a67|z ?a| = ρ(ρ > 0)a75a45a64a67a70a223a198z = aa82a48a66a208 a81a224 a75f(z)a225 a91a73 a208 a81 a51 a154 a36 a159a226a227 a163 a49a50a38a39a42a213 a223 a217a218 a215a159a226 a51a228a229 a154 a36 a159a226 a75 a161a162 a116a121a49a50a38a39a213 a223 a42a230a231a232a36a214 a134a215a216 a122 a233a234 1 a61f(z)a64G : |z ?a| < R a70a73a74 a51 a211a64G a70a182a64f(z)a69a225a235a143a252a208 a81{z n}a75 a3 limn→∞zn = a, a110z n negationslash= aa75a76f(z)a64G a70a219a660a51 a116a2361 a135a42a237a238 lim n→∞zn = a a161a162a239a240 a163a241a242{zn}a42a35a36a243a244a223a163aa51 a233a234 2 a61f(z)a64G : |z ?a| < R a70a73a74 a51 a211a64G a70a182a64a245aa81a69a108a246a247 la88a221a109aa81a69a108 a252a248a113a114ga75a64la72a88g a70f(z) ≡ 0a75a76a64a251a252a113a114 G a70f(z) ≡ 0 a51 a116a2362a42a56a218a186 a167 a170 a162z = a a223a163 a45a164a42a45 a249 a75 a122 a170a250a123a124a116a251a112a35a217a252a253a42a254 a249 a51 a233a234 3 a61f(z)a64G a70a73a74 a51 a211a64G a70a182a64a108 a81z = a a71a245aa81a69a108a246a247 la88a221a109aa81a69a108 a252a248a113a114ga75a64la72a88g a70f(z) ≡ 0a75a76a64a251a252a113a114G a70f(z) ≡ 0a51 a0a1a2 a3a4a5a6a7a8a9a10a11a12 a139 a14 a2555.1 a250a123a124 a54 a116a236 1 a0a1 a56a49a50a38a39a42a2a35 a215a159a226 a51 a59a60 5.3 a61a64a113a114 G a70a109a16a252a73a74a62a63 f1(z) a174 f2(z) a75 a3 a64 G a70a182a64a108a252a3a63 {zn}a75 f1(zn) = f2(zn)a51a211{zn}a69a108a252a18a19 a81z = a(negationslash= z n) a47a4 a64G a70a75a76a64G a70a109f1(z) ≡ f2(z)a51 a5a6 a75 a161a162a54 a116a236 3 a0a1a163 a116a236 4a51 a233a234 4 a61f1(z)a174f2(z)a254a64a113a114G a70a73a74a75 a3 a64G a70a69a108a246a247a88a108a252a248a113a114a70a177a39a75a76a64 G a70f1(z) ≡ f2(z)a51 a165 a163 a41a42a7a8a9a252 a75a10 a181 a122 a233a234 5 a64a175a253a72a85a11a69a219a39a106a75a64 z a12a13a72a97a14a85a11a75a129a130a171a252a219a39a106a16a15a69a62a63a64 z a12 a13a72a254 a111 a73a74a69 a51 §5.4 a16a17a18a19a20Laurenta21a22 a2310a24 §5.4 a28a29a30a31a32 Laurent a33a34 a108a252a62a63a223a198a82a64a73a74 a81a153 Taylor a86a87 a224 a75a109a95a183a25a130a26a12a64a194 a81a27a203a28a29 a85a30a31a32a33 a34 a35a36a37a38 Laurenta28a29 a33 a39a40 5.4 (Laurent) a41a42a32f(z)a43a44ba45a46a47a48a49a50a51a52R1 ≤ |z ?b| ≤ R2 a53a54a55a56a57a58 a59a60a61 a49a52a62a48a63a64 z a65 a58 f(z)a66a44a67a30a31a32 a28a29 a45 f(z) = ∞summationdisplay n=?∞ an(z ?b)n, R1 < |z ?b| < R2, a68a69 an = 12pii contintegraldisplay C f(ζ) (ζ ?b)n+1dζ, C a70a49a52a62a71a62a46a72a73a48a63a74a72a75a76a77a78a79 (a80a815.2)a33 a825.2 Laurent a83a84 a85 a86 a49a52a48a62a87a88a89a90a91a92a45 C1 a93 C2 a58 a59a94a95a96a97a98 a51a52a48 Cauchya99a90a100a101 a58a102 f(z) = 12pii contintegraldisplay C2 f(ζ) ζ ?zdζ ? 1 2pii contintegraldisplay C1 f(ζ) ζ ?zdζ. a60a61C 2 a53 a48a99a90 a58 a66a44a103a104a105a67a44a106a48a107a108 a58 1 2pii contintegraldisplay C2 f(ζ) ζ ?zdζ = ∞summationdisplay n=0 an(z ?b)n, |z ?b| < R2, an = 12pii contintegraldisplay C2 f(ζ) (ζ ?b)n+1dζ. a60a61C 1 a53 a48a99a90 ? 12pii contintegraldisplay C1 f(ζ) ζ ?zdζ = 1 2pii contintegraldisplay C1 f(ζ) (z ?b)?(ζ ?b)dζ = 12pii contintegraldisplay C1 f(ζ) z ?b ∞summationdisplay k=0 parenleftbiggζ ?b z ?b parenrightbiggk dζ a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12211 a123 = ∞summationdisplay k=0 (z ?b)?k?1 · 12pii contintegraldisplay C1 f(ζ)(ζ ?b)kdζ. a124?k?1 = n, k = ?(n + 1) a58 a59 ? 12pii contintegraldisplay C1 f(ζ) ζ ?zdζ = ?∞summationdisplay n=?1 (z ?b)n · 12pii contintegraldisplay C1 f(ζ) (ζ ?b)n+1dζ = ?∞summationdisplay n=?1 an(z ?b)n, |z ?b| > R1, a68a69 an = 12pii contintegraldisplay C1 f(ζ) (ζ ?b)n+1dζ. a125a126a127 a90a77a128a129a130 a58 a36 a102 f(z) = ∞summationdisplay n=?