a0 a1 star a2a3a4a5a6a7a8a9a10 6 star a1122–27 a12a13a14a8a15a16a17a18a19a20a6a7 a0a1a2 a3a4a5a6a7a8a9a10 a111 a12 a13a14a15 a16a17a18a19a20a21a22a23 §7.1 a24a25a26a27 a28a29a30a31 a32a33a34G a35a36a37C a38a39a40a41a42a43a35a44a45a46a47a48a49a50 a51a52a53a54a55a56a57a58a59 b k, k = 1,2,3,···,na60a61a62a63f(z)a64G a65a45a66a67a68a61a64 G a69a70a71a61a72a64C a73a74 a53f(z) a35 a58a59 a61a75 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 a867.1 a87a88a89a90 a91 a92a937.1 a61a94a95a96 a55a58a59b k a97a46a47a48a49γk a61a98γk a99a64G a65a61a72a100a101a102a103a61a75a104a105a106a70 a107a33a34 Cauchy a108a109a110a62a63 a97 Laurenta83a84a111a35a85a63a112a113a61a114 a53 contintegraldisplay 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)a64 a58a59b a77 a35a78a63a61 a150 a75a73a151a61a114a152a149f(z)a64z = ba35a82 a34 a65Laurenta83a84a69(z?b)?1 a153 a35a85a63a50 star a64a154 a59 a35a155a156a157a61a158a159 a107a160a161a162a163a164 a149a78a63a50 star a165a166a167a168a169a170a171 a32ba59 a152f(z)a35a39a172a154 a59 a61a75a64ba59a35a82 a34 a65a61 f(z) = a?1(z ?b)?1 + a0 + a1(z ?b) + a2(z ?b)2 +···. §7.1 a3a4a5a6 a112a12 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 a64ba59a110a184a82a34a65a67a68a61b a152Q(z)a35a39a172a185 a59 a61Q(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 7.1 a149 1 z2 + 1 a64 a58a59 a77 a35a78a63a50 a187 z = ±i a152a79a35a39a172a154 a59 a50 resf(±i) = 12z vextendsinglevextendsingle vextendsingle z=±i = ?i2. a186 7.2 a149 eiaz ?eibz z2 a64 a58a59 a77 a35a78a63a50 a187 z = 0 a152a79a35a39a172a154 a59 a50 resf(0) = limz→0z · e iaz ?eibz z2 . = limz→0 eiaz ?eibz z = i(a?b). star a188a166a167a168a169a170a171 a32z = b a152f(z)a35ma172a154 a59 a61m ≥ 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 7.3 a1491/(z2 + 1)3 a64 a58a59 a77 a35a78a63a50 a187 z = ±i a152a79a35a191a172a154 a59 a50 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 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(a199a114a152a200a111a201a202 a198) a39a203a35a94a204a61a64a94a204a65 a52 ∞a59 a158a205a152f(z)a35 a58a59 a60a178a206 a58a59 a50 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 = 0a59a82a34a65a208a209a63a83a84a69t?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. a191 a55a16 a108a179a63a61A, Ba183C a61 a197a17 a114a152a62a63f(z)a64a39a172a154 a59z = 1, z = 2 a183z = 3a59 a77 a35a78a63a50 a18a19 A = res 1(z ?1)(z ?2)(z ?3) vextendsinglevextendsingle vextendsinglevextendsingle z=1 = 12, §7.1 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. a92a211 a62a63f(z)a20 a53a21 a172a154 a59 a61a199a158a159a22a23a24 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 a78a63a108a109a32a94a204a33a40a35 a163a164a34a35 a38a78a63a35 a163a164 a61 a36a37 a205a32a108a33a40a183a39a108a67a68a62a63a35a94a204a33a40 a38 a85a39a40a61a114 a53 a158a205a41a42a44a43a24 a163a164a28a189a44 a108a33a40a50 a0a1a2 a3a4a5a6a7a8a9a10 a115 a12 §7.2 a45a27a46a47a48a25a49a50a51 a53 a109a191a52a62a63a35a33a40a35a53a113a152 I = integraldisplay 2pi 0 R(sinθ,cosθ)dθ, a184a69Ra152sinθ, cosθa35 a53 a109a62a63a61a64a33a40 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 . a53 a109a191a52a62a63R(sinθ,cosθ)a64a33a40 a33a54[0,2pi] a73a70a71a61a114a65a66a67 a53 a109a62a63R parenleftbiggz2 ?1 2iz , z2 + 1 2z parenrightbigg a64a45a63a64a35a64a203a73a206 a58a59 a50 a186 7.4 a163a164 a33a40 I = integraldisplay 2pi 0 1 1 + εcosθdθ, |ε| < 1a50 a187 a68a69 a73a62a35a202a70a71a72a61a73a74 a53 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. a189a196 a64 a163a164 a78a63a111a61 a37a75a76 a62a63 2/(εz2 + 2z + ε)a53a174 a55 a154 a59 a61 z = ?1± √1?ε2 ε , a77a78 a81a79a74a35a173a33a38 1a61 a176 a159a101a79a80a81a61a39a108 a36a53 a39 a55 a154 a59 a61 z = (?1 +√1?ε2)/εa61 a77 a81a45a63a64 a65a50 §7.3 a82a83a84a85 a116 a12 §7.3 a86a87a50a51 a206a88a33a40a35a108a195a38 integraldisplay ∞ ?∞ f(x)dx = limR 1 → +∞R 2 → +∞ integraldisplay R2 ?R1 f(x)dx. a53 a111 a189a89 a154 a54 a101a90a64a61 a77 limR→+∞ integraldisplay R ?R f(x)dxa90a64a61a76a38a33a40a91a66a61a92a38 v.p. integraldisplay ∞ ?∞ f(x)dx = lim R→+∞ integraldisplay R ?R f(x)dx. a93a94 a61a95 a189 a174 a89 a154 a54a96 a90a64a111a61a79a74a97a108 a57 a80a50 a64a106a61a62a73 a27 a61a33a40 integraldisplay ∞ ?