Wu Chong-shi a0a1a2a3 a4a5a6a7a8a9a10a11 ( a12) a131a14 a15a16a17a18 a19a20a21a22a23a24a25a26 ( a27) §11.1 a28a29a30a31a32a33a34a35a36a37 a38a39a40a41a42a43a44a45a46a47a48a49a50a51 I = integraldisplay ∞ ?∞ f(x)cospxdx a52 I = integraldisplay ∞ ?∞ f(x)sinpxdx. a38a53a54a55a56a57p > 0 a58 a59a60a61a62a63a64a65a66a67a68a69a70a71a72a73a74a75a76a65a77a78 a58 a79a80a81a66a82a83a84a85a86a87a88a89a90f(z)cos pz a91f(z)sinpz a58 a61a80a92a90z = ∞a80a82a83 sinza91cosz a65a93a94a95a96(a61a97a98a99a100z a71a84a101 a102a103a104a105∞ a106 a68sinza91cosza70a71a107a108a105a84a101a65a83a109) a68a84a110a105a111a112a113a114 lim R→∞ integraldisplay CR f(z)cospzdz a91 lim R→∞ integraldisplay CR f(z)sinpzdz. a115a116a43a117a118a119a120a121a41a122a123a124a125 f(z)eipz a58a126a127 a122a123f(z)eipz a128a129a130a131a132a133a134a135a135a136a137a138a139 a68a140 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 a141a142a143a144 res braceleftbigf(z)eipzbracerightbig. a134a145a146a147a148a149a150 lim R→∞ integraldisplay CR f(z)eipzdz, a42a151a152a153a154a155a156a157a155a68a158a48a49a159a160 integraldisplay ∞ ?∞ f(x)cospxdx a156 integraldisplay ∞ ?∞ f(x)sinpxdx. a90a161a68a162a163a164a165a166a60 a58 a167a168 11.1(Jordana167a168) a57 a1280 ≤ argz ≤ pi a43a169a170 a133 a68a171|z|→ ∞ a172 a68Q(z) a173a174a175a176a177a178 0a68a140 lim R→∞ integraldisplay CR Q(z)eipzdz = 0, a179a180p > 0a68C R a119a49a181 a139 a125a182a183a68Ra125 a130a184 a43 a129a130 a182 a58 a185 a171z a128CR a129 a172 a68z = Reiθ a68 vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay CR Q(z)eipzdz vextendsinglevextendsingle vextendsinglevextendsingle = vextendsinglevextendsingle vextendsinglevextendsingle integraldisplay pi 0 QparenleftbigReiθparenrightbigeipR(cosθ+isinθ)Reiθidθ vextendsinglevextendsingle vextendsinglevextendsingle Wu Chong-shi §11.1 a186a187a188a189 a5a190a191a192a193a194 a132a14 ≤ integraldisplay pi 0 vextendsinglevextendsingleQparenleftbigReiθparenrightbigvextendsinglevextendsinglee?pRsinθRdθ <εR integraldisplay pi 0 e?pRsinθdθ = 2εR integraldisplay pi/2 0 e?pRsinθdθ. a195a196a43a197a198 a128 a178a199 a116a200 a148sinθ a201a58a202a20311.1a48a204a68a1710 ≤ θ ≤ pi/2a172 a68 a135 sinθ ≥ 2θ/pia68a205a49vextendsingle vextendsinglevextendsingle vextendsingle integraldisplay CR Q(z)eipzdz vextendsinglevextendsingle vextendsinglevextendsingle < 2εR integraldisplay pi 0 e?pR·2θ/pidθ = 2εR pi2pR parenleftbig1?e?pRparenrightbig = εpip parenleftbig1?e?pRparenrightbig. a20611.1 a38a207a68a158a195a196a208 lim R→∞ integraldisplay CR Q(z)eipzdz = 0. square a178 a119a68 a128a209a210Jordan a211a212 a43a213a214a215a68 integraldisplay ∞ ?