Wu Chong-shi a0a1a2 a3 a4 a5 a6( a7) a81a9 a10a11a12 a13 a14 a15 a16 ( a17) §5.1 a18 a19 a20 a21a22a23a24a25a26a27a24a28a29a21a30a23a31a30a23a28a22a23a32 ∞summationdisplay n=0 cn(z ?a)n = c0 + c1(z ?a) + c2(z ?a)2 +···+ cn(z ?a)n +···. a33a26a34a35a36a37a38a39a31a30a23a28a22a23a32a40a26a41a42a43a44a41a25a45a31a34a35a30a23a28a22a23a46 a47a48 5.1 Abel( a49a50) a47a48 a51a52a22a23 ∞summationtext n=0 cn(z ?a)n a53a54a55 z0 a56a57 a32a58 a53a59a a55 a29a60a61a32 |z0 ?a|a29a62a63a31a60a64a65a66 a56a57 a32a67 a53 |z ?a| ≤ r(r < |z0 ?a|) a68 a34a69 a56a57 a46 a70 a71a29 ∞summationtext n=0 cn(z ?a)n a53z0 a56a57 a32a72a34a73a74a75a76a77a78a79 limn→∞cn(z0 ?a)n = 0. a71a80a81 a53a82 a23 q a32a83|c n(z0 ?a)n| < q a46a84 a59 a32 |cn(z ?a)n| = |cn(z0 ?a)n|· vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle n < q vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle n . a71a29 vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle < 1 a85|z ?a| < |z0 ?a|a86 a32 ∞summationtext n=0 vextendsinglevextendsingle vextendsinglevextendsingle z ?a z0 ?a vextendsinglevextendsingle vextendsinglevextendsingle n a56a57 a32a72 ∞summationdisplay n=0 cn(z ?a)na53a60|z ?a| < |z0 ?a|a64a65a66 a56a57 a46 a67a87|z ?a| ≤ r < |z 0 ?a|a86 a32 |cn(z ?a)n| ≤ q r n |z0 ?a|n, a25a23a28a22a23 ∞summationtext n=0 rn |z0 ?a|n a56a57 a32a72 ∞summationdisplay n=0 cn(z ?a)na53a60|z ?a| ≤ r (r < |z0 ?a|)a68 a34a69 a56a57 a46 a88a89 a90a22a23 ∞summationtext n=0 cn(z ?a)n a53a54a55z1 a91a92 a32a58 a53 a60|z ?a| = |z 1 ?a|a93a94a94a91a92 a46 a70 a45a95a96a97a46a90a22a23 ∞summationtext n=0 cn(z ? a)n a53a60|z ? a| = |z1 ? a|a93 a54 a34 a55 z2 a56a57 a32a58a98 Abel a73 a99a32a22a23a76a100 a53 a60|z ?a| = |z 2 ?a|(|z2 ?a| > |z1 ?a|) a64 a56a57 a32a101a102a103a104a105a46a72a22a23 ∞summationtext n=0 cn(z ?a)n a53 a60|z ?a| = |z 1 ?a|a93a94a94a91a92 a46 square Wu Chong-shi §5.1 a106 a5 a6 a82a9 a107a108a109a110a107a108a111a112 a113a114a34a115a22a23 a53z a116a117a118 a31a119a120a34 a55 a32a121a26a77a122 a56a57 a32a77a122 a91a92 a46a71a80a32 a66a114a21a22a23a123a124a32a125a126a127a128a33a129a31a130a131a132 a53z a116a117a118 a34a133a134 a55 a21a22a23 a56a57 a32 a53a135 a93 a34a133a134 a55 a21a22 a23 a91a92 a46a33a136 a56a57 a55 a101 a91a92 a55a137a138 a81 a53 a34a115a134a139a140a46 star a141a142Abela73a99a32a33a115a134a139a140a34a73a26a60a46a33a115a60a32a125a143a29a21a22a23a31 a107a108a109a46 star a56a57 a60a31a60a61a132 z = a a55 a46 star a56a57 a60a31a62a63a143a29a107a108a111a112a46 a56a57 a62a63a144 a59 a260a46a33 a86 a32 a56a57 a60a145a146a29a34a115 a55 a46a147z = a a55 a93 a32a21a22a23 a53a148 a116a117a94a94 a91a92 a46 a56a57 a62a63a40a144 a59 a26∞a46a33 a86 a56a57 a60a125a26 a148 a116a117 a46a21a22a23 a53a148 a116a117 a56a57 a32a149∞ a55a150 