∞ f(x)dx a152a98a99a100a101a102a103a35a61a207a101a104a105a106 a55 a62a63a35a94a204a33a40a50 ? a73a74a158a159 a25a26 a24a14a100a62a63 f(x)a106a107a38a106a62a63f(z) ? a38a67a205a104a105a94a204a33a40a207 a58a108 a78a63a108a109 a163a164 a61a109a97a110 a215 (1) a111a73a112a95a35a33a40a59a60a113a53a105a46a47a94a204a61 a163a164 contintegraldisplay f(z)dz a114 (2) a64a111a73a35a59a60a73a35a33a40a61 a115a116a117a176a37 a149 a163a164 a35a206a88a33a40a118a119 a57a120 a61 a115a116 a158a159a44a45a202a43a24 a163a164 a28 a40a50 a121a122a94 a35a123a70a95 a94 a152a111a73a159 a150a59 a38a64a124a61 Ra38a125a60a35a73a125a64CR a61contintegraldisplay C f(z)dz = integraldisplay R ?R f(z)dz + integraldisplay CR f(z)dz. a86 7.2 a113a126a127R → ∞a50 a189a128 a61a73a74a43a129 a37a163a164 integraldisplay CR f(z)dz a35a154 a54 a66a50 a36a37f(z) a130a131a112a95a35a132a133a61 a189 a152a158a159a123a134a35a50 a101a135a136 a32 a62a63f(z)a130a131a157a137a132a133 a215 a0a1a2 a3a4a5a6a7a8a9a10 a117 a12 1. f(z)a64a73a125a61a62 a52 a67 a53a54a55a56a57a58a59 a60a152 a77a77 a67a68a35a61 a64a100a101a73a74 a53a58a59 a114 2. a64 0 ≤ argz ≤ pia138a94a65a61a95|z| → ∞a111a61zf(z)a39a139a24 a140 a81 0 a61a141 a194 a81a142a143a35 ε > 0 a61a90a64 M(ε) > 0 a61a98a95 |z|≥ M a610 ≤ argz ≤ pia111a61|zf(z)| < εa50 a234a144 a238a145a146a147 a254a148a149 a50a1501 a238a145a146a151a152 a142 a153a154a126a155a156a129a130a254 a244a157 a129a130 a61 a147a158 a0 a233a159a160 a115a116a117a118a143a144a127a128a129a130 contintegraldisplay C f(z)dz = integraldisplay R ?R f(z)dz + integraldisplay CR f(z)dz = 2pii summationdisplay a219a161a162a163 resf(z). a1502a238a145a146a61a164a165a244a166a141a155a156a167a168a129a130a126a169a170a145a146 limx→±∞xf(x) = 0 a126a171a172a173a174 a61a175a176a61a177a178a179 a118 3.1 a61a180 a151a152 a142 lim R→∞ integraldisplay CR f(z)dz = 0. a181 a154 a54R →∞ a61a114a182a134 integraldisplay ∞ ?∞ f(x)dx = 2pii summationdisplay a73a125a61a62 resf(z). a186 7.5 a163a164 a108a33a40I = integraldisplay ∞ ?∞ dx (1 + x2)3 a50 a187 a19 a111 a93a94a183 a47a73a184 a37 a149a35a132a133a61a190 I = integraldisplay ∞ ?∞ dx (1 + x2)3 = 2pii·res 1 (1 + z2)3 vextendsinglevextendsingle vextendsinglevextendsingle z=i =2pii· parenleftbigg ?3i16 parenrightbigg = 38pi. a121 a126a61a38a67 a194a58a108 a78a63a108a109 a163a164 a108a33a40a35a185a186a187a188 a53 a39 a55 a41a42a189a190a35a109a67a61a101a135a191a192a106a39a157 a193 a62a35a194a184 a215 a38a67a205a195 a58a108 a78a63a108a109 a163a164 a206a88a33a40a61a73a74a97a110 a215 1. a111a73a112a95a35a33a40a59a60a113a53a105a46a47a94a204a61 a163a164 contintegraldisplay f(z)dza114 2. a64a111a73a35a59a60a73a35a33a40a61 a115a116a117a176a37 a149 a163a164 a35a206a88a33a40a118a119 a57a120 a61 a115a116 a158a159a44a45a202a43a24 a163a164a28 a40a50 §7.3 a82a83a84a85 a118 a12 a185a81 a189a128 a35a109a67a61a114a158a159a196a197a198a199a24a200 a108 a78a63a108a109 a163a164 a108a33a40a50 star a201a218 a244f(x) a202a125 a116 a61 a248a203 a222 a129a130 integraldisplay ∞ 0 f(x)dxa61a243a222 integraldisplay ∞ 0 f(x)dx = 12 integraldisplay ∞ ?∞ f(x)dx, a204a233a205a172 a0 a233a206a160a2077.2a126a127a128 a61 a147a208a209 a219a163 a126a210a211 a61a212a213a214integraldisplay ∞ 0 f(x)dx = 12 integraldisplay ∞ ?∞ f(x)dx = pii summationdisplay a219a161a162a163 resf(z). star a201a218 a133a129a130 integraldisplay ∞ 0 f(x)dx a229a61a215 a129 a125 a116f(z) a216 a139a217a218a203a219a220a221 a61a222a201 f(z) = f(zeiθ), a223a224 a61a255a0 a233a206a160a2077.3 a229a126a127a128a154a143a144 a50 a86 7.3 a186 7.6 a163a164 a108a33a40 integraldisplay ∞ 0 dx 1 + x4 a50 a86 7.4 a187 a78 a81 a189a196 a35a225a33a62a63 f(x) = 11 + x4 a152x4 a35a62a63a61 a176 a159a61a73a74a158a159a226 a108a93 7.4 a35a94a204 a215 a98 a197 a100a101 a780 a134Ra61a98a64a227a134a228 a197a229 a101a61a191a98 a197a229 a101 a78iR a230a134 a150a59 a50 a189a128 a61a104a105a78a63a108a109a61 a53 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 a0a1a2 a3a4a5a6a7a8a9a10 a119 a12 =2pii res 11 + z4 vextendsinglevextendsingle vextendsinglevextendsingle z=eipi/4 = pi2 1?i√2 . a181 a154 a54R →∞ a61 a18 a38 limz→∞z · 11 + z4 = 0, a176 a159a61a104a105a231a109 3.1a61 a53 lim R→∞ integraldisplay CR dz 1 + z4 = 0. a81a152a114a182a134 integraldisplay ∞ 0 dx 1 + x4 = √2 4 pi. a133 a234 a238 a222a232 a229 a61a233 a172a205a172 a0 a233a206a160 a161a234a235 a126a127a128 a50a234a176a215 