∞ an(z ?b)n, R1 < |z ?b| < R2, an = 12pii contintegraldisplay C f(ζ) (ζ ?b)n+1 dζ. a34a131 a107a108a132a45a42a32 f(z) a43a49a52 R1 < |z ?b| < R2 a62a48 Laurent a28a29 a58 a68a69 a48a31a32a132a45 Laurenta31 a32a33 square a133a134a135a136a137 a n a138a139a140a141a142a143a144a145a146a147a148a149a150C a33 (a151a152a153a154 a133a155a156a157) a158a159a160 star Laurenta28a29a48a75a161a162a66a44a163a164a45 f(z)a43R1 < |z ?b| < R2 a62 a54a55a56a57 a33 star a60a61 Laurenta28a29a130a165 a58a166 a32 (a167a168a70a169a30a170a48 a166 a32) an negationslash= 1n!f(n)(b). star f(z)a43a62a46C1 a69a171 a56a57 a33a72a172a165a130 a58 a43C1 a53 a70 a102a173 a65a48a33 a174a61b a65 a58 a66a175a70f(z)a48 a173 a65 a58 a162a66a175a70f(z)a48 a56a57 a65a33 ? a176a108ba65a70C2 a62a48a177a72 a173 a65 a58 a59C 1 a66a44a178a179a180a181a58 a182a183a184a185a36a186a1870 < |z?b| < R a33 a34a35a36a37a38f(z) a43a188a189 a173 a65 ba48a190a52a62a48 Laurenta28a29a33 ? a87a46C2 a48a191a192a162a66a44a45∞ a58a193 a174 a43∞a65a162 a182a183 a33 star Laurenta28a29a194 a102 a169a30a170 a58a195a102a196 a30a170a33 §5.4 a112a113a114a115a116Laurenta120a121 a12212a123 ? a169a30a170a43a46C2 a62(|z?b| < R2)a197 a60a182a183 a58 a43C2 a62a48a63a74a72 a131 a76a51a52 a69 a72a198 a182a183 a58 a132 a45Laurenta31a32a48a169 a59a127 a90a199 ? a196 a30a170a43a46C1 a87(|z?b| > R1)a197 a60a182a183 a58 a43C1 a87a48a63a74a72 a131 a76a51a52 a69 a72a198 a182a183 a58 a132 a45Laurenta31a32a48a200a201 a127 a90a33 ? a126a127a90a77a129a130 a58 a36a202a187 Laurent a31a32 a58 a43a49a52 R1 < |z ?b| < R2 a62a197 a60a182a183 a58 a43a49a52a62a48a63a74a72 a131 a76a51a52 a69 a72a198 a182a183 a33 ? a203R1 = 0a35 a58 Laurenta31a32a48a200a201 a127 a90 a36a204a205a206a207a208 f(z) a43z = ba65a48 a173a209a210 a33 star Laurent a211a212a213a214a215a216 a41f(z)a43a49a52R1 < |z ?b| < R2 a62 a102 a126a131 Laurent a31a32 a58 f(z) = ∞summationdisplay n=?∞ an(z ?b)n = ∞summationdisplay n=?∞ aprimen(z ?b)n. a126a217a218a219 a44 (z ?b)?k?1 a58a220 a49a52a62a71a62a46a72a73a48a63a72 a185a221 C a99a90 (a34a126a131a31a32a43 a185a221 a53a222 a223 a72a198 a182a183 a58a224a225 a66a44a226a170a99a90) a58 a59a227a61 contintegraldisplay C (z ?b)n?k?1dz = 2piiδnk, a228 a102 ak = aprimek a33 a224 a45ka63a74 a58 a228 a102 ak = aprimek, k = 0,±1,±2,···. a167a229 a37 Laurenta28a29 a48a177a72 a210 a33 a230a231a232a233 Laurent a234a235a236a237a215a238a239a240a241a241a242a243 a58 a244a245a246a247a248 a235a242a243 (a249a250a251a252a253 a248 a235) a33 a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12213 a123 §5.5 Laurent a254a255a0a1a2a3 a4 Laurent a5a6 a58a7a8a9a10a11a12a138a139 a4a136a137 (a133a13a14a15a16 a144a145a142a143a58a147a17a18a19a20a21a22 a141 )a33a23a24a25a26 a58a27a28 a4 Laurent a5a6 a141a29a30a31a32 a33 a33a34a35a137a36a37a38a39a40a41 a141 Laurenta5a6 a18a42a147a141a58a43 a24 a58 a44a45 a12 a152a153 a31a32a58a46 a14a47a48 a150 a36a133a49a39a40a41a50a51a48 f(z) a141a52a53 a137 a58a54a55a56a147 a38 a18 f(z) a141 Laurenta5a6a33 Taylora5a6 a140a57a58a141a31a32a58a8a59a28a60a141a61a62a58a63a7a8a64a12a65 a4 Laurent a5a6a33 a66 5.4 a67 1 z(z ?1) a430 < |z| < 1 a62a93|z| > 1 a62a48a68 a29 a101a33 a69 1 z(z ?1) a430 < |z| < 1 a62a48a68 a29 a50a101a72a70a70 ∞summationtext n=?∞ anzn a33a71a44 1 z(z ?1) = ? 1 z 1 1?z = ? 1 z ∞summationdisplay n=0 zn = ? ∞summationdisplay n=0 zn?1 = ? ∞summationdisplay n=?1 zn, 0 < |z| < 1. a162a66a44a67 a127 a90a90a101a48a72a73 a160 1 z(z ?1) = ? 1 z ? 1 1?z = ? 