a73a26a151 a55 a46 a152a21a22a23a31 a56a57 a62a63a31a153a97a32a25a45a31a154a155a115a132 1. a141a142Cauchya156a157 a97a32a87 limn→∞|cn(z ?a)n|1/n < 1 a85 |z ?a| < 1lim n→∞|cn| 1/n a86 a22a23a65a66 a56a57a158 a87 limn→∞|cn(z ?a)n|1/n > 1 a85 |z ?a| > 1lim n→∞|cn| 1/n a86 a22a23 a91a92 a46a71a80a32a21a22a23 ∞summationtext n=0 cn(z ?a)n a31 a56a57 a62a63a26 R = 1lim n→∞|cn| 1/n = limn→∞ vextendsinglevextendsingle vextendsinglevextendsingle 1 cn vextendsinglevextendsingle vextendsinglevextendsingle 1/n . 2. a141a142d’Alemberta156a157 a97a32a51a52 limn→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1(z ?a)n+1 cn(z ?a)n vextendsinglevextendsingle vextendsinglevextendsingle = |z ?a| lim n→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1 cn vextendsinglevextendsingle vextendsinglevextendsingle a81 a53 a32a58a87 limn→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1(z ?a)n+1 cn(z ?a)n vextendsinglevextendsingle vextendsinglevextendsingle < 1 a85 |z ?a| < limn→∞ vextendsinglevextendsingle vextendsinglevextendsingle cn cn+1 vextendsinglevextendsingle vextendsinglevextendsingle a86 a22a23a65a66 a56a57a158 a87 limn→∞ vextendsinglevextendsingle vextendsinglevextendsinglecn+1(z ?a)n+1 cn(z ?a)n vextendsinglevextendsingle vextendsinglevextendsingle > 1 a85 |z ?a| > limn→∞ vextendsinglevextendsingle vextendsinglevextendsingle cn cn+1 vextendsinglevextendsingle vextendsinglevextendsingle a86 a22a23 a91a92 a46a71a80a32a21a22a23 ∞summationtext n=0 cn(z ?a)n a31 a56a57 a62a63a26 R = limn→∞ vextendsinglevextendsingle vextendsingle cnc n+1 vextendsinglevextendsingle vextendsingle. a159a160a161a162a163a164a165a166a167a168a169a170a171a172a173a174a46Cauchya168a169a175a176a177a178a179a167a32a180d’Alemberta168a169 a181a175a171a182a183a167( a184 a162a185a186 lim n→∞|cn/cn+1|a187a188) a46 a189a190a191a192a193a194a195a196 a32a197a197a198a199a200a201a202a203a46 Wu Chong-shi a0a1a2 a3 a4 a5 a6( a7) a83a9 a113a114a21a22a23 ∞summationtext n=0 cn(z ?a)n a31a204a34a28a205a26 z a31a206a207a30a23a32Abel a73a99a208a209a210a211a32a21a22a23a53a212 a56 a57 a60a64a119a34a213a214a215 a68 a34a69 a56a57 a32a71a80a32 a141a142 4.2a216 a32 a53 a56a57 a60a64a32a21a22a23a217a218a128a34a115a206a207a30a23 ( a219 a220a124a32a21a22a23a31a221a30a23 a53 a56a57 a60a64a206a207) a32a144 a59 a66a21a22a23a222a28a223a134 a219 a222a28a152a224a23a32 integraldisplay z z0 ∞summationdisplay n=0 cn(z ?a)n dz = ∞summationdisplay n=0 cn integraldisplay z z0 (z ?a)n dz = ∞summationdisplay n=0 cn n + 1 bracketleftbig(z ?a)n+1 ?