a129 a125 a116 1/(1+z4)a133a127a128a134a139 a144 a238 a135a136 a215 z = eipi/4 a236z = ei3pi/4 a50 a143a144a237 a233 a172a238a239a240a241 a237a242 a50a0 a233a243a244 a61a201a218 a238a143a144 a117a129a130 integraldisplay ∞ 0 dx 1 + x100, a206a160a245a246 a141pi/50a126a247a235 a127a128 a61 a127a128a134 a145 a139 a237a238 a135a136 a114a212 a206a160 a161a234a235 a127a128 a61 a127a128a134a248a139 50a238a135a136a50a144a248 a133a143a144a237 a219 a126a249a250a251a252 a0a253a50 a201a218a220a61 a133 a219a163a234 a242 a222a232 a229 a61 a247 a235 a127a128 a236 a161a234a235 a127a128 a144a248a254a255a0a0a1a2 a126a3 a61 a223a224 a61 a133 a4 a163a234 a238 a222a232 a229 a61 a247 a235 a127a128a5 a145a6 a244a7a237 a126 a1a2a50 a186 7.7 a163a164 a33a40 integraldisplay ∞ 0 dx 1 + x3 a50 a187 a93a94 a61 a189 a111 a58a8a9a10a11 a52a38 2pi/3a35a12a53a94a204 (a937.5)a50 a86 7.5 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. a181 a154 a54R →∞ a61 a18 a38 limz→∞z · 11 + z3 = 0, a176 a159 lim R→∞ integraldisplay CR dz 1 + z3 = 0. §7.3 a82a83a84a85 a1310a14 a121 a126a114a182a134 integraldisplay ∞ 0 dx 1 + x3 = 2pi 3 e?ipi/6 1?ei2pi/3 = pi 3cos pi6 = 2pi3√3. a15a16a17 a18a19a20a21a22a23a24a25 a1311a14 §7.4 a26a27a28a29a30a31a32a33a34a35 a36a37a38a39a40a41a42a43a44a45a46a47a48a49 I = integraldisplay ∞ ?∞ f(x)cospxdx a50 I = integraldisplay ∞ ?∞ f(x)sinpxdx. a36a51a52a53a54a55p > 0 a56 a57a58a59a60a61a62a63a64a65a66a67a68a69a70a71a72a73a74a63a75a76 a56 a77a78a79a64a80a81a82a83a84a85a86a87a88f(z)cos pz a89f(z)sinpz a56 a59a78a90a88z = ∞a78a80a81 sinza89cosz a63a91a92a93a94(a59a95a96a97a98z a69a82a99 a100a101a102a103∞ a104 a66sinza89cosza68a69a105a106a103a82a99a63a81a107) a66a82a108a103a109a110a111a112 lim R→∞ integraldisplay CR f(z)cospzdz a89 lim R→∞ integraldisplay CR f(z)sinpzdz. a113a114a41a115a116a117a118a119a39a120a121a122a123 f(z)eipz a56a124a125 a120a121f(z)eipz a126a127a128a129a130a131a132a133a133a134a135a136a137 a66a138 contintegraldisplay C f(z)eipzdz = integraldisplay R ?R f(x)eipxdx + integraldisplay CR f(z)eipzdz = integraldisplay R ?R f(x)[cospx + isinpx]dx + integraldisplay CR f(z)eipzdz = 2pii summationdisplay a139a140a141a142 res braceleftbigf(z)eipzbracerightbig. a132a143a144a145a146a147a148 lim R→∞ integraldisplay CR f(z)eipzdz, a40a149a150a151a152a153a154a155a153a66a156a46a47a157a158 integraldisplay ∞ ?∞ f(x)cospxdx a154 integraldisplay ∞ ?∞ f(x)sinpxdx. a88a159a66a160a161a162a163a164a58 a56 a165a166 7.1(Jordan a165a166) a55 a1260 ≤ argz ≤ pi a41a167a168 a131 a66a169|z| → ∞ a170 a66Q(z) a171a172a173a174a175a176 0a66a138 lim R→∞ integraldisplay CR Q(z)eipzdz = 0, a177a178p > 0a66C R a117a47a179 a137 a123a180a181a66Ra123 a128a182 a41 a127a128 a180a183 a56 a184 a169z a126CR a127 a170 a66z = Reiθ a66 vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay CR Q(z)eipzdz vextendsinglevextendsingle vextendsinglevextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay pi 0 QparenleftbigReiθparenrightbigeipR(cosθ+isinθ)Reiθidθ vextendsinglevextendsingle vextendsinglevextendsingle a183a185a186a187a188a189a190a191a192a193a194a195a196a197p > 0 a198a199a200 a190a191a201a202a203Rep > 0 a204 §7.4 a205a206a207a208 a19a209a210a211a212a213 a21412 a215 ≤ integraldisplay pi 0 vextendsinglevextendsingleQparenleftbigReiθparenrightbigvextendsinglevextendsinglee?pRsinθRdθ <εR integraldisplay pi 0 e?pRsinθdθ =2εR integraldisplay pi/2 0 e?pRsinθdθ. a2167.6 a217a218a41a219a220 a126 a176a221 a114a222 a146 sinθ a223a56a224a2257.6a46a226a66a1690 ≤ θ ≤ pi/2a170 a66 a133sinθ ≥ 2θ/pi a66a227a47 vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay CR Q(z)eipzdz vextendsinglevextendsingle vextendsinglevextendsingle < 2εR integraldisplay pi 0 e?pR·2θ/pidθ = 2εR pi2pR parenleftbig1?e?pRparenrightbig = εpip parenleftbig1?e?pRparenrightbig. a36a228a66a156a217a218a229 lim R→∞ integraldisplay CR Q(z)eipzdz = 0. square a176 a117a66 a126a230a231Jordan a232a233 a41a234a235a236a66 integraldisplay ∞ ?∞ f(x)eipxdx = 2pii summationdisplay a139a140a141a142 resbraceleftbigf(z)eipzbracerightbig. a40a149a122a152a153a154a155a153a66a237a158 integraldisplay ∞ ?∞ f(x)cospxdx =Re ? ? ?2pii summationdisplay a139a140a141a142 resbracketleftbigf(z)eipzbracketrightbig ? ? ? = ?2piIm ?? ? summationdisplay a139a140a141a142 resbracketleftbigf(z)eipzbracketrightbig ?? ?, integraldisplay ∞ ?