1 z ? ∞summationdisplay n=0 zn = ? ∞summationdisplay n=?1 zn. 1 z(z ?1) a43|z| > 1 a62a48Laurenta68 a29 a50a101a162a70 ∞summationtext n=?∞ anzn a58 1 z(z ?1) = 1 z2 1 1? 1z = 1z2 ∞summationdisplay n=0 parenleftbigg1 z parenrightbiggn = ?∞summationdisplay n=?2 zn, |z| > 1. a133a134a74a75a76a48 a58a77a147 a49a35a137a36a44 a77a141a78 a40a41 a141 Laurenta5a6 a18a79 a44a80 a77a141 a33 1/z(z?1) a360 < |z| < 1 a41 a141 Laurenta5a6 a46a28a147 a49a81 a52a82a58 a83a36 |z| > 1 a41 a141 Laurenta5a6 a28a84a85 a86a49a81 a52a82a58a87a88a27a28a89a52a82 a33 a66 5.5 a67a90a70 a166 a32a73a67 cotz a43z = 0a190a52a62a48 Laurenta68a91a33 a69 a90a70 a166 a32a73a92a175a67 a61 a102 a179 a131 a196 a30a170 (a169a30a170)a48a93a50a33 cotz = ∞summationdisplay n=?1 b2n+1z2n+1. (a151a152a153 a46a28a147 a49a81 a52a82a58 a133a49 a145a94 a36 5.6 a95a96 a45 a33) cosz = sinz ∞summationdisplay n=?1 b2n+1z2n+1, ∞summationdisplay n=0 (?)n (2n)!z 2n = ∞summationdisplay k=0 (?)k (2k + 1)!z 2k+1 ∞summationdisplay l=0 b2l?1z2l?1 §5.5 Laurenta97 a115a98a99a100a101 a12214 a123 = ∞summationdisplay k=0 ∞summationdisplay l=0 (?)k (2k + 1)!b2l?1z 2(k+l) = ∞summationdisplay n=0 bracketleftBigg nsummationdisplay l=0 (?)n?l (2n?2l + 1)!b2l?1 bracketrightBigg z2n. a227a102a37a38a103a104a105 a166 n summationdisplay l=0 (?)l (2n?2l + 1)!b2l?1 = 1 (2n)!. a226a106a67 a56a58 a167 a37 n = 0 : b?1 = 1; n = 1 : 13!b?1 ? 11!b1 = 12!, b1 = ?13; n = 2 : 15!b?1 ? 13!b1 + 11!b3 = 14!, b3 = ? 145; n = 3 : 17!b?1 ? 15!b1 + 13!b3 ? 11!b5 = 16!, b5 = ? 2945; ... a71a44 cotz = 1z ? 13z ? 145z3 ? 2945z5 ?···. a94a95cotz a48 a173 a65a90a107 a58 a66a108a109 a102 a31a32a48 a182a183a184a185 a45 0 < |z| < pia33 a110a111a112 a66a44a113a67a31a32a114a73a33 cotz = 1tanz = 1 z + 13z3 + 215z5 + 17315z7 +··· = 1z 1 1 + 13z2 + 215z4 + 17315z6 +··· = 1z bracketleftBig 1? parenleftbigg1 3z 2 + 2 15z 4 + 17 315z 6 +··· parenrightbigg + parenleftbigg1 3z 2 + 2 15z 4 + 17 315z 6 +··· parenrightbigg2 ? parenleftbigg1 3z 2 + 2 15z 4 + 17 315z 6 +··· parenrightbigg3 + parenleftbigg1 3z 2 + 2 15z 4 + 17 315z 6 +··· parenrightbigg4 ?+··· bracketrightBig = 1z bracketleftBig 1? 13z2 + parenleftbigg ? 215 + 19 parenrightbigg z4 a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12215 a123 + parenleftbigg ? 17315 + 2× 13 × 215 ? 127 parenrightbigg z6 +··· bracketrightBig = 1z bracketleftbigg 1? 13z2 ? 145z4 ? 2945z6 ?··· bracketrightbigg = 1z ? 13z ? 145z3 ? 2945z5 +···. a115a116a117 a235a213 Laurent a211a212 a66 5.6 a67a42a32ln z ?2z ?1 a431 < |z| < 2a118 2 < |z| < ∞ a62a48a30a31a32a68a91a33 a69 a110a111a69a119 a70a48a68a91a51a52a70a49a50a51a52 a58 a71a44 a58 a176a108a175a120a30a31a32a68a91a48a121 a58 a37a38 a48a72a70a70Laurent a31a32a33 a42a32ln z ?2z ?1 a102 a126a131a122 a65 a160 z = 1 a93 z = 2 a58 a228 a43a49a52 1 < |z| < 2 a62 a171 a66a175a120Laurenta68a91a33 a43a49a522 < |z| < ∞ a62 a58 a42a32ln z ?2z ?1 a70 a54a55a56a57 a48 a58 a123a124a125a126a127a128a129 a70 a54a55 a90 a122a130 a58 a72a66a120 Laurenta68a91a33a131a176 a58a132 a129 a70a43a133a79 a53a134 arg(z ?2)?arg(z ?1) = pi a58 a59 ln z ?2z ?1 vextendsinglevextendsingle vextendsinglevextendsingle z=∞ = 0. a61 a70 a102 ln z ?2z ?1 = ln 1?2/z1?1/z = ln parenleftbigg 1? 2z parenrightbigg ?ln parenleftbigg 1? 1z parenrightbigg = bracketleftBigg ?2z ? 12 parenleftbigg2 z parenrightbigg2 ? 13 parenleftbigg2 z parenrightbigg3 ?··· bracketrightBigg ? bracketleftBigg ?1z ? 12 parenleftbigg1 z parenrightbigg2 ? 13 parenleftbigg1 z parenrightbigg3 ?··· bracketrightBigg = ? 1z ? 32 1z2 ? 73 1z3 ?···? 2 n ?1 n 1 zn ?