∞ f(x)eipxdx = 2pii summationdisplay a141a142a143a144 resbraceleftbigf(z)eipzbracerightbig. a42a151a124a154a155a156a157a155a68a216a160 integraldisplay ∞ ?∞ f(x)cospxdx =Re ?? ?2pii summationdisplay a141a142a143a144 resbracketleftbigf(z)eipzbracketrightbig ?? ? = ?2piIm ?? ? summationdisplay a141a142a143a144 resbracketleftbigf(z)eipzbracketrightbig ?? ?, integraldisplay ∞ ?∞ f(x)sinpxdx =Im ? ? ?2pii summationdisplay a141a142a143a144 resbracketleftbigf(z)eipzbracketrightbig ? ? ? = 2piRe ? ? ? summationdisplay a141a142a143a144 resbracketleftbigf(z)eipzbracketrightbig ? ? ?. a21711.1 a148a149 a41a42 integraldisplay ∞ 0 xsinx x2 + a2 dx, a > 0a58 a218 a219a220 a129a132 a43a221a222a68 a135 integraldisplay ∞ ?∞ xeix x2 + a2dx = 2pii· 1 2e i·ia = piie?a. a205a49 integraldisplay ∞ ?∞ xsinx x2 + a2dx = pie ?a, integraldisplay ∞ 0 xsinx x2 + a2dx = pi 2e ?a. a223a224a225 a172 a68a226a160a227 integraldisplay ∞ ?∞ xcosx x2 + a2dx = 0. a38a119a228a229a43a68a230a125a121a41a122a123a119 a138 a122a123 a58 Wu Chong-shi a0a1a2a3 a4a5a6a7a8a9a10a11 ( a12) a133a14 §11.2 a231a232a233a234a235a236a33a237a238 a239a41a42 ( a57a239 a139 a125 c) a43a40a240a119 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. a126a127 a38a241 a137a242a136a243a244a245 a54a246 a128 a68a247a119 lim δ→0 bracketleftbigg integraldisplay c?δ a f(x)dx + integraldisplay b c+δ f(x)dx bracketrightbigg a246 a128 a68a140a248a125a239a41a42a43a249 a201 a246 a128 a68a250a125 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 . a171a229a68 a126a127 a239a41a42a251a179a249 a201 a245 a246 a128 a68a252a253a254a255 a173 a40a0a1 a58 a92a161a68a2a3a4a5a66a67a80a164a165a6a66a67a68a7a59a60a8a9a65a10a5a66a67 contintegraldisplay C f(z)dza106 a68a4a11a12a65a6 a96a13a80a81a66a82a83a65a95a96a68a14a15a16a17a95a96a18a19a20a21a22a65a66a67a77a78 a58 a23a24a25a26a27a28a29a165a30a31 a32a33a34a35a36a59a60a61a63a66a67a65a37a93a38a39 a58 a21711.2 a148a149 a41a42 integraldisplay ∞ ?∞ dx x(1 + x + x2) a58 a218 a38a119 a173 a137a40a41 a41a42a68 a40a41a42a43a44a45a128 a41a42a46a47a125a48a49a46a47a68a50 a44a45 a125a121a41a122a123 a128x = 0a139 a54a51a52 (x = 0 a139 a125a239 a139) a58 a224a41a42 a128 a249 a201a53 a240a215a246 a128 a68 v.p. integraldisplay ∞ ?∞ dx x(1 + x + x2) = limR1→∞ 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 . a230a224a68 a128a54a55a56 a123a40 a212 a148a149 a224a41a42 a172 a68 a54a57a58a59a60a61 a41a42 contintegraldisplay C dz z(1 + z + z2), a179a180a43a41a42a170a62 C a126a20311.2a205a63a68a202 a49a181 a139 a125a182a183a64 δ a125 a130a184 a43a65 a130 a182a66 C δ a156a49a181 a139 a125a182a183a64Ra125 a130a184 a43a67 a130 a182a66 C R a49a251 a68a69a70?R →?