∞ f(x)sinpxdx =Im ?? ?2pii summationdisplay a139a140a141a142 resbracketleftbigf(z)eipzbracketrightbig ?? ? a238a239a240 a18a19a20a21a22a23a24a25 a21413 a215 =2piRe ?? ? summationdisplay a139a140a141a142 resbracketleftbigf(z)eipzbracketrightbig ?? ?. a2417.8 a146a147 a39a40 integraldisplay ∞ 0 xsinx x2 + a2dx, a > 0a56 a242 a243a244 a127a130 a41a245a246a66 a133 integraldisplay ∞ ?∞ xeix x2 + a2dx = 2pii· 1 2e i·ia = piie?a. a227a47 integraldisplay ∞ ?∞ xsinx x2 + a2dx = pie ?a, integraldisplay ∞ 0 xsinx x2 + a2dx = pi 2e ?a. a247a248a249 a170 a66a250a158a251 integraldisplay ∞ ?∞ xcosx x2 + a2dx = 0. a36a117a252a253a41a66a254a123a119a39a120a121a117 a136 a120a121 a56 §7.5 a255a0a1a2a3a4 a209a5a6 a21414 a215 §7.5 a7a8a9a10a11a12a31a13a14 a15a39a40 ( a55a15 a137 a123 c) a41a38a16a117 integraldisplay b a f(x)dx = lim δ1→0 integraldisplay c?δ1 a f(x)dx + lim δ2→0 integraldisplay b c+δ2 f(x)dx. a124a125 a36a17 a135a18a134a19a20a21 a52a22 a126 a66a23a117 lim δ→0 bracketleftbigg integraldisplay c?δ a f(x)dx + integraldisplay b c+δ f(x)dx bracketrightbigg a22 a126 a66a138a24a123a15a39a40a41a25 a223 a22 a126 a66a26a123 v.p. integraldisplay b a f(x)dx = lim δ→0 bracketleftbigg integraldisplay c?δ a f(x)dx + integraldisplay b c+δ f(x)dx bracketrightbigg . a169a253a66 a124a125 a15a39a40a27a177a25 a223 a21 a22 a126 a66a28a29a30a31 a171 a38a32a33 a56 a90a159a66a34a35a36a37a64a65a78a162a163a38a64a65a66a39a57a58a40a41a63a42a37a64a65 contintegraldisplay C f(z)dza104 a66a36a43a44a63a38 a94a45a78a79a64a80a81a63a93a94a66a46a47a48a49a93a94a50a51a52a53a54a63a64a65a75a76 a56 a55a56a57a58a59a60a61a163a62a63 a64a65a66a67a68a57a58a59a61a64a65a63a69a91a70a71 a56 a2417.9 a146a147 a39a40 integraldisplay ∞ ?∞ dx x(1 + x+ x2) a56 a242 a36a117 a171 a135a72a73 a39a40a66 a72a73a74a75a76a77a126 a39a40a78a79a123a80a81a78a79a66a82 a76a77 a123a119a39a120a121 a126x = 0a137 a52a83a84 (x = 0 a137 a123a15 a137) a56 a248a39a40 a126 a25 a223a85 a16a236a22 a126 a66 v.p. integraldisplay ∞ ?∞ dx x(1 + x + x2) = lim R1→∞ integraldisplay ?1 ?R1 dx x(1 + x + x2) + limR2→∞ integraldisplay R2 1 dx x(1 + x + x2) + lim δ→0 bracketleftBiggintegraldisplay ?δ ?1 dx x(1 + x + x2) + integraldisplay 1 δ dx x(1 + x + x2) bracketrightBigg . a254a248a66 a126a86a87a88 a121a38 a233 a146a147 a248a39a40 a170 a66 a86a89a90a91a92a93 a39a40 contintegraldisplay C dz z(1 + z + z2), a177a178a41a39a40a168a94C a124a2257.7a227a95a66a224 a47a179 a137 a123a180a181a96δ a123 a128a182 a41a97 a128 a180a98 C δ a154a47a179 a137 a123a180a181a96 Ra123a128a182a41a99a128a180a98 CR a47a27a100a101a102?R →?δa154δ → Ra103 a49 a56a176 a117a66a243a244 a88 a121a38 a233 a66 a133 a238a239a240 a104a105a20a21a22a23a24a25 a21415 a215 a2167.7 contintegraldisplay C dz z(1 + z + z2) = integraldisplay ?δ ?R dx x(1 + x + x2) + integraldisplay Cδ dz z(1 + z + z2) + integraldisplay R δ dx x(1 + x + x2) + integraldisplay CR dz z(1 + z + z2) =2pii·res 1z(1 + z + z2) vextendsinglevextendsingle vextendsinglevextendsingle z=ei2pi/3 = ? pi√3 ?ipi. a254a123 limz→∞z · 1z(1 + z + z2) = 0, a227a47a66a243a244 a232a2333.1 a66 a133 lim R→∞ integraldisplay CR dz z(1 + z + z2) = 0. a82a254a123 a126z = 0a137 a41a106a107 a131 a66 1 z(1 + z + z2) = 1 z + a0 + a1z +···, a227a47 integraldisplay Cδ dz z(1 + z + z2) = integraldisplay Cδ dz z + integraldisplay Cδ [a0 + a1z +···]dz. a126a18a134a108 a44δ → 0a236a66 a133 lim δ→0 integraldisplay Cδ dz z(1 + z + z2) = ?pii. a36a228a66a122 a18a134R →∞ a66δ → 0a66a156a158a251 v.p. integraldisplay ∞ ?∞ dx x(1 + x + x2) = ? pi√ 3. a109a110a111 a124a125 a39a40a168a94a178a41a97 a128 a180a98a117a112a236 a128a129a130a113a114 z = 0a137 a66a254a115a116 z = 0a137a117 a168 a126 a168a94 a131 a66a117a118a119a158a251a52a249a41a120 a125a121 a123a122a29 a121 a123a44a56 a63a111a112a124a68a69a125a126a66a127a103a64a65a128a129a44a130a93a94a63a131a74a66a132a133a111a112a75a48a93a94a63a134a73 a135a44a63a64a65a107( a136a137 a86a67a66a133a111a112a138a63a139a140a107) a56 §7.5 a255a0a1a2a3a4 a209a5a6 a21416 a215 a39a130a141a131a142 (a62a34a66a143a144a145a80a81a63a146a147a64a65) a55a66a91a64a36a37a64a65a148a82a78a38a64a65a66a77a149a103 a39a40a41a63a42a37a64a65a124a66a148a82a78a84a85a86a150a79a64a80a81 f(x) a151 a52f(z)a66a90a50a39a42a37a80a81a63a75 a76a64a65a124a66a64a65a128a129a44a152a68a69a126a153a93a94 a56 a59a123a55a56a63a62a63a124a68a69a125a126 a56 a2417.10 a146a147 a39a40 integraldisplay ∞ ?