···. a66 5.7 a67exp braceleftbiggz 2 parenleftbigg t? 1t parenrightbiggbracerightbigg a430<|t|<∞ a62a48Laurenta68a91a33 a69 a67a31a32 a219 a73a33 a224 a45 ezt/2 = ∞summationdisplay k=0 parenleftBigz 2 parenrightBigk tk k!, |t| < ∞, e?z/2t = ∞summationdisplay l=0 parenleftBigz 2 parenrightBigl (?)l l! parenleftbigg1 t parenrightbiggl , vextendsinglevextendsingle vextendsingle1t vextendsinglevextendsingle vextendsingle< ∞a167|t| > 0, §5.5 Laurenta97 a115a98a99a100a101 a12216 a123 a71a44 exp braceleftbiggz 2 parenleftbigg t? 1t parenrightbiggbracerightbigg = ∞summationdisplay k=0 parenleftBigz 2 parenrightBigk tk k! ∞summationdisplay l=0 parenleftBigz 2 parenrightBigl (?)l l! parenleftbigg1 t parenrightbiggl = ∞summationdisplay k=0 ∞summationdisplay l=0 (?)l k!l! parenleftBigz 2 parenrightBigk+l tk?l = ∞summationdisplay n=0 bracketleftBig ∞summationdisplay l=0 (?)l l!(l + n)! parenleftBigz 2 parenrightBig2l+nbracketrightBig tn + ?∞summationdisplay n=?1 bracketleftBig ∞summationdisplay l=?n (?)l l!(l + n)! parenleftBigz 2 parenrightBig2l+nbracketrightBig tn = ∞summationdisplay n=?∞ Jn(z)tn, a68a69 Jn(z) = ? ??? ??? ??? ??? ∞summationdisplay l=0 (?)l l!(l + n)! parenleftBigz 2 parenrightBig2l+n , n = 0,1,2,···; ∞summationdisplay l=?n (?)l l!(l + n)! parenleftBigz 2 parenrightBig2l+n , n = ?1,?2,?3,··· a132a45na135Bessela42a32a33 a176a108a178a136a137a65a70a42a32 f(z) a48 a173 a65 a58a225 a43a178a136a137a65a48a190a52a62 a54a55a56a57 a48a121 a58 a59 a66 a86 f(z) a43∞ a65a48a190a52a62a120 Laurenta68a91( a102 a35a36a138 a54a139 a165 a187 a43 ∞a65a120 Laurenta68a91)a33 star a140a141f(z)a36∞a142 a141a143 a40a41(∞ a142a23a26)a144a145a146a147 a58a56a148a149a150a151a152a153 t = 1/z a58 a35a137f(1/t)a36 t = 0a142 a141a143 a40a41(t = 0 a23a26)a144a145a146a147 a58a43 a83 f parenleftbigg1 t parenrightbigg = ∞summationdisplay n=?∞ antn, 0 < |t| < r, f(z) = ∞summationdisplay n=?∞ anz?n, 1r < |z| < ∞. a133a134 a141 a50a51a154 a144a7a8 a76 a149a18a8 ∞a142a151a155a156 a141a147 a49a39a40 a33 star f(1/t) a141 Laurent a53 a137 a140a89a52a82 (a157a158a159 a137 a82 )a160 a143a18a89a161 a160 a143a58 a81 a52a82a18a162 a14 a160 a143 a33 a43 a24 a58 a163a164a165 a64a166a58 a74a75a135 f(z)a36z = ∞ a142 a143 a40a41 a141 Laurent a53 a137 a140a58 z a141 a81 a52a82a167 a151 a89a161 a160 a143a58 a83 a89a52a82a167 a151 a162 a14 a160 a143 a33 a89a52a82 a163a164a168a169 a150 a35a137 f(z)a36∞ a142 a141a170a171a172 a33 a53a173 a48a131 4 a93 a1316a48a174a175a176a93a50a44a118a131 7 a58 a162a177a66a44a178 a187 a70a43∞a65a190a52a62 Laurenta68a91a33 a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12217 a123 §5.6 a179a180a181a255a182a183a184a185a186 a39a187 a41f(z)a45 a54a55 a42a32(a188a189 a55 a42a32a48a72 a131 a54a55 a90 a122) a58 ba65a70a190a48 a173 a65a33a176a108a43ba65a191 a43a72 a131 a190a52 a58 a43a192a190a52a62(a114ba65a87) a58 f(z)a193a193a66a194 a58 a59 a132 ba45f(z)a48a188a189 a173 a65a33 a195 a188a189 a173 a65a48a131a196a33 a60a61 a42a321/sin(1/z) a58a222 a223 a58 1/z = npi a58 a167z = 1/npi a58 n = 0,±1,±2,···a70a190a48 a173 a65a33 a59z = 0 a70 a34a197 a173 a65a48a198a65(a199a179a65)a160a43z = 0a48a63a74a72 a131 a190a52 a69 a58 a200 a191a43a178a136a189 a131 a173 a65 a58 a228z = 0 a70 a195 a188 a189 a173 a65a33 a176a108z = ba70 a54a55 a42a32f(z)a48a188a189 a173 a65 a58 a59 a72a70a191a43a72 a131 a49a520 < |z?b| < R a58 a43a192a49a52a62 a58 f(z)a66a44a68a91 a187 Laurent a31a32 a58 f(z) = ∞summationdisplay n=?