(z 0 ?a)n+1 bracketrightbig, d dz bracketleftbigg ∞summationdisplay n=0 cn(z ?a)n bracketrightbigg = ∞summationdisplay n=0 cnd(z ?a) n dz = ∞summationdisplay n=0 cn+1(n + 1)(z ?a)n. star a21a22a23a53 a56a57 a60 a118 a31 a56a57a225a226 ? a144a59a94a94 a56a57 a32 ? a144a59a94a94 a91a92 a32 ? a40a144a59a53a34a133a134a55 a56a57 a32 a53a135 a34a133a134 a55 a91a92 a46 1 + z + z2 +···+ zn +··· a53|z| = 1a118a94a94 a91a92a158 z 1 + z2 2 + z3 3 +···+ zn n +··· a53|z| = 1 a118 a147z = 1 a93a227 a56a57 a32a67 a53 z = 1a55 a91a92a158 z2 1·2 + z3 2·3 + z4 3·4 +···+ zn n(n?1) +··· a53|z| = 1 a118a94a94 a56a57 a46 a228a229a230a35a130a131a32a21a22a23a31 a56a57 a60 a118 a121 a150 a73a154a151 a55 a46 a149 a85 a83 a53 a151 a55 a32a21a22a23a231a100a144a232a26 a56a57 a31 ( a85 a154a233a73a31a30a23a234) a46 a103a21a22a23 ∞summationdisplay n=0 cn(z ?a)n a53 a56a57 a60a64 a56a57a235 f(z)a32a51a52a22 a23 a53 a56a57 a60a236 a118 a54a55 z0 a40 a56a57 a32a221a29S(z 0) a32a58 a237a238a239 a49a240 a47a48(a228a96) a208a209a210a211a32a87 z a113 a56a57 a60a64a241a114 z 0 a86 a32a242a77a243a244 a53a59z0 a29a245 a55 a44a246a247a292φ < pia31a248a249a64( a250a2515.1)a32f(z)a125 a34a73a241a114S(z 0) a46 a252 5.1 a253a254a255a8a0a1a2 Wu Chong-shi §5.2 a3a4a5a6a7a8a9a10a6a11a12a13 a84a9 §5.2 a14a15a16a17a18a19a20a21a17a22a23a24 §5.3 a68 a154a25a30a23a22a23a206a207 a225 a31a26a229a32a40a144 a59 a45a123a27a229a28a29a30a31a95a25a223a134a31a206a207 a225 a46 a47a48 5.2 a103 1. f(t,z)a175ta31z a167a32a33a34a35a32 t > aa32z ∈ Ga32 2. a36a37a38a39t ≥ aa32f(t,z)a175Ga40 a167a202a41a42a43a34a35a32 3. a44a45 integraldisplay ∞ a f(t,z)dta188Ga40a46a47a163a164a32a48?ε > 0a32?T(ε)a32a190T2 > T1 > T(ε)a196a32a171 vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle integraldisplay T2 T1 f(t,z)dt vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle < ε, a58F(z) = integraldisplay ∞ a f(t,z)dta53G a64a26a206a207a31a32a49 Fprime(z) = integraldisplay ∞ a ?f(t,z) ?z dt. a70 a119a50a34a115a51a52a53a54{a n} a0 = a < a1 < a2 < a3 < ··· < an < an+1 < ···, limn→∞an = ∞. a55u n(z) = integraldisplay an+1 an f(t,z)dta32a58a141a142 3.7a216 a25a114a28a29a30a31a73a223a134a31a206a207 a225 a31a73a99a32a144a56u n(z) a26G a64 a31a57a234a206a207a30a23a46a58a71a29 F(z) = ∞summationdisplay n=0 un(z) a53G a118 a34a69 a56a57 a32a72 a141a142 Weierstrassa73a99a32a56 F(z) = ∞summationdisplay n=0 un(z) = integraldisplay ∞ a f(t,z)dt a53G a64a206a207a32a49 Fprime(z) = ∞summationdisplay n=0 uprimen(z) = integraldisplay ∞ a ?f(t,z) ?z dt. square a66a114a28a29a30a31a59a223a134a40a144 a59a60a61a62 a94 a99a46 a53a63 a45a33a115a73a99 a86 a32a64a77 a156a65a66a67 a223a134( a219 a59a223a134)a26a68a34a69 a56a57 a46a25a45a31 a156a157 a97a26a132a69a70 a187 a188 a34a35φ(t)a32a71a72|f(t,z)| < φ(t)a32z ∈ Ga32a180a73 integraldisplay ∞ a φ(t)dta163a164a32a181 integraldisplay ∞ a f(t,z)dta188Ga40a74a36 a180a73 a46a47 a163a164a46 a75a29a28a29a30a31 a66a67 a223a134a31a34a115a76a77a32a78 a117 a27a229a223a134 