∞ sinx x dxa56 a242 a154a155a253 a173 a66 a86 a169 a90a91 a39a40 contintegraldisplay C eiz z dz a66a39a40a168a94C a154a1569a32a249( a2257.7)a56 contintegraldisplay C eiz z dz = integraldisplay ?δ ?R eix x dx + integraldisplay Cδ eiz z dz + integraldisplay R δ eix x dx + integraldisplay CR eiz z dz. a126 a39a40a168a94 a117 a168a41a78a107 a131 a66a119a39a120a121a157a158a66a159a168a94a39a40a1230 a56 a243a244Jordan a232a233 a154 a232a2333.2a66a40a149a133 lim R→∞ integraldisplay CR eiz z dz = 0, limδ→0 integraldisplay Cδ eiz z dz = ?pii. a254a248 integraldisplay ∞ ?∞ eix x dx = pii. a150a151a17a160a41a152a153a154a155a153a66a237a158 v.p. integraldisplay ∞ ?∞ cosx x dx = 0, integraldisplay ∞ ?∞ sinx x dx = pi. a219 a176 a36a161a37a162a41a39a40a66a250a46a47a163 a148 integraldisplay ∞ ?∞ sin2x x2 dx =pi;integraldisplay ∞ ?∞ sin3x x3 dx = 3 4pi; integraldisplay ∞ ?∞ sin4x x4 dx = 2 3pi;integraldisplay ∞ ?∞ sin5x x5 dx = 115 192pi;integraldisplay ∞ ?∞ sin6x x6 dx = 11 20pi; a50a164 a66a165a166a167a41a120 a125a168 integraldisplay ∞ ?∞ sinnx xn dx = pi (n?1)! [n/2]summationdisplay k=0 (?)k parenleftbiggn k parenrightbiggparenleftbiggn?2k 2 parenrightbiggn?1 . a146a147 a36a169a39a40a66a219a220 a126 a176 a113a114 a173a170a171 a92a93 a39a40a41a119a39a120a121 a56 a156 a124 a66a123a229 a146a147 a39a40 integraldisplay ∞ ?∞ sin2x x2 dx, a156 a86a89a90a91a92a93 a39a40 contintegraldisplay C 1?ei2z z2 dz, a238a239a240 a104a105a20a21a22a23a24a25 a21417 a215 a39a40a168a94C a172a124a2257.7a56 a36a51a66a156 a92a93 a39a40a115a173a66 a126 a152a174 a127 a46a47 a133a136a137 a56 a23a36a161 a136a137 a66 a171a175a176a177 a66 a132a144 a117a46a178 a136a137 a50a171a179 a18a137 a56 a36a112 a232a2333.2 a156a46a47a180 a148 a56a124a125 a117a181 a179a50 a181 a179 a47 a127 a41 a18a137 a66 a50 a117a182 a74 a136a137 a66a183a97a180a98 C δ a41a39a40a156a46 a144 a174a176∞a56 §7.6 a184a185a208 a105a209a212a213 a21418 a215 §7.6 a186a187a29a30a31a34a35 a43a114 a173a176 a66a36a51a227 a176 a41a188 a223 a120a121a41a39a40a117a112 a92a93 a120a121a41a189a190 a176 a41 a56 a112 a92 a121a107 a177 a180a66a152 a93 a38a39a40a178 a41a39a40 a93a191xa126x > 0 a170 a86a89 a233 a157a123 argx = 0a56 a171 a161 a73 a226a41a188 a223 a120a121a39a40a117 I = integraldisplay ∞ 0 xs?1Q(x)dx, a177a178sa123a152a121a66Q(x) a19 a223 a66 a126 a113a152a174 a127a192a133a136a137 a56 a123a229a193a217a39a40a194a195a66 a143 a157 limx→∞x·xs?1Q(x)dx = limx→∞xsQ(x) = 0. a196a31 a90a91 a32 a86 a41 a92a93 a39a40 contintegraldisplay C zs?1Q(z)dz = integraldisplay R δ xs?1Q(x)dx + integraldisplay CR zs?1Q(z)dz + integraldisplay δ R parenleftbigxei2piparenrightbigs?1 Q(x)dx +integraldisplay Cδ zs?1Q(z)dz. a149a103z = 0 a197z = ∞a78a79a64a80a81a63a198a94a66a199a69a200a133a150a201a56a202a203a36a43a204a49a66a148a205a206a202a204a207 a44a208argz = 0 a56 a59 a104 a63a64a65a128a129a149a204a49a63a209a134a73a135 (a72a129a65a210a88R a211δ)a197 a204a207a44a55a208 a212a52( a213a2147.8)a56 a202a204a207a44a55a208a63a64a65a215a216a109a110a217a199a133a111a112a63a36a37a64a65a130a218 a56a219a220 a78a34 a221a111a112a202a209a134a73a135a63a64a65a107 a56 a2167.8 a224a232a2333.1a154a232a2333.2a46a47a180a148a66a124a125 a1260 ≤ argz ≤ 2pi a41a167a168 a131 a66 limz→0zsQ(z) = 0, limz→∞zsQ(z) = 0, a238a239a240 a104a105a222a21a22a23a24a25 a21419 a215 a138 lim δ→0 integraldisplay Cδ zs?1Q(z)dz = 0, lim R→∞ integraldisplay CR zs?1Q(z)dz = 0. a165a223 a171a224 a66 a124a125Q(z)a126a225a129a130a127a226a229a133a134a135a227a228a136a137(a52a126a113a152a174a127)a229 a66a117 a19 a223 a157a158a41a66a254a115a46 a47 a86a87a88 a121a38 a233a56 a126 a122 a18a134δ → 0, R → ∞ a230 a66a156a158a251 parenleftbig1?ei2pisparenrightbigintegraldisplay ∞ 0 xs?1Q(x)dx = 2pii summationdisplay a231a141a142 res braceleftbigzs?1Q(z)bracerightbig. a227a47 integraldisplay ∞ 0 xs?1Q(x)dx = 2pii1?ei2pis summationdisplay a231a141a142 res braceleftbigzs?1Q(z)bracerightbig. a232 a143a233 a85 a66 a126a146a147a88 a121 a170 a66 a143a234a235a127a130a236 a176 a188 a223 a120a121 zsa227a237a41 a134a238 a66a237 0 ≤ argz ≤ 2pi a56 a109a110a111 a124a125a239 a38 a126a240 a101 a127a241argz = 2pi a66a117a118a242a243a244 a230 a120 a125a121 a109a110a111 a124a125Q(x)a245 a133 a171 a38a41 a236 a24 a74a246 a66a156 a124 a117 x a41 a136 a120a121 a50a247 a120a121a66a117a118a46a47a122a177a248a44a45 a41a168a94 a121 a2417.11 a146a147 a39a40 integraldisplay ∞ 0 xα?1 x + ei?dx a660 < α < 1, ?pi < ? < pi a56 a242 a36a51a41a119a39a120a121a252a253 a230a231a127a249 a245a246a178a41 a143 a157a66a254a248 integraldisplay ∞ 0 xα?1 x+ ei?dx = 2pii 1?ei2piαe i(?