δa156δ → R a71 a51 a58a178 a119a68a219a220 a56 a123a40 a212 a68 a135 a20611.2contintegraldisplay 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. a230a125 limz→∞z · 1z(1 + z + z2) = 0, Wu Chong-shi §11.2 a72a73a74a75a76a77 a190a78a79 a134a14 a205a49a68a219a220 a211a2123.2a68a135 lim R→∞ integraldisplay CR dz z(1 + z + z2) = 0. a50a230a125 limz→0z · 1z(1 + z + z2) = 1, a205a49a68a219a220 a211a2123.1a68a135 lim δ→0 integraldisplay Cδ dz z(1 + z + z2) = ?pii. a38a207a68a124 a242a136R →∞ a68δ → 0a68a158a160a227 v.p. integraldisplay ∞ ?∞ dx x(1 + x + x2) = ? pi√ 3. a80a81a82 a126a127 a41a42a170a62a180a43a65 a130 a182a66a119a83a215 a130a131a132a84a85 z = 0a139 a68a230a86a87 z = 0a139a88 a170 a128 a170a62 a133 a68a119a89a90a160a227a54a225a43a91 a127a92 a125a93a253 a92 a94a12a24 a65a113a114a95a70a71a96a97a68a98a105a66a67a99a100a12a101a95a96a65a102a76a68a103a104a113a114a77a16a95a96a65a105a75 a106a12a65a66a67a109( a107a108 a88a35a68a104a113a114a109a65a110a111a109) a58 a7a101a112a102a113 (a30a2a68a114a115a116a82a83a65a117a118a66a67) a23a68a93a32a4a5a66a67a119a84a80a6a66a67a68a79a120a105 a7a8a9a65a10a5a66a67a95a68a119a84a80a86a87a88a121a81a66a82a83 f(x) a122 a20f(z)a68a92a18a7a10a5a82a83a65a77 a78a66a67a95a68a66a67a99a100a12a123a70a71a97a124a95a96 a58 a61a94a23a24a65a30a31a95a70a71a96a97 a58 a21711.3 a148a149 a41a42 integraldisplay ∞ ?∞ sinx x dxa58 a218 a125a126a229 a175 a68 a54 a171 a58a59 a41a42 contintegraldisplay C eiz z dz a68a41a42a170a62C a156a127 9a0a225( a20311.2)a58 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. a128 a41a42a170a62 a88 a170a43a46a128 a133 a68a121a41a122a123a129a130a68a131a170a62a41a42a1250 a58 a219a220Jordan a211a212 a156 a211a2123.1a68a42a151a135 lim R→∞ integraldisplay CR eiz z dz = 0, limδ→0 integraldisplay Cδ eiz z dz = ?pii. a230a224 integraldisplay ∞ ?∞ eix x dx = pii. a152a153a241a132a43a154a155a156a157a155a68a216a160 v.p. integraldisplay ∞ ?∞ cosx x dx = 0, integraldisplay ∞ ?∞ sinx x dx = pi. a197 a178 a38a133a39a134a43a41a42a68a226a48a49a135 a150 integraldisplay ∞ ?∞ sin2x x2 dx =pi; Wu Chong-shi a0a1a2a3 a4a5a6a7a8a9a10a11 ( a12) a135a14 integraldisplay ∞ ?∞ sin3x x3 dx = 3 4pi;integraldisplay ∞ ?∞ sin4x x4 dx = 2 3pi; integraldisplay ∞ ?∞ sin5x x5 dx = 115 192pi;integraldisplay ∞ ?∞ sin6x x6 dx = 11 20pi; a52a136 a68a137a138a139a43a91 a127a140 integraldisplay ∞ ?∞ sinnx xn dx = pi (n?1)! [n/2]summationdisplay k=0 (?)k parenleftbiggn k parenrightbiggparenleftbiggn?2k 2 parenrightbiggn?1 . a148a149 a38a141a41a42a68a197a198 a128 a178 a115a116 a175a142a143 a60a61 a41a42a43a121a41a122a123 a58 a127 a126 a68a125a208 a148a149 a41a42 integraldisplay ∞ ?