F(z) = integraldisplay ∞ 0 e?t2 cos2ztdt. (5.1) a33a115a223a134 a68 a31a79a223a30a23a80a100a74a75a73a99a31a81a155a115a78a79a32a67a49a71a29a66a114a82a23 z = x + iy a32a154 |cos2zt| = radicalBig cosh22yt?cos22xt ≤ cosh2|yt| ≤ e2|yt|. Wu Chong-shi a0a1a2 a3 a4 a5 a6( a7) a85a9 a84 a59 a32a66a114z a116a117a118 a31a119a120a34a115a213a214a215 a118 a32 |Imz| < y 0 a32a114a26a32 vextendsinglevextendsingle vextendsinglee?t2 cos2zt vextendsinglevextendsingle vextendsingle < e?t2+2y0t, a67a223a134 integraldisplay ∞ 0 e?t2+2y0tdt a56a57 a32a84 a59 a28a29a30a31 a66a67 a223a134 (5.1)a34a69 a56a57 a32a71a80a32a33a115a223a134a75a29 z a31a30 a23a32 a53z a116a117a118 a31a119a120a34a115a214a215a64a206a207a46a83a84a34a85a32a125a154 Fprime(z) = ? integraldisplay ∞ 0 e?t22t sin2ztdt = e?t2 sin2zt vextendsinglevextendsingle vextendsingle ∞ 0 ?2z integraldisplay ∞ 0 e?t2 cos2ztdt = ?2zF(z). a206a33a115a86a134a87a88a32a125a144 a59a89 a235 F(z) = Ce?z2 a32a212a68 a25a23C a26 C = F(0) = integraldisplay ∞ 0 e?t2dt = 12√pi, a33a129a32a41a90a125 a89 a235 integraldisplay ∞ 0 e?t2 cos2ztdt = 12√pie?z2. Wu Chong-shi §5.3 a11a12a91 a6 a6 Taylora92a93 a86a9 §5.3 a22a23a94a20a17 Taylor a95a96 a46 a161a97a34a35 a188a98 a167a163a164a99a100a101a102 a46 a161a42a43a34a35a46 a69 a39a103a46 a161a42a43a34a35a102a104a178a97a105a35 a226 a47a48 5.1 (Taylor) a103a30a23 f(z) a53a59 a a29a60a61a31a60C a64a106C a118 a206a207a32a58a66a114a60a64a31a119a107 z a55 a32f(z)a144a45a21a22a23a108a109a29 ( a219 a220a124a32 f(z)a144 a53aa55 a108a109a29a21a22a23) f(z) = ∞summationdisplay n=0 an(z ?a)n, a212 a68 an = 12pii contintegraldisplay C f(ζ) (ζ ?a)n+1dζ = f(n)(a) n! , C a50a110a86a111 a87a112a113a46 a70 a141a142Cauchya223a134a114a39a32a66a114a60C a64a119a120a34a55 z a32a154 f(z) = 12pii contintegraldisplay C f(ζ) ζ ?zdζ. a149a26a32 1 ζ ?z = 1 (ζ ?a)?(z ?a) = 1 ζ ?a ∞summationdisplay n=0 parenleftbiggz ?a ζ ?a parenrightbiggn . a80a22a23 a53 vextendsinglevextendsingle vextendsinglevextendsinglez ?a ζ ?a vextendsinglevextendsingle vextendsinglevextendsingle ≤ r < 1a31a214a215 a68 a34a69 a56a57 a32a71a80a144 a59 a222a28a223a134a32 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+1dζ bracketrightbigg (z ?a)n = ∞summationdisplay n=0 an(z ?a)n, an = 12pii contintegraldisplay C f(ζ) (ζ ?a)n+1dζ = f(n)(a) n! . square a124a115a132 1. a73a99a31a78a79a144a59a116a117a32a242a77 f(z)a53C a64a206a207a85 a144a46 a159a196 a36a37a118a119 a167 z a32a120a121a122a122a a123 a99a124a125 a46 a99Cprime a32 a103z a126a127a188 a99a100a46 f(z) a188Cprime a100a128 Cprime a40 a175a42a43a167a46 a113 a129a130a131a132a133a134a135a136a137a138a139a140a141a131a129a142a136a143a144a145a146a147a148a149a150 Wu Chong-shi a0a1a2 a3 a4 a5 a6( a7) a87a9 2. a33a151Taylora108a109a31a38a39a221a152a153a30a23a68 