+pi)(α?1) = pisinpiα ·ei?(α?1) (star) a112a36 a135 a39a40a250a46a47a250 a148 a171 a169a165a223 a171a224 a41a120 a125a56 a156 a124 a66a237a123a36 a135 a39a40a41a251a252 a108 a44a66 ? = 0a66a138 integraldisplay ∞ 0 xα?1 1 + xdx = pi sinpiα. a126Γ a120a121 a171a253 a178 a143 a100a254 a86a87 a251a36 a135 a120 a125a56 a82 a124 a66a150a151 (star)a45a17a160a41a155a153a66a250a46a47a158a251 integraldisplay ∞ 0 xα?1 x2 + 2xcos?+ 1dx = pi sinpiα sin(1?α)? sin? . a59a163a255a35a78a39 0 < α < 1a63a0a1a55a2a3a63a66a77a78a68a69a4a5a6a7a30 < α < 2 a56a8a9 (star) a101a61a10a63a36a11a66a45a68a69a2a3a99a12a63a255a35 a56 a13 a171 a161a188 a223 a120a121a41a39a40a14a27 a236 a121a120a121 a56a15 a245a246a236 a130 a41a156a16 a56 a2417.12 a146a147 a39a40 integraldisplay ∞ 0 lnx 1 + x + x2dxa56 a242 a122a168a94 a124a2257.8a66a146a147a92a93a39a40contintegraldisplay C lnz 1 + z + z2dz = integraldisplay R δ lnx 1 + x + x2 dx + integraldisplay CR lnz 1 + z + z2dz §7.6 a184a185a208 a105a209a212a213 a21420 a215 + integraldisplay δ R lnparenleftbigxei2piparenrightbig 1 + x + x2dx + integraldisplay Cδ lnz 1 + z + z2dz =2pii summationdisplay a231a141a142 res braceleftbigg lnz 1 + z + z2 bracerightbigg =2pii parenleftbigg 2pi 3√3 ? 4pi 3√3 parenrightbigg = ?4pi 2i 3√3. a254a123 limz→∞z· lnz1 + z + z2 = 0, limz→0z · lnz1 + z + z2 = 0, a243a244 a232a2333.1a154a232a2333.2a66a133 lim R→∞ integraldisplay CR lnz 1 + z + z2dz = 0, limδ→0 integraldisplay Cδ lnz 1 + z + z2dz = 0. a227a47a66a122 a18a134R →∞, δ → 0 a66a237a158 integraldisplay ∞ 0 lnx 1 + x+ x2dx? integraldisplay ∞ 0 lnx + 2pii 1 + x + x2 dx = ? 4pi2i 3√3. a17a18a153a39a202a204a207a44a55a208a63a64a65a19a217a199a133a111a112a63a64a65a130a218a66a77a78a66a20a21a82a22a66a138a58a152a40a23 a24a25a26a27a66a50a28a29a55a162a163a148a20a57a58a199a133a111a112a63a206a64a65 integraldisplay ∞ 0 1 1 + x + x2 dx = 2pi 3√3. a59a60a75a76a64a65 contintegraldisplay C lnz 1 + z + z2dz a111a112 integraldisplay ∞ 0 lnx 1 + x + x2 dx a63a30a31a32a33a34a27a35 star a36a37 a41a179a254a117a66a154a243a45a120a121a52a249a66 a236 a121a120a121 lnza41a188 a223 a74a76a77a126 a155a153 a127 a66a254a248a183 a240 a101 a127 a236 a241 a39a40 a170 a66a177a152a153 (a237lnx) a38 a32a39a40 a56 star a41a36a37a42a43a44a45 a247a194a46 a168 ? a47a48a49 a236a50a51a52a53 integraldisplay ∞ 0 f(x)dxa49a54a55f(x)a56a57a58a59a60 (a61a62a63a64a65a66a67 7.3a68a69a43a70 a66a71a72)a49a63a73a74a75a76a67a77a60 a51a78 a71a72a79a80 a52a53 contintegraldisplay C f(z) lnzdz a81a82a83a84 ? a47a85a49a54a55a86a71a72 a52a53 integraldisplay ∞ 0 f(x)lnxdxa49a87a63a73a88a89a90a91 a52a53 contintegraldisplay C f(z)ln2zdz a84 a61a92a93a94a95a96a97a98a99 ln2z a43a59a60a100 ln2x a101 (lnx+ 2pii)2 a102a103a104a105a49a106a98a43a107 a108a109a110a111a112a113 a86a43 lnxa114a84 a115a116a117 a81a118a119a120 12 a69 a112 a86a82a43 a52a53 a71a72a84a92a62a49a88a89 a52a53 contintegraldisplay C ln2z 1 + z + z2dz a49a79a80C a56a91a84 a121a122a123 a124a125a126a127a128a129a130a131 a13221 a133 a134 a90a97a135a43a71a72a136a137a49 a117 a83a138 integraldisplay ∞ 0 ln2x 1 + x + x2 dx ? integraldisplay ∞ 0 (lnx + 2pii)2 1 + x+ x2 dx = 2pii summationdisplay a139a140a141 res braceleftbigg ln2z 1 + z + z2 bracerightbigg = 2pi√3 bracketleftbigg16 9 pi 2 ? 4 9pi 2 bracketrightbigg = 83√3pi3. a50 a57 ?4pii integraldisplay ∞ 0 lnx 1 + x + x2dx + 4pi 2 integraldisplay ∞ 0 1 1 + x + x2dx = 8 3√3pi 3. a112 a73a49 a117 a83a138 a110a111a112 a86a82a43 a52a53 integraldisplay ∞ 0 lnx 1 + x + x2dx = 0. a142 a62a143a144a49a145a146a63a73a147a148a83a138 integraldisplay ∞ 0 1 1 + x + x2 dx = 2pi 3√3. a73a97a149a150a151a77a60 a51a78a152 a48a153a154a155a156 a152 a76a67 a71a72 a51a52a53 a84a157 a50a158a159a152a160a161 a49a93a162a163a164a165a151 a154a166a167 a152a168 a153a169a170 a152a51a52a53 a84 a142 a151a93 a168 a153a169a170a143a144a49a146 a109a171a172 a48a173a169a170 a152a51a52a53 a49 a174a175a1764.5 a68a69 a152a177a178a179a152 a65a180 a52a53 a49a145a63a73a67a77a60 a51a78 a71a72a84 a77a60 a51a78 a145a56a57a181a64 a152 a49 a109 a173a169a170 a152a51a52a53 a117a182 a73a183a67a77a60 a51a78 a81a71a72a84a156a184a69a146a164a165a151a71 a72 a51a52a53a152 a171a172 a70a66a84 a185a186a187a188 a13222 a133 a189a190a191a192a193a194a195 integraldisplay ∞ ?