∞ sin2x x2 dx, a158 a54a57a58a59a60a61 a41a42 contintegraldisplay C 1?ei2z z2 dz, a41a42a170a62C a144a126a20311.2a58 a38a53a68a158 a60a61 a41a42a86a145a68 a128 a154a146 a129 a48a49 a135a138a139 a58 a247a38a133 a138a139 a68 a173a147a148a149 a68 a134a146 a119a48a150 a138a139 a52a173a151 a242a139 a58 a38a83 a211a2123.1a158a48a49a152a150a58a126a127 a119a153 a151a52 a153 a151 a49 a129 a43 a242a139 a68 a52 a119a154 a42 a138a139 a68a155a65a182a66 C δ a43a41a42a158a48 a146 a176a178∞a58 Wu Chong-shi §11.3 a156a157a158a159a160a161a162 a1636a164 §11.3 a165a166a31a32a33a36a37 a45a116 a175a148 a68a38a53a205 a148 a43a167 a201 a122a123a43a41a42a119a83 a60a61 a122a123a43a168a169 a148 a43 a58 a83 a60 a123a128 a149 a152a68a154 a61 a40a41a42a180 a43a41a42 a61a170xa128x > 0 a172 a54a57 a212 a129a125 argx = 0a58 a173 a133 a41 a204a43a167 a201 a122a123a41a42a119 I = integraldisplay ∞ 0 xs?1Q(x)dx, a179a180sa125a154a123a68Q(x) a243 a201 a68 a128 a115a154a146 a129a171a135a138a139 a58 a125a208a172a195a41a42a173a174a68 a145 a159 limx→∞x·xs?1Q(x)dx = limx→∞xsQ(x) = 0. a58a59 a0 a54 a43 a60a61 a41a42contintegraltext C z s?1Q(z)dz a58 a120a105z = 0 a175z = ∞a80a81a66a82a83a65a176a96a68a177a71a178a104a121 a179a24a180a181a4a11a182a17a68a119a183a184a180a182a185a12a186 argz = 0 a58 a61 a106 a65a66a67a99a100a120a182a17a65a187a105a75a106 (a74a100a67a188a90R a189δ) a175 a182a185a12a23a186a190a20 ( a191a19211.3)a58 a180a182a185a12a23a186a65a66a67 a193a194a111a112a195a177a104a113a114a65a4a5a66a67a101a196 a58a197a198 a80a2a199a113 a114a180a187a105a75a106a65a66a67a109 a58 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. a20611.3 a202a211a2123.1a156a211a2123.2a48a49a152a150a68a126a127 a1280 ≤ argz ≤ 2pi a43a169a170 a133 a68 limz→0zsQ(z) = 0, limz→∞zsQ(z) = 0, a140 lim δ→0 integraldisplay Cδ zs?1Q(z)dz = 0, lim R→∞ integraldisplay CR zs?1Q(z)dz = 0. a137a200 a173a201 a68 a126a127Q(z)a128a202a131a132a129a203a208a135a136a137a204a205a138a139(a54a128a115a154a146a129)a206 a68a119 a243 a201 a129a130a43a68a230a86a48 a49 a54a55a56 a123a40 a212a58 a128 a124 a242a136δ → 0, R → ∞ a207 a68a158a160a227 parenleftbig1?ei2pisparenrightbigintegraldisplay ∞ 0 xs?1Q(x)dx = 2pii summationdisplay a208a143a144 res braceleftbigzs?1Q(z)bracerightbig. a205a49 integraldisplay ∞ 0 xs?1Q(x)dx = 2pii1?ei2pis summationdisplay a208a143a144 res braceleftbigzs?1Q(z)bracerightbig. Wu Chong-shi a209a210a211a212 a213a214a215a216a217a218a219a220 ( a221) a2227 a223 a224 a145a225 a53 a68 a128a148a149a56 a123 a172 a68 a145a226a227a129a132a228 a178 a167 a201 a122a123 zs a205a229a43 a136a230 a68a216 0 ≤ argz ≤ 2pi a58 a80a81a82 a126a127a231 a40 a128a232 a69 a129a233 argz = 2pi a68a119a89a234a235a236 a207 a91 a127a92 a80a81a82 a126a127Q(x)a237 a135 a173 a40a43 a228 a248 a42a238 a68a127 a126 a119 x a43 a138 a122a123 a52a239 a122a123a68a119a89a48a49a124a179a240a46a47 a43a170a62 a92 a21711.4 a148a149 a41a42 integraldisplay ∞ 0 xα?1 x + ei?dx a680 < α < 1, ?pi < ? < pi a58 a218 a38a53a43a121a41a122a123a228a229 a209a210a129a241 a221a222a180a43 a145 a159a68a230a224 integraldisplay ∞ 0 xα?1 x + ei?dx = 2pii 1?ei2piαe i(?