a31 Taylora114a39a154a155a32a149a26a78a79a228a155a46 star a53a152a153a30a23a68 a32 f(x) a31a119a107a156a224a23a81 a53 a32a157a228a75 a59 a243a96 Taylora114a39a81 a53 ( a219 Taylora114a39 a56 a57 )a46 star a53a82a153a30a23a68a32a206a207a31a77a152 (a34a156a224a23a81a53) a125a75a59a243a96 Taylora22a23 a56a57 a46 3. a107a108a158a159 a30a23f(z)a31a151a55a160a148a161a73a128 Taylora22a23a31 a56a57 a62a63a46a103 ba26f(z)a31a162 a a55 a41 a163a31a151 a55 a32a58a34a164a124a123a32 a56a57 a62a63 R = |b?a|a46 f(z)a188 a99|z ?a| < |b ?a| a100a165a165a42a43a32f(z)a121a122 a188 a99a100a166a167 a123Taylora105a35(a168 a192a169 a32 Taylora105a35a188 a99|z ?a| < |b?a| a100a163a164)a46a159a170a175a169a32f(z)a167Taylora105a35a163a164a165a166a171a172 a37|b?a| a46 a163a164a165a166 a46a173a174 a171a193a175 a37|b?a|a46a176a181a32ba174a170a126a177a188 a163a164a99a100a32a178a180a97a105a35 a188 a163a164 a99a100a165a165a42a43a32a179 ba174 a123a180 a174a167a181a182a183a184 ( a185a186ba174a175a121a187a180 a174a32a188 5.5 a189)a46 1 1 + z2 = ∞summationdisplay n=0 (?)nz2n, |z| < 1. a30a23a31a151 a55 z = ±i a125 a161 a73a128Taylora22a23a31 a56a57 a62a63 R = |±i| = 1a46 a67 a53 a152a23a248a249a64a32 Taylora22a23a31 a56a57 a62a63a101a30a23 a225a190 a137a138 a31a191a192a125a193 a59 a27a229a46 1 1 + x2 = ∞summationdisplay n=0 (?)nx2n, ?1 < x < 1, a125a193 a59 a99a206 a56a57 a62a63a29a107a261a32a71a29a30a231/(1+x2) a53a194 a115a152a195 a118 a205a26a196a197a144a224a44a198a49a119a107a156a224a23 a205a26a81 a53 a31a199 4. Taylor a200a201a202a203a50a204 a205 a73a34a115 a53 a60C a64a206a207a31a30a23a32a58a206a31 Taylora108a109a26a207a34a31a32 a85 a108a109a192a23a n a26 a160a148 a233a73a31a46 a70 a208a73a154a155a115 Taylora22a23 a53 a60C a64a205 a56a57a235 a155a34a115a206a207a30a23 f(z)a32 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 +···. a50a209a210z→aa32a58a113a114a22a23 a53 C a64a31a119a34a213a214a215 a68 a34a69 a56a57 a32a72a154 a0 = aprime0. a222a28a86a211a32a212a50a209a210 z → aa32a58 a89 a1 = aprime1. a51a80a213a197a32 a85 a144a96 a89 an = aprimen, n = 0,1,2,···. square Taylora108a109a31a207a34 a225 a208a209a210a211a132 Wu Chong-shi §5.3 a11a12a91 a6 a6 Taylora92a93 a88a9 star a228a229a45a214a122a87a97a32a89 a235 a31f(z) a53 a155a34a115a60a64a31Taylora108a109a26a207a34a31a46a71a80a32a228a34a73a77a45a152 a224a23a31a153a97a73a108a109a192a23a46 star a51a52a53a155a34a55a108a109a31a155a115Taylora22a23a154a215a32a58a144a59a222a28a216a217a192a23a46 ? a76a218a26a53a155a34a55a108a109a31a155a115 Taylora22a23a154a215a32a219a144a59a222a28a216a217a192a23a46 ? a155a34a115a30a23a53a228a155a55a108a109a89 a235 a31a155a115Taylora22a23a32 a85 a83a154a114a220a31 a56a57 a214a215a32a40a228a232a221 a222a216a217a108a109a192a23a46