∞ sin2n+1 x x2n+1 dx, n = 1,2,··· a196a197a198 a199a200Euler a201a202a49 a109 sin2n+1 x = parenleftbiggeix ?e?ix 2i parenrightbigg2n+1 = parenleftbigg 1 2i parenrightbigg2n+1 2n+1summationdisplay k=0 parenleftbigg2n + 1 k parenrightbiggparenleftbig eixparenrightbig2n+1?kparenleftbig?e?ixparenrightbigk = parenleftbigg 1 2i parenrightbigg2n+1 2n+1summationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg ei(2n+1?2k)x = parenleftbigg 1 2i parenrightbigg2n+1 nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbiggbracketleftBig ei(2n+1?2k)x ?e?i(2n+1?2k)x bracketrightBig = (?) n 22n nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg sin(2n + 1?2k)x a61a62a49a88a89a90a91 a52a53 contintegraldisplay C 1 z2n+1f(z)dz, a52a53a203a204C a54a205a206a49a207 f(z) = nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg ei(2n+1?2k)z ?Q2n?1(z), Q2n?1(z) a57a56a208a75 2n? 1 a148a152a209a114a202a49a210 z = 0 a92a211a52a59a60 f(z)/z2n+1 a152a48a212a213a214a49a215 z = 0 a92 f(z)a1522na212a216a214a49 nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbiggbracketleftbig i(2n + 1?2k)bracketrightbigl ?Ql2n?1(0) = 0, l = 0,1,2,···,2n?1. a50 a57 Q2n?1(0) = nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg Qprime2n?1(0) = i nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg (2m + 1?2k) = 0 parenleftbigg a61a92 d dx sin 2n+1 xvextendsinglevextendsingle x=0 = 0 parenrightbigg Qprimeprime2n?1(0) = ? nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg (2m + 1?2k)2 ... a121a122a123 a124a125a126a127a128a129a130a131 a13223 a133 Q(2n?2)2n?1 (0) = (?)n?1 nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg (2m + 1?2k)2n?2 Q(2n?1)2n?1 (0) = (?)n?1i nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg (2m + 1?2k)2n?1 = 0 parenleftbigg a61a92 d2n?1 dx2n?1 sin 2n+1 xvextendsinglevextendsingle x=0 = 0 parenrightbigg a157a62a215a63 a51a217 Q2n?1(z) = n?1summationdisplay l=0 (?)l (2l)! bracketleftBigg nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg (2n + 1?2k)2l bracketrightBigg z2l, a215Q2n?1(z)a572n?2a148 a152 a58a148 a209 a114a202a49a218a60a92a219a60a84 a199a200 a77a60 a51a78 a109 integraldisplay ?δ ?R 1 x2n+1f(x)dx + integraldisplay Cδ 1 z2n+1f(z)dz + integraldisplay R δ 1 x2n+!f(x)dx + integraldisplay CR 1 z2n+1f(z)dz = 0. a61a92 limz→∞ 1z2n+1 = 0, a112 a73 lim R→∞ integraldisplay CR 1 z2n+1e i(2n+1?2k)zdz = 0; a220 a61a92 limz→∞z · 1z2n+1 = 0, a112 a73 lim R→∞ integraldisplay CR 1 z2n+1Q2n?1(z)dz = 0. a221a222a223 a81 a117 a83a138 lim R→∞ integraldisplay CR 1 z2n+1f(z)dz = 0. a224 a48a70a135a49 limz→0z· 1z2n+1f(z) = limz→0 1z2nf(z) = limz→0 1z2n braceleftBigg nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg ei(2n+1?2k)z ?Q2n?1(z) bracerightBigg = 1(2n)! nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg [i(2n + 1?2k)]2n = (?) n (2n)! nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg (2n + 1?2k)2n, a185a186a187a188 a13224 a133 a112 a73 lim δ→0 integraldisplay Cδ 1 z2n+1f(z)dz = ?pii× (?)n (2n)! nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg (2n + 1?2k)2n. a225 a213 a160δ → 0, R →∞ a49a215a83integraldisplay ∞ ?∞ 1 x2n+1f(x)dx = pii (?)n (2n)! nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg (2n + 1?2k)2n. a226a227a228a229 a49 a222a230a231 Q 2n?1(x)a152a218a60a92a219a60a49 a117 a83a138integraldisplay ∞ ?∞ 1 x2n+1 braceleftBigg nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg sin(2n + 1?2k)x bracerightBigg dx = (?)n22n integraldisplay ∞ ?∞ sin2n+1 x x2n+1 dx = pi(?) n (2n)! nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg (2n + 1?2k)2n. a154a232 a117 a82 a217 a151 integraldisplay ∞ ?∞ sin2n+1 x x2n+1 dx = pi (2n)! nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbiggparenleftbigg2n + 1 2 ?k parenrightbigg2n . a121a122a123 a124a125a126a127a128a129a130a131 a13225 a133 a189a190a191a192a193a194a195 integraldisplay ∞ ?