+pi)(α?1) = pi sinpiα ·e i?(α?1) (star) a83a38 a137 a41a42a226a48a49a242 a150 a173 a141a137a200 a173a201 a43a91 a127a58 a127 a126 a68a229a125a38 a137 a41a42a43a243a244a245a46a68 ? = 0a68a140 integraldisplay ∞ 0 xα?1 1 + xdx = pi sinpiα. a128Γ a122a123 a173a246 a180 a145 a68a247 a54a55 a227a38 a137 a91 a127a58 a50 a126 a68a152a153 (star)a47a241a132a43a157a155a68a226a48a49a160a227 integraldisplay ∞ 0 xα?1 x2 + 2xcos? + 1dx = pi sinpiα sin(1?α)? sin? . a61a165a248a3a80a7 0 < α < 1a65a249a250a23a251a252a65a68a79a80a70a71a253a254a255a0a2520 < α < 2 a58a1a2 (star) a103a29a3a65a4a4a68a13a70a71a251a252a101a5a65a248a3 a58 a6 a173 a133a167 a201 a122a123a43a41a42a7a251 a228 a123a122a123 a58a8 a221a222a215 a132 a43a127a9 a58 a21711.5 a148a149 a41a42 integraldisplay ∞ 0 lnx 1 + x + x2 dxa58 a218 a124a170a62 a126a20311.3a68a148a149a60a61a41a42contintegraldisplay C lnz 1 + z + z2dz = integraldisplay R δ lnx 1 + x + x2 dx + integraldisplay CR lnz 1 + z + z2dz + integraldisplay δ R lnparenleftbigxei2piparenrightbig 1 + x + x2dx + integraldisplay Cδ lnz 1 + z + z2dz =2pii summationdisplay a208a143a144 res braceleftbigg lnz 1 + z + z2 bracerightbigg = 2pii parenleftbigg 2pi 3√3 ? 4pi 3√3 parenrightbigg = ?4pi 2i 3√3. a230a125 limz→∞z · lnz1 + z + z2 = 0, limz→0z · lnz1 + z + z2 = 0, a219a220 a211a2123.2 a156 a211a2123.1 a68 a135 lim R→∞ integraldisplay CR lnz 1 + z + z2dz = 0, limδ→0 integraldisplay Cδ lnz 1 + z + z2dz = 0. a205a49a68a124 a242a136R →∞, δ → 0 a68a216a160 integraldisplay ∞ 0 lnx 1 + x + x2dx? integraldisplay ∞ 0 lnx + 2pii 1 + x + x2 dx = ? 4pi2i 3√3. Wu Chong-shi §11.3 a10a11a189 a5a190a193a194 a138a14 a12a13a124a7a180a182a185a12a23a186a65a66a67a14a195a177a104a113a114a65a66a67a101a196a68a79a80a68a15a16a84a17a68a109a26a123a8a18 a19a20a21a22a68a18a23a24a25a26a27a28a29a30a31a32a33a34a35a36a37a38a39 integraldisplay ∞ 0 1 1 + x + x2 dx = 2pi 3√3. a40a41a42a43a38a39 contintegraldisplay C lnz 1 + z + z2dz a34a35 integraldisplay ∞ 0 lnx 1 + x + x2 dxa44a45a46 star a47a48a49a50a51a52a53a54a55a56a57a58a59a60a53a61a58a57a58 lnz a49a62a63a64a65a66a67a68a69a70a53a51a71a72a73a74 a70a75a76a77a78a79a53a80a81a69 (a82lnx)a83a84a85a86a87 star a88a47a48a89a49a90a91a92a93a94a95 ? a96a97a53a61a98a99a77a78 integraldisplay ∞ 0 f(x)dxa53 a100a101f(x) a59a52a102a57a58 (a51a71a103a104a105a106a10710.3a108a109a49a110 