Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10 a111 a12 a13a14a15 a16a17a18a19a20a21a22a23 §10.1 a24a25a26a27 a28a29a30a31 a32a33a34 G a35a36a37 C a38a39a40a41a42a43a35a44a45a46a47a48a49a50a51a52a53a54a55a56a57a58a59 bk, k = 1,2,3,···,n a60a61a62a63 f(z) a64 G a65a45a66a67a68a61a64 G a69a70a71a61a72a64 C a73a74a53 f(z) a35a58 a59a61a75 contintegraldisplay C f(z)dz = 2pii nsummationdisplay k=1 resf(bk). resf(bk) a76a38 f(z) a64 bk a77 a35a78a63a61a79a80a81 f(z) a64 bk a35a82 a34 a65Laurent a83a84a69(z ?bk)?1 a35a85a63 a(k)?1 a50 a8610.1 a87a88a89a90 a91 a92a93 10.1 a61a94a95a96a55a58a59bk a97 a46a47a48a49γk a61a98γk a99 a64G a65a61a72a100a101a102a103a61a75a104a105a106 a70a107 a33a34 Cauchy a108a109a110a62a63 a97 Laurenta83a84a111a35a85a63a112a113a61a114a53contintegraldisplay C f(z)dz = nsummationdisplay k=1 contintegraldisplay γk f(z)dz = 2pii nsummationdisplay k=1 a(k)?1 = 2pii nsummationdisplay k=1 resf(bk). square a115a116a117a118a119a120a121a122 a61a123a124a125 a116a126a127a128a129a130a131a132 a125 a116a133a127a128a134a126a135a136a137a138a139a140 a50a141a142 a143a144 a123a124a125 a116a126a127a128a129a130a131 a61a145a146 a143a144a147 a125 a116a133a127a128a134a135a136a148a126a115a116 a50 star a149f(z)a64a58a59b a77 a35a78a63a61 a150 a75a73a151a61a114a152a149f(z)a64z = ba35a82 a34 a65Laurenta83a84a69(z?b)?1 a153 a35a85a63a50 star a64a154a59a35a155a156a157a61a158a159a107a160a161a162a163a164a149a78a63a50 Wu Chong-shi §10.1 a3a4a5a6 a112a12 star a165a166a167a168a169a170a171 a32b a59a152f(z)a35a39a172a154a59a61a75a64ba59a35a82 a34 a65a61 f(z) = a?1(z ?b)?1 + a0 + a1(z ?b) + a2(z ?b)2 +···. a159(z ?b)a173a83a84a113a174a175a61 (z ?b)f(z) = a?1 + a0(z ?b) + a1(z ?b)2 + a2(z ?b)3 +···. a176 a159 a?1 = lim z→b (z ?b)f(z). star a177a178a179a180a35a155a156a152f(z) a158a159a181a182a38P(z)/Q(z)a61P(z)a183Q(z) a99 a64ba59a110a184a82 a34 a65a67a68a61b a152Q(z)a35a39a172a185a59a61Q(b) = 0a61Qprime(z) negationslash= 0a61P(b) negationslash= 0a61a75 a?1 = lim z→b (z ?b)f(z) = lim z→b (z ?b)P(z)Q(z) = P(b)Qprime(b). a186 10.1 a149 1 z2 + 1 a64a58a59a77a35a78a63a50 a187 z = ±i a152a79a35a39a172a154a59a50 resf(±i) = 12z vextendsinglevextendsingle vextendsingle z=±i = ?i2. a186 10.2 a149 eiaz ?eibz z2 a64a58a59a77a35a78a63a50 a187 z = 0 a152a79a35a39a172a154a59a50 resf(0) = limz→0z · e iaz ?eibz z2 . = limz→0 eiaz ?eibz z = i(a?b). star a188a166a167a168a169a170a171 a32z = b a152f(z)a35ma172a154a59a61m ≥ 2a61 f(z) = a?m(z ?b)?m + a?m+1(z ?b)?