∞ sin2n x x2n dx, n = 1,2,···a196a197a198 a199a200Euler a201a202a49 a109 sin2n x = parenleftbiggeix ?e?ix 2i parenrightbigg2n = (?) n 22n 2nsummationdisplay k=0 parenleftbigg2n k parenrightbiggparenleftbig eixparenrightbig2n?kparenleftbig?e?ixparenrightbigk = (?) n 22n 2nsummationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg ei(2n?2k)x = (?) n 22n braceleftBiggn?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg ei(2n?2k)x + (?)n parenleftbigg2n n parenrightbigg + 2nsummationdisplay k=n+1 (?)k parenleftbigg2n k parenrightbigg ei(2n?2k)x bracerightBigg = (?) n 22n braceleftBiggn?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbiggbracketleftBig ei(2n?2k)x + e?i(2n?2k)x bracketrightBig + (?)n parenleftbigg2n n parenrightbiggbracerightBigg = (?) n 22n braceleftBigg 2 n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg cos(2n?2k)x + (?)n parenleftbigg2n n parenrightbiggbracerightBigg a61a62a49a88a89a90a91 a52a53 contintegraldisplay C 1 z2nf(z)dz, a52a53a203a204C a54a205a206a49a207 f(z) = n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg ei(2n?2k)z + (?) n 2 parenleftbigg2n n parenrightbigg ?Q2n?2(z), Q2n?2(z)a57a56a208a752n?2a148 a152a209 a114a202a49a210z = 0a92a211 a52 a59a60f(z)/z2na152a48a212a213a214a49a215z = 0a92 f(z) a1522n?1 a212a216a214a49 n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg + (?) n 2 parenleftbigg2n n parenrightbigg ?Q2n?2(0) = 0, n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbiggbracketleftbig i(2n?2k)bracketrightbigl ?Ql2n?2(0) = 0, l = 1,2,···,2n?2. a50 a57 Q2n?2(0) = n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg + (?) n 2 parenleftbigg2n n parenrightbigg = 0 parenleftBig a61a92sin2nvextendsinglevextendsinglex=0 = 0 parenrightBig Qprime2n?2(0) = i n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2m?2k) a185a186a187a188 a13226 a133 Qprimeprime2n?2(0) = ? n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2m?2k)2 = 0 parenleftbigg a61a92 d2 dx2 sin 2nvextendsinglevextendsingle x=0 = 0 parenrightbigg ... Q(2n?3)2n?2 (0) = (?)ni n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2m?2k)2n?3 Q(2n?2)2n?2 (0) = (?)n+1 n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2m?2k)2n?2 = 0 parenleftbigg a61a92 d2n?2 dx2n?2 sin 2nvextendsinglevextendsingle x=0 = 0 parenrightbigg a157a62a215a63 a51a217 Q2n?2(z) = i n?2summationdisplay l=0 (?)l (2l + 1)! bracketleftBiggn?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2n?2k)2l+1 bracketrightBigg z2l+1, a215Q2n?2(z)a572n?3a148 a152a233 a148 a209 a114a202a49a218a60a92a234 a228 a60a84 a199a200 a77a60 a51a78 a109 integraldisplay ?δ ?R 1 x2nf(x)dx + integraldisplay Cδ 1 z2nf(z)dz + integraldisplay R δ 1 x2nf(x)dx + integraldisplay CR 1 z2nf(z)dz = 0. a61a92 limz→∞ 1z2n = 0, a112 a73 lim R→∞ integraldisplay CR 1 z2ne i(2n?2k)zdz = 0; a220 a61a92 limz→∞z · 1z2n = 0, a112 a73 lim R→∞ integraldisplay CR 1 z2ndz = 0; a147a61a92 limz→∞z · 1z2nQ2n?2(z) = 0, a112 a73 lim R→∞ integraldisplay CR 1 z2nQ2n?2(z)dz = 0. a93 a168 a229 a53 a221a222a223 a81 a117 a83a138 lim R→∞ integraldisplay CR 1 z2nf(z)dz = 0. a121a122a123 a124a125a126a127a128a129a130a131 a13227 a133 a224 a48a70a135a49 limz→0z · 1z2nf(z) = limz→0 1z2n?1f(z) = limz→0 1z2n?1 braceleftBiggn?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg ei(2n?2k)z + (?) n 2 parenleftbigg2n n parenrightbigg ?Q2n?2(z) bracerightBigg = 1(2n?1)! n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg [i(2n?2k)]2n?1, a112 a73 lim δ→0 integraldisplay Cδ 1 z2nf(z)dz = ?pii× i2n?1 (2n?1)! n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2n?2k)2n?1 = (?)n+1 pi(2n?1)! n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2n?2k)2n?1. a225 a213 a160δ → 0, R →∞ a49a215a83integraldisplay ∞ ?∞ 1 x2nf(x)dx = (?) n pi (2n?1)! n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2n?2k)2n?1. a226a227 a219 a229 a49 a222a230a231 Q 2n?2(x)a152a218a60a92a234 a228 a60a49 a117 a83a138 integraldisplay ∞ ?∞ 1 x2n braceleftBiggn?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg cos(2n?2k)x + (?) n 2 parenleftbigg2n n parenrightbiggbracerightBigg dx = (?)n22n?1 integraldisplay ∞ ?∞ sin2n x x2n dx = (?)n pi(2n?1)! n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2n?2k)2n?1. a154a232 a117 a82 a217 a151 integraldisplay ∞ ?∞ sin2n x x2n dx = pi (2n?1)! n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (n?k)2n?1 = pi(2n?1)! nsummationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (n?k)2n?1.