∞ an(z ?b)n. a34a35 a66a175a201a202a203a176a93a204 a160 star a31a32a68a91a101 a171a205 a196a206 a170 a160 b a65a132a45f(z)a48a66a207 a173 a65a33 z = 0a36a70a42a32 sinz z = ∞summationdisplay n=0 (?)n (2n + 1)!z 2n, |z| < ∞ a93 1 z ?cotz = 1 3z + 1 45z 3 + 2 945z 5 +···, |z| < pi a48a66a207 a173 a65a33 star a31a32a68a91a101a92 a205 a102 a179 a131 a196a206 a170 a160 b a65a132a45f(z)a48a199a65a33 star a31a32a68a91a101 a205 a102 a178a136a189 a131 a196a206 a170 a160 b a65a132a45f(z)a48 a110 a210a173 a65a33 a208 a173 a90a91 a158a159 a42a32a43a203a176 a173 a65a193a48a209a45a33 a250a210a211a212 a227a61 a43a66a207 a173 a65a193 a58 a31a32a68a91a101 a69a171a205 a196a206 a170 a58 a228 a31a32 a171 a92a70a43a49a52a62 a182a183 a58a225 a213 a43a49a52a48 a69 a47 a58 a167a66a207 a173 a65 z = ba193a162a70 a182a183 a48a33arrowvertexdbl arrowvertexdblarrowdblbt a34a35 a48 a182a183 a51a52a70a72 a131 a46 a58 a46a47a43a66a207 a173 a65 z = b a58 a31a32a43 a182a183 a46a62a48a63a72a76a51a52 a69 a72a198 a182a183 a58arrowvertexdbl arrowvertexdblarrowdblbt a93 a42a32 a97a214 a58 lim z→b f(z) = lim z→b ∞summationdisplay n=0 an(z ?b)n = a0. §5.6 a215a216 a114a115a116a217a218a219a220 a12218 a123 a42a32a43a66a207 a173 a65a193a48a199a179 a55 a70 a102 a179a48a33 arrowvertexdbl arrowvertexdblarrowdblbt a67 a102 a199a179 a55 a120a45 f(z)a48a70a221 a58 f(z) = ?? ? f(z), z negationslash= b; lim z→b f(z), z = b, a34a222a37a38 a48 f(z)a43ba65a162 a36 a70 a56a57 a48 a208 a33 a34 a169a70a66a207 a173 a65 a34 a72a132a223a48 a227 a130a33 a206a224 a58 a176a108z = ba70a42a32f(z)a48a188a189 a173 a65 a58a225 a213f(z) a43z = ba48a190a52a62 a102 a89 a58 a59z = b a70f(z) a48a66a207 a173 a65a33 a85 a86f(z) a43z = ba48a190a52a62a120 Laurenta68a91 a58 f(z) = ∞summationdisplay n=?∞ an(z ?b)n, 0 < |z ?b| ≤ ρ. a224 a45a43a46C : |z ?b| = ρ a53a58 |f(z)| < M a58 a71a44 |an| = vextendsinglevextendsingle vextendsinglevextendsingle 1 2pii contintegraldisplay C f(z) (z ?b)n+1dz vextendsinglevextendsingle vextendsinglevextendsingle≤ 1 2pi contintegraldisplay C |f(z)| |z ?b|n+1|dz| < M ρn. a124ρ → 0 a58 a167 a37 an = 0, n = ?1,?2,?3,···. square a225 a212 a42a32a43a199a65a190a52a62a48 Laurenta68a91 a102a102 a179 a131 a196a206 a170 a58 f(z) = ∞summationdisplay n=?m an(z ?b)n = a?m(z ?b)?m + a?m+1(z ?b)?m+1 +··· +a?1(z ?b)?1 + a0 + a1(z ?b) +··· = (z ?b)?mbracketleftbiga?m + a?m+1(z ?b) + a?m+2(z ?b)2 +···bracketrightbig = (z ?b)?mφ(z), φ(z)a43z = ba65a118 a68 a190a52a62a70 a56a57 a48 a58 a?m negationslash= 0a33 ba65 a36 a132a45f(z)a48 ma135a199a65a33 a92a201|z ?b|a226a227a181 a58 |f(z)|a66a44a228 a61 a63a64a169a32 a58 a167 lim n→b f(z) = ∞. a117 a235a236 a225 a212a241a213 a225a229a116a230 ∞ a58a231a232a233a58 a117 a235a236 a225 a212a234a235 a230a236a237 a213a33 a206a224 a58 a176a108 ba70f(z)a48a188a189 a173 a65 a58 a213 lim z→b f(z) = ∞ a58 a59b a70f(z)a48a199a65a33 a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12219 a123 a85 a224 a45 lim z→b f(z) = ∞ a58 a228 a203|z ?b| < δ a35 a58 |f(z)| > M, |f(z)|?1 < 1M = ε, a167 lim z→b 1 f(z) = 0. a61 a70a66 a124 1 f(z) = (z ?b) mg(z), a68a69lim z→b g(z) negationslash= 0 a58 a213g(z) a43z = ba118 a68 a190a52a62 a56a57 a33a71a44 f(z) = (z ?b)?m · 1g(z) = (z ?b)?mφ(z). square a238 a62 z = b a18 f(z) a141 ma239a240a142 a58a161a241 a38 a18 1/f(z) a141 ma239a242a142a33 a168 a25a243a244a33 a105a67 a34a131a105 a166a58 a66a44a245a246a247a248a249a250a199a65a33 ? z = npia701/sinz a48a72a135a199a65a199 ? z = 2kpii, k = 0,±1,±2,···a701/parenleftbigez ?1parenrightbiga48a72a135a199a65a199 ? z = 1a701/(z ?1)2 a48a175a135a199a65a33 a251 a216a211a212 a42a32a43 a110 a210a173 a65a190a52a62a48 Laurenta68a91a252 a102 a178a136a189 a131 a196a206 a170a33 ? a238 a62 z = b a18 a35a137f(z) a141a253a172a170 a142 a58a161a254 z → ba13 a58 f(z) a141 a240a255 a44a0a36 a33 ? a1a2a3 a166a4a58 z → b