(m?1) +···+ a?1(z ?b)?1 + a0 + a1(z ?b) +···. a174a175a173a73 (z ?b)m a61 (z ?b)mf(z) = a?m + a?m+1(z ?b) +···+ a?1(z ?b)m?1 + a0(z ?b)m + a1(z ?b)m+1 +···. a189 a111a?1 a152(z ?b)mf(z)a35a83a84a113a69(z ?b)m?1 a153a35a85a63a61a190 a?1 = 1(m?1)! d m?1 dzm?1(z ?b) mf(z) vextendsinglevextendsingle vextendsinglevextendsingle z=b . a186 10.3 a1491/(z2 + 1)3 a64a58a59 a77 a35a78a63a50 a187 z = ±i a152a79a35a191a172a154a59a50 resf(±i) = 12! d 2 dz2(z ?i) 3 · 1 (z2 + 1)3 vextendsinglevextendsingle vextendsinglevextendsingle z=±i = 12! d 2 dz2 1 (z ±i)3 vextendsinglevextendsingle vextendsinglevextendsingle z=±i Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10 a113 a12 = 12!(?3)(?4)(z±i)?5 vextendsinglevextendsingle vextendsinglevextendsingle z=±i = ? 316i. star a192 a29a193∞ a168a169 a28a29 a194 a81∞a59a61a108a195 resf(∞) = 12pii contintegraldisplay Cprime f(z)dz, a189a196 a35Cprime a152a95∞a59a197a198(a199a114a152a200a111a201a202a198) a39a203a35a94a204a61a64a94a204a65a52 ∞a59a158a205a152f(z)a35a58a59 a60a178a206a58a59a50 star resf(∞)a207a101a152f(z)a64∞a82 a34 a65Laurenta83a84a69z1 a153a35a85a63a50 resf(∞) = 12pii contintegraldisplay Cprime f(z)dz = ? 12pii contintegraldisplay C f parenleftbigg1 t parenrightbigg dt t2 =? 1t2f parenleftbigg1 t parenrightbigg a64t = 0a59a82 a34 a65a208a209a63a83a84a69t?1a153a35a85a63 =?f parenleftbigg1 t parenrightbigg a64t = 0a59a82 a34 a65a208a209a63a83a84a69t1a153a35a85a63 =?f(z)a64z = ∞a59a82 a34 a65a208a209a63a83a84a69z?1a153a35a85a63a50 star a189a55a210a211a183a53a54a212 a77 a101a213a214 a77a215 1. a216a217a218a219a220a61a125 a116f(z)a133∞a136a126a115a116 a61a221a222f(z) a133∞a136a223a224a134a225a226a116a227a228a229z?1 a230 a126 a231a116a232a233?1 a61a234a235a236a142 a237a238a239a240 a50 2. a216a241a242a219a220a61a243a222z?1 a230a244a245a222f(z) a133∞a136a223a224a134a225a226a116a227a228a246a126a247a248a249a130a61a250a251a61a252 a253∞a136a254 a244f(z) a126a135a136 a61resf(∞)a255a0 a233a254 a141 0a50 a1a2 a61a252 a253∞a136 a244f(z) a126a135a136 a61 a3a4 a244a237a5 a6a136 a61a255a0 a233 a141 0a50 star a28a29a169a7a8 a9 a31 a192 a29 a169a10a11a11a12a50 a13 a50a14a62a63 f(z) = 1(z ?1)(z ?2)(z?3) a15 a40a40a113a50 1 (z ?1)(z ?2)(z ?3) = A z ?1 + B z?2 + C z ?3. a191a55a16a108a179a63a61A, Ba183C a61a197a17a114a152a62a63f(z)a64a39a172a154a59z = 1, z = 2a183z = 3a59 a77 a35a78a63a50 a18a19 A = res 1(z ?1)(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=1 = 12, Wu Chong-shi §10.1 a3a4a5a6 a114a12 B = res 1(z ?1)(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=2 = ?1, C = res 1(z ?1)(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=3 = 12. a92 a211a62a63f(z)a20a53a21a172a154a59a61a199a158a159a22a23a24 a77 a109a50 a13a92 a61 1 (z ?1)2(z ?2)(z ?3) = A (z ?1)2 + B z ?1 + C z?2 + D z ?3. a25a26a27a28 A = res 1(z ?1)(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=1 = 12, B = res 1(z ?1)2(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=1 = 34, C = res 1(z ?1)2(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=2 = ?1, D = res 1(z ?1)2(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=3 = 14. star a28a29a169a7a8 a29a30 a30a31 a11a50 a78a63a108a109a32a94a204a33a40a35a163a164a34a35a38a78a63a35a163a164a61 a36a37 a205a32a108a33a40a183a39a108a67a68a62a63a35a94a204a33a40 a38 a85a39a40a61a114a53a158a205a41a42a44a43a24a163a164 a28a189a44 a108a33a40a50 Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10 a115 a12 §10.2 a45a27a46a47a48a25a49a50a51 a53a109a191a52a62a63a35a33a40a35a53a113a152 I = integraldisplay 2pi 0 R(sinθ,cosθ)dθ, a184a69Ra152sinθ, cosθa35a53a109a62a63a61a64a33a40 a33a54 a73a152a70a71a35a50 a97a55a56 z = eiθ a61a75 sinθ = z 2 ?1 2iz , cosθ = z2 + 1 2z , dθ = dz iz , a57a58 a35a33a40a59a60a75 a55 a38 z a61a62a73a35a45a63a64a35a64a203|z| = 1a50a81a152a61 I = contintegraldisplay |z|=1 R parenleftbiggz2 ?1 2iz , z2 + 1 2z parenrightbigg dz iz = 2pi summationdisplay |z|<1 res braceleftbigg1 zR parenleftbiggz2 ?1 2iz , z2 + 1 2z parenrightbiggbracerightbigg . a53a109a191a52a62a63R(sinθ,cosθ)a64a33a40 a33a54[0,2pi] a73a70a71a61a114a65a66a67a53a109a62a63R parenleftbiggz2 ?1 2iz , z2 + 1 2z parenrightbigg a64a45a63a64a35a64a203a73a206a58a59a50 a186 10.4 a163a164a33a40 I = integraldisplay 2pi 0 1 1 + εcosθdθ, |ε| < 1a50 a187 a68a69 a73a62a35a202a70a71a72a61a73a74a53 I = integraldisplay 2pi 0 1 1 + εcosθdθ = contintegraldisplay |z|=1 1 1 + εz 2 + 1 2z dz iz = contintegraldisplay |z|=1 2 εz2 + 2z + ε dz i = 2pi summationdisplay |z|<1 res braceleftbigg 2 εz2 + 2z + ε bracerightbigg = 2pi· 22εz + 2 vextendsinglevextendsingle vextendsinglevextendsingle z=(?1+√1?ε2)/ε = 2pi√1?ε2. a163a164a78a63a111a61a75a76a62a63 2/(εz2 + 2z + ε)a53a174a55a154a59a61 z = ?1± √1?ε2 ε , a77a78 a81a79a74a35a173a33a38 1a61 a176 a159a39a108 a36 a53a39a55a154a59a61 z = (?1 +√1?ε2)/εa61 a77 a81a45a63a64a65a50 Wu Chong-shi §10.3 a79a80a81a82 a116 a12 §10.3 a83a84a50a51 a206a85a33a40a35a108a195a38 integraldisplay ∞ ?∞ f(x)dx = limR 1 →+∞R 2 →+∞ integraldisplay R2 ?R1 f(x)dx. a53a111 a189a86 a154a54a101a87a64a61 a77 limR→+∞ integraldisplay R ?R f(x)dxa87a64a61a76a38a33a40a88a66a61a89a38 v.p. integraldisplay ∞ ?∞ f(x)dx = lim R→+∞ integraldisplay R ?R f(x)dx. a90a91 a61a92 a189 a174 a86 a154a54a93a87a64a111a61a79a74a94a108 a57 a80a50 a64a106a61a62a73 a27 a61a33a40 integraldisplay ∞ ?∞ f(x)dx a152a95a96a97a98a99a100a35a61a207a101a101a102a106 a55 a62a63a35a94a204a33a40a50 ? a73a74a158a159 a25a26 a24a14a97a62a63 f(x)a103a104a38a106a62a63f(z) ? a38a67a205a101a102a94a204a33a40a207 a58a105 a78a63a108a109a163a164a61a106a94a107 a215 (1) a108a73a109a92a35a33a40a59a60a110a53a102a46a47a94a204a61a163a164 contintegraldisplay f(z)dza111 (2) a64a108a73a35a59a60a73a35a33a40a61a112a113a114 a176a37 a149a163a164a35a206a85a33a40a115a116 