a141a31a139 a44 a77a58 f(z) a7a8a5a6 a44 a77a141 a137 a145a33 a131a176 a58 z = 0a70a42a32 e1/z = ∞summationdisplay n=0 1 n! parenleftbigg1 z parenrightbiggn , 0 < |z| < ∞ a48 a110 a210a173 a65a33a203 z a44 a171a218 a72a101a7 a61 0a35 a58 a36 a102 a171a218 a48a107a108 a160 ? a254 z a8 a89a9a10a11 a34 0a13 a58 e1/z → ∞a199 ? a254 z a8 a81 a9a10a11 a34 0a13 a58 e1/z → 0a199 ? a254 z a8a12 a10a11 a34 0a13 a58 e1/z a44 a11 a34 a147 a49 a3 a38 a141 a137 a33 a13 a91a70 a58 ? a254 z a8a14a15 ±i/2npi, n = 1,2,3,··· a11 a340a13 a58 e1/z a16a1511 ( a43 a83 a8 1a151a17a18a142) a199 ? a254 z a8a14a15 ±i/(2n+1)pi, n = 0,1,2,3,··· a11 a340a13 a58 e1/z a16a151?1 ( a43 a83 a8 ?1a151a17a18a142)a199 ? a254 z a8a14a15 ±i/(2n+ 1/2)pi, n = 0,1,2,··· a11 a34 0a13 a58 e1/z a16a151?i ( a43 a83 a8 ?ia151a17a18a142) a33 a66a44a229 a127 a58 a60a61a110 a210a173 a65z = ba130a165 a58 a63a74a19a70a72 a131 a32A( a102 a179a188∞) a58 a200 a66a44a250 a38 a72 a131 a20a21z n → ba58 a168 a37f(z n) → A( a171 a229)a33 §5.6 a215a216 a114a115a116a217a218a219a220 a12220 a123 a22a23a128 a139 a165 a58 a43 a110 a210a173 a65a48a63a74a72 a131 a181a190a52a62 a58 a42a32 f(z) a66a44a24 (a128 a213 a24a178a136a189a106) a63a74a48 a102 a179a32 a55a58a25 a189a66a175 a102 a72 a131 a131a87a33 a236a26a27 a212 a98a28a186a29z = 1/t a58 a125f(z) a30 a187f(1/t) a130 a158a159a31 star a132 t = 0a65a70f(1/t)a32a66a207 a173 a65 a58 a59 z = ∞ a65a70f(z)a32a66a207 a173 a65a199 star a132 t = 0a65a70f(1/t)a32a199a65 a58 a59z = ∞ a65a70f(z)a32a199a65a199 star a132 t = 0a65a70f(1/t)a32 a110 a210a173 a65 a58 a59 z = ∞ a65a70f(z)a32 a110 a210a173 a65 a31 ? z = ∞a701/(1 + z)a32a66a207 a173 a65a199 ? z = ∞a701 + z2 a32a175a135a199a65a199 ? z = ∞a70ez, sinz, cosz, ···a32 a110 a210a173 a65 a31 a109a110a111 a112a113a114a115a116a117a118a119a120a121 a12221 a123 §5.7 a33 a34 a35 a36 a72 a131 a131a196 a31 a206a37a38 ∞summationdisplay n=0 zn = 1 + z + z2 +··· ? a39a40z = 0a65a41a42a43a32a44a45a42g1 : |z| < 1 a46a47a48a49a50a51a52a53a54a55a56 a38 a49a57a41 f1(z)a31 ? a39a42a58a49 a37a38a59a60a61 a32 a31 a62a63a64 a53a65 a37a38 a51a66a67a49a68a40a69a70 f1(z)a39a44a45a42a46a71a72a52a73a74a56a75a76a77a78a79a80a75a76a81 a82a83 a49a84z = i/2a73a49a85 f1 parenleftbiggi 2 parenrightbigg = 1 + i2 + parenleftbiggi 2 parenrightbigg2 +···, fprime1 parenleftbiggi 2 parenrightbigg = 1 + 2· i2 + 3· parenleftbiggi 2 parenrightbigg2 +···, fprimeprime1 parenleftbiggi 2 parenrightbigg = 2·1 + 3·2· i2 + 4·3· parenleftbiggi 2 parenrightbigg2 +···, ... f(n)1 parenleftbiggi 2 parenrightbigg = n! + (n + 1)!1! i2 + (n + 2)!2! parenleftbiggi 2 parenrightbigg2 +···, ... a86a87 a49f1(z)a84z = i/2a73a74Taylora88a89 a59 ∞summationdisplay n=0 1 n!f (n) 1 parenleftbiggi 2 parenrightbigg · parenleftbigg z ? i2 parenrightbiggn . a64 a53a90a75a91a92a93a84a94a74a47a48a42g2 : |z ?i/2| < r a46a47a48a49a93a50a51a95a52a53a54a55a56a75a49a57a41 f2(z)a81 a965.3 a97a98a99a100 f1(z)a101f2(z)a102a103a104a105a106a107 §5.7 a108 a109 a110 a111 a11222a113 star a114g1 a115 g2 a116a117a118a119a120 g1intersectiontextg2 a121 a49f1(z) ≡ f2(z)a81 a122a123a124 a49 f1(z) = 11?z, a125a126 f(n)1 parenleftbiggi 2 parenrightbigg = n!parenleftbigg 1? i2 parenrightbiggn+1, a127a128 a49a93a129a69a130 f2(z) = ∞summationdisplay n=0 1parenleftbigg 1? i2 parenrightbiggn+1 parenleftbigg z ? i2 parenrightbiggn = 11?z, vextendsinglevextendsingle vextendsinglevextendsinglez ? i 2 vextendsinglevextendsingle vextendsinglevextendsingle< √5 2 . star f1(z)a101f2(z)a131a132a133a134a135a136a137a138a139 (a1401/(1?z))a114a132a135 a119a120a121a116a141a142a143 a81 ? a64a144a53a51a66a67a145a85a78a146a74a85a147a148a149a150 g1 : |z| < 1 a151 g2 : |z ?(i/2)| < √5/2. ? a84a152a153a154a155 g1intersectiontextg2 a49f1(z) ≡ f2(z)a81 star a64a156a157a158a159a160a84a154a155 g1 a46a74a65a90a75a70 a60 a49a130a161a95a84a162a52a154a155 g2 a46a74a162a52a65a90a75a51a66a67a81 star a84 a144 a53a154a155a74a152a153a163a164 g1intersectiontextg2 a46a165a166a167a168a81 star a169a170 a64 a53a171a172a49 a157 a68a129a173a70a174a175a74 a159a160 a148a149a49a176a177a68a129a178a88a161a179a53 z a180a181a81 a182a183 a184 a56a75f1(z) a84a154a155g1 a46a54a55a49a56a75 f2(z)a84a154a155 g2 a46a54a55a49a185a84 g1 a186 g2 a74a152a153a154 a155g1intersectiontextg2 a46a49f1(z) ≡ f2(z)a49a187a188f2(z)a189f1(z)a84g2 a46a74a54a55a190a191a192a193a194a49 f1(z) a59 f2(z)a84g1 a46a74a54a55a190a191a81 ? a195a196a197a198a199a200 a116a201a202 a49a203a204a205a206a138a139 a116 a182a183 a120 a101a197a198a207a208a81 a82a83 a49 a63 a90a75a209a210a164 a159a160 a74a56a75a49a211a175a145a85a52 a159 a74a212 a63 a148a149a49a213a214a54a55a190a191a49a68a129 a159a160 a70a84 a215a216 a148a149a46a74a54a55a56a75a81 a124 a181a74 a82a217a218 a49 a157 a130a161a95a84a154a155 g1 uniontextg 2 a46a54a55a74a56a75 f(z)a49 f(z) = braceleftBigg f1(z), z ∈ g1; f2(z), z ∈ g2. a219a220a221 a108a109a222a223a224a225a226a227a228a229 a11223a113 a162a52a53 a123a230a82a217 a49 a157 a59 Γa56a75a81 a231a232a63 a74a210a164 a159a160( a233a84a234a235a180a181a54a55)a70 a60 a49 a236 a214a54a55a190a191a49 a158 a185a130a161a94a84a237a180a181a74 a159a160 ( a238a239 9.1a240)a81 ? a241a136a242a243a196a244a197a198a199a200 a116a245a246 a134a247a248a249a250a251 a116a252 a197 a245a246 a81 a82a83 a49a253a254a165a79 a232a255 a164a0a1 d2w dz2 + p(z) dw dz + q(z)w = 0, a213 a232 a93a233a129a69a130a84a52 a159 a148a149a46a74a54a67a81a213a214a54a55a190a191a49a68 a128a158a64 a53a54a67a2a3a70a84a4a5a148a149a46a74a51 a66a67a81 a6 5.1 a184w 1 a59 a0a1 d2w dz2 + p(z) dw dz + q(z)w = 0, (5.1) a74a54a49a84a154a155 G1 a46a54a55a81a7 tildewidew1 a59 w1 a84a154a155G2 a46a74a54a55a190a191a49a8 w1 ≡ tildewidew1, z ∈ G1intersectiontextG2, a187 tildewidew1 a9a59 a0a1 (5.1)a74a54a81 a10 a184 d2tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = g(z), g(z)a84G2 a46a54a55a81 a86 a189w1 a59 a0a1(5.1)a84a154a155G1 a46a74a54a49 a11 a84a4 a217 a154a155G1intersectiontextG2 a46a49 a9a12a13 a0a1 d2w1 dz2 + p(z) dw1 dz + q(z)w1 = 0. a185a84 a87a217 a154a155a46a49 w1(z) ≡ tildewidew1(z)a49 a11 d2tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = 0, z ∈ G1 intersectiontextG 2, a8g(z) ≡ 0, z ∈ G1intersectiontextG2 a81a14a15a54a55a56a75a74a16a52a17a49a18a8a19a130 g(z) ≡ 0, z ∈ G2, a20 a8 tildewidew1 a84G2 a46 a12a13 a0a1 d2tildewidew1 dz2 + p(z) dtildewidew1 dz + q(z)tildewidew1 = 0. square a6 5.2 a184w 1 a151w2 a145a59 a0a1 (5.1)a74 a144 a53a21a17a22a23a54a49 a126a24 a84a154a155 G1 a46a54a55a81a7 tildewidew1 a151 tildewidew2 a164a25 a59 w1 a151w2 a84a154a155 G2 a46a74a54a55a190a191a49a8a84 z ∈ G1intersectiontextG2 a218 w1 ≡ tildewidew1, w2 ≡ tildewidew2. a187 tildewidew1 a151 tildewidew2 a9 a21a17a22a23a81 §5.7 a108 a109 a110 a111 a11224a113 a10 a231a821 a26a49 tildewidew1 a151 tildewidew2 a9a59 a0a1 (a84G2 a46)a74a54a81 a231 a254w1 a151w2 a74a21a17a22a23a17a27a49 ?[w1,w2] ≡ vextendsinglevextendsingle vextendsinglevextendsinglew1 w2 wprime1 wprime2 vextendsinglevextendsingle vextendsinglevextendsinglenegationslash= 0, z ∈ G1. a184 ?