a57a117 a61a112a113a158a159a44a45a202a43a24 a163a164 a28 a40a50 a118a119a91 a35a120a70a92 a91 a152a108a73a159 a150 a59a38a64a121a61 Ra38a122a60a35a73a122a64CR a61contintegraldisplay C f(z)dz = integraldisplay R ?R f(z)dz + integraldisplay CR f(z)dz. a86 10.2 a110a123a124R → ∞a50 a189a125 a61a73a74a43a126 a37 a163a164 integraldisplay CR f(z)dz a35a154a54a66a50 a36a37f(z) a127a128a109a92a35a129a130a61 a189 a152a158a159a120a131a35a50 a101a132a133 a32 a62a63f(z)a127a128a157a134a129a130 a215 Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10 a117 a12 1. f(z)a64a73a122a61a62a52a67a53a54a55a56a57a58a59a60a152 a77a77 a67a68a35a61a64a97a98a73a74a53a58a59a111 2. a640 ≤ argz ≤ pia135a94a65a61a92|z|→∞a111a61zf(z)a39a136a24a137a81 0a61 a138a194 a81a139a140a35ε > 0a61a87a64 M(ε) > 0a61a98a92 |z|≥ M a610 ≤ argz ≤ pia111a61|zf(z)| < εa50 a234a141 a238a142a143a144 a254a145a146 a50a1471 a238a142a143a148a149 a142 a150a151a126a152a153a129a130a254 a244a154 a129a130 a61 a144a155 a0 a233a156a157 a115a116a117a118a143a144a127a128a129a130 contintegraldisplay C f(z)dz = integraldisplay R ?R f(z)dz + integraldisplay CR f(z)dz = 2pii summationdisplay a219a158a159a160 resf(z). a1472a238a142a143a61a161a162a244a163a141a152a153a164a165a129a130a126a166a167a142a143 limx→±∞xf(x) = 0 a126a168a169a170a171 a61a172a173a61a174a175a176 a118 3.2 a61a177 a148a149 a142 lim R→∞ integraldisplay CR f(z)dz = 0. a178 a154a54R →∞a61a114a179a131 integraldisplay ∞ ?∞ f(x)dx = 2pii summationdisplay a73a122a61a62 resf(z). a186 10.5 a163a164a108a33a40I = integraldisplay ∞ ?∞ dx (1 + x2)3 a50 a187 a19 a111 a90a91a180 a47a73a181 a37 a149a35a129a130a61a190 I = integraldisplay ∞ ?∞ dx (1 + x2)3 = 2pii·res 1 (1 + z2)3 vextendsinglevextendsingle vextendsinglevextendsingle z=i =2pii· parenleftbigg ?3i16 parenrightbigg = 38pi. a118 a123a61a38a67 a194a58a105 a78a63a108a109a163a164a108a33a40a35a182a183a184a185a53a39a55a41a42a186a187a35a109a67a61a101a132a188a189a106a39a157 a190 a62a35a191a181 a215 a38a67a205a192 a58a105 a78a63a108a109a163a164a206a85a33a40a61a73a74a94a107 a215 1. a108a73a109a92a35a33a40a59a60a110a53a102a46a47a94a204a61a163a164 contintegraldisplay f(z)dza111 2. a64a108a73a35a59a60a73a35a33a40a61a112a113a114 a176a37 a149a163a164a35a206a85a33a40a115a116 a57a117 a61 a112a113a158a159a44a45a202a43a24a163a164 a28 a40a50 Wu Chong-shi §10.3 a79a80a81a82 a118 a12 a182a81 a189a125 a35a109a67a61a114a158a159a193a194a195a196a24a197 a105 a78a63a108a109a163a164a108a33a40a50 star a198a218 a244f(x) a199a125 a116 a61 a248a200 a222 a129a130 integraldisplay ∞ 0 f(x)dxa61a243a222 integraldisplay ∞ 0 f(x)dx = 12 integraldisplay ∞ ?∞ f(x)dx, a201a233a202a169 a0 a233a203a157a20410.2a126a127a128 a61 a144a205a206 a219a160 a126a207a208 a61a209a210a211integraldisplay ∞ 0 f(x)dx = 12 integraldisplay ∞ ?