[tildewidew1, tildewidew2] ≡ vextendsinglevextendsingle vextendsinglevextendsingle tildewidew1 tildewidew2 tildewidewprime1 tildewidewprime2 vextendsinglevextendsingle vextendsinglevextendsingle= g(z), g(z)a84G2 a46a54a55a81 a231 a254a84 z ∈ G1intersectiontextG2 a218a49 w1 ≡ tildewidew1, w2 ≡ tildewidew2, a11g(z) negationslash= 0, z ∈ G 1 intersectiontextG 2 a81a9 a92a14a15a54a55a56a75a74a16a52a17a49 a157 a19a130 g(z) negationslash= 0, z ∈ G2. a127a128 a49 tildewidew1 a151 tildewidew2(a84G2 a46) a9 a21a17a22a23a81 square a54a55a190a191 a59 a170a28a56a75a29a30 a218a31 a169a32a74a33a34a194a52a81 a197a198a199a200 a116a35a36a37 a243a38a39 a245a246 star a197a198a199a200a40a41a42a43a49a44a45a46a47a138a139 a116a48a49 a249a50a81 a83a51 a84a65a90a75f1(z)a74a47a48a52g1 a74a53a54 a124 a55a56 a12 a95a57 a58 a73a49a8a84a47a48a52a59 a124 a71a72a52a73a49a4a71 a72a60a74a61a155a46a145a85f1(z)a74 a58 a73a49 a62a63 a49a84g1 a46a169a64a65Taylora88a89a49a4a47a48a148a149a66a67a68a129a173 a70g1 a81 star a197a198a199a200 a116a68a69 a134a41 a115a70a71 a102a105a107 a72a73 a67a74a75a76a190a191 (a161a74a52a154a155) a74a77 a51 a59a78 a167a74a107 a213a214a54a55a190a191a130a161a74a56a75 a59a79 a76a74a49a80 a59a81 a76a74a107 a27 a82a83a84a85a86a87a88 ?[w1,w2]a89G1 a90a91a92a93a94a95a96a97a98 a85a99a100a101 a97 ?(z) ≡ ?[w1(z),w2(z)] = ?(z0) · exp bracketleftBig ?integraltextzz0p(ζ)dζ bracketrightBig , a102a103 ?[w 1(z),w2(z)]a89a104a93a94 z0 a97a98 a85 ?[w1(z),w2(z)]a105 a97a98a106 a219a220a221 a108a109a222a223a224a225a226a227a228a229 a11225a113 §?5.8 Bernoulli a107a108 Euler a107 a109a110 a30a56a75 f(z) = zez ?1 a84z = 0a73a74a65a90a75a88a89a81 a231 a254 a87 a56a75a164a111a74a112a73a189 2npi, n = 0,±1,±2,···a49 a126a24 a189a52a79a112a73a192a74 a113z = 0 a114 a59 a56a75a164 a217 a74a52a79a112a73a49 a86a87 z = 0 a73 a59 f(z)a74a68a115 a58 a73a49 limz→0 zez ?1 = 1. f(z)a68a84a52a155|z| < 2pi a116a65a117a118a88a89a49 z ez ?1 = ∞summationdisplay n=1 Bn n! z n, Bn a188a189a119a120a62a75a81a121a181a157a175a69 Bn a81a86a189 z ez ?1 = z 2 parenleftbiggez + 1 ez ?1 ?1 parenrightbigg = z2 e z/2 + e?z/2 ez/2 ?e?z/2 ? z 2, a122 a52a123a189z a74a124a56a75a49a88a89 a113 a233a85 z a74a124a125a65a49 a127a128 B2n+1 = ?12δn0, n = 0,1,2,···. a126a63a127a159a128 a75a129a69a70 B2n a81a189 a87 a49a130 (5.41)a67a131a132a133 ez ?1 z bracketleftbigg ? z2 + ∞summationdisplay n=0 B2n (2n)!z 2n bracketrightbigg = ?12 ∞summationdisplay k=1 1 k!z k + ∞summationdisplay k=0 bracketleftbigg[k/2]summationdisplay n=0 1 (k?2n + 1)! B2n (2n)! bracketrightbigg z2k = 1, a157 a68 a128 a69a130 B0 = 0a128a77B2n a74a134a2a23 a128 [k/2]summationdisplay n=0 k! (k?2n + 1)! B2n (2n)! = 1 2. a158 a185a2a130 B2 = 16, B4 = ? 130, B6 = 142, B8 = ? 130, B10 = 566, B12 = ? 6912730, B14 = 76, B16 = ?3617510 , ···. a67a135a136a70 B4n?2 > 0, B4n < 0, n = 1,2,3,···a81 a137a63a124 a181a74a77 a51 a80a68 a128 a130a161a138 a81 a85 a63 a74a88a89a67a49 a82a83 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; 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; §?5.8 Bernoullia223a139Eulera223 a11226a113 ln sinzz = ∞summationdisplay n=1 (?)n 2 2n?1 n(2n)!B2nz 2n, |z| < pi; lncosz = ∞summationdisplay n=1 (?)n2 2n?1(22n ?1) n(2n)! B2nz 2n, |z| < pi 2; ln tanzz = ∞summationdisplay n=1 (?)n?1 2 2n(22n?1 ?1) n(2n)! B2nz 2n, |z| < pi 2. a63 a74 a156 a74a140a129a80a68 a128 a130a161 2ez/2 ez + 1 = sech z 2 = ∞summationdisplay n=0 En n! parenleftBigz 2 parenrightBign , |z| < pi, a4 a218E n a188a189a141a142a75a49 E2n+1 = 0, n = 0,1,2,···; E0 = 1, E2 = ?1, E4 = 5, E6 = ?61, ···. a94a143 a12a13 a134a2a23 a128 ksummationdisplay l=0 (2k)! (2l)!(2k?2l)!E2l = 0, k ≥ 1. a124 a181a74a77 a51 a80a68 a128 a132a133 2ez/2 ez + 1 = sech z 2 = ∞summationdisplay n=0 E2n (2n)! parenleftBigz 2 parenrightBig2n , |z| < pi.