∞ f(x)dx = pii summationdisplay a219a158a159a160 resf(z). star a198a218 a133a129a130 integraldisplay ∞ 0 f(x)dx a229a61a212 a129 a125 a116f(z) a213 a139a214a215a200a216a217a218 a61a219a198 f(z) = f(zeiθ), a220a221 a61a255a0 a233a203a157a20410.3 a229a126a127a128a151a143a144 a50 a86 10.3 a86 10.4 a186 10.6 a163a164a108a33a40 integraldisplay ∞ 0 dx 1 + x4 a50 a187 a78 a81 a189a196 a35a222a33a62a63f(x) = 11 + x4 a152x4 a35a62a63a61 a176 a159a61a73a74a158a159a223 a105a93 10.4 a35a94a204 a215 a95a197a97a98 a780 a131Ra61a95a64a224a131a225a197a226a98a61a188a95a197a226a98 a78iR a227a131 a150 a59a50 a189a125 a61a104a105a78a63a108a109a61a53 contintegraldisplay C dz 1 + z4 = integraldisplay R 0 dx 1 + x4 + integraldisplay CR dz 1 + z4 + integraldisplay 0 R idy 1 + (iy)4 =(1?i) integraldisplay R 0 dx 1 + x4 + integraldisplay CR dz 1 + z4 =2pii res 11 + z4 vextendsinglevextendsingle vextendsinglevextendsingle z=eipi/4 = pi2 1?i√2 . a178 a154a54R →∞a61 a18 a38 limz→∞z · 11 + z4 = 0, a176 a159a61a104a105a228a109 3.2a61a53 lim R→∞ integraldisplay CR dz 1 + z4 = 0. Wu Chong-shi a0a1a2 a3a4a5a6a7a8a9a10 a119 a12 a81a152a114a179a131 integraldisplay ∞ 0 dx 1 + x4 = √2 4 pi. a133 a234 a238 a219a229 a229 a61a230 a169a202a169 a0 a233a203a157 a158a231a232 a126a127a128 a50a234a173a212 a129 a125 a116 1/(1+z4)a133a127a128a134a139 a141 a238 a135a136 a215 z = eipi/4 a233z = ei3pi/4 a50 a143a144a234 a230 a169a235a236a237a238 a237a239 a50a0 a233a240a241 a61a198a218 a235a143a144 a117a129a130 integraldisplay ∞ 0 dx 1 + x100, a203a157a242a243 a141pi/50 a126a244 a232 a127a128 a61 a127a128a134 a145 a139 a237a238 a135a136 a111a209 a203a157 a158a231a232 a127a128 a61 a127a128a134a248a139 50a238a135a136a50a141a245a133a143a144a234a219a126a246a247a248a249a0a250a50 a198a218a220a61 a133 a219a160a234 a239 a219a229 a229 a61 a244 a232 a127a128 a233 a158a231a232 a127a128 a141a245a251a252a0a253a254a255 a126a0 a61 a220a221 a61 a133 a1 a160a234 a238 a219a229 a229 a61 a244 a232 a127a128a2 a145a3 a244a4a237 a126 a254a255a50 a186 10.7 a163a164a33a40 integraldisplay ∞ 0 dx 1 + x3 a50 a187 a90a91 a61 a189 a111 a58a5a6a7a8 a52a38 2pi/3a35a9a53a94a204 ( a9310.5) a50 a86 10.5 contintegraldisplay C dz 1 + z3 = integraldisplay R 0 dx 1 + x3 + integraldisplay CR dz 1 + z3 + integraldisplay 0 R ei2pi/3dx 1 + x3 = parenleftBig 1?ei2pi/3 parenrightBigintegraldisplay R 0 dx 1 + x3 + integraldisplay CR dz 1 + z3 = 2pii res 11 + z3 vextendsinglevextendsingle vextendsinglevextendsingle z=eipi/3 = 2pi3 e?ipi/6. a178 a154a54R →∞a61 a18 a38 limz→∞z · 11 + z3 = 0, a176 a159 lim R→∞ integraldisplay CR dz 1 + z3 = 0. a118 a123a114a179a131 integraldisplay ∞ 0 dx 1 + x3 = 2pi 3 e?ipi/6 1?ei2pi/3 = pi 3cos pi6 = 2pi3√3.