a106a111a112) a53a103a113a114a115a116a107a117a58a99a118a111a112a119a120a77a78 contintegraldisplay C f(z) lnz dz a121a122a123a87 ? a96a124a53 a100a101a125 a111a112a77a78 integraldisplay ∞ 0 f(x)lnxdxa53a126a103a113a127a128a129a130a77a78 contintegraldisplay C f(z)ln2z dz a87 a51a131a132a79a73a74a70a75a76 ln 2z a49a57a58a63 ln2x a54 (lnx + 2pii)2 a84a83a85a86a53a133a75a49a134 a135a136a137a138a139a140a125 a49 lnxa141a87 a66a67a142a121a143a144a145 12 a109 a139a125 a122a49a77a78a111a112a87a131a71a53a127a128a77a78 contintegraldisplay C ln2z 1 + z + z2dz a53a119a120 C a59a130a87 a146 a129a70a147a49a111a112a148a149a53a142a123a150integraldisplay ∞ 0 ln2x 1 + x + x2dx? integraldisplay ∞ 0 (lnx + 2pii)2 1 + x + x2 dx = 2pii summationdisplay a151a152a153 res braceleftbigg ln2z 1 + z + z2 bracerightbigg = 2pi√3 bracketleftbigg16 9 pi 2 ? 4 9pi 2 bracketrightbigg = 83√3pi3. a98a52 ?4pii integraldisplay ∞ 0 lnx 1 + x + x2dx + 4pi 2 integraldisplay ∞ 0 1 1 + x + x2dx = 8 3√3pi 3. a139 a113a53a142a123a150 a137a138a139a125 a122a49a77a78 integraldisplay ∞ 0 lnx 1 + x + x2dx = 0. a154 a71a155a156a53a157a158a103a113a159a160a123a150 integraldisplay ∞ 0 1 1 + x + x2 dx = 2pi 3√3. a113a70a161a162a163a117a58a99a118a49a97a164a165a166a167a49a116a107 a111a112a99a77a78a87a168a98a169a170a49a171a172a53a132a173a174a175a176a163 a165a177a178a49a179a164a180a181a49a99a77a78a87 a154 a163a132a179a164a180a181a155a156a53a158 a136 a80a182a97a183a180a181a49a99a77a78a53 a184a185a1864.5 a108a109 a49a187a188a189a49a105a190a77a78a53a157a103a113a107a117a58a99a118a111a112a87 Wu Chong-shi a191a192a193a194 a195a196a197a198a199a200a201a202 ( a203) a2049a205 a206a207a208a209a210a211a212 integraldisplay ∞ ?∞ sin2n+1 x x2n+1 dx, n = 1,2,··· a213a214a215 a55a216Eulera217a56a53 a136 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 a51a71a53a127a128a129a130a77a78 contintegraldisplay C 1 z2n+1f(z)dz, a77a78a218a219C a100a220a221a53a222 f(z) = nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg ei(2n+1?2k)z ?Q2n?1(z), Q2n?1(z) a52a59a223a115 2n? 1 a160a49a62a141a56a53a224 z = 0 a131a225a77a57a58 f(z)/z2n+1 a49a97a226a227a228a53a82 z = 0 a131 f(z)a492na226a229a228a53 nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbiggbracketleftbig i(2n + 1?2k)bracketrightbigl ?Ql2n?1(0) = 0, l = 0,1,2,···,2n?1. a98a52 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 a51a131 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 ... Wu Chong-shi a230a231a232a233 a20410a205 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 a51a131 d2n?1 dx2n?1 sin 2n+1 xvextendsinglevextendsingle x=0 = 0 parenrightbigg a168a71a82a103a99a234 Q2n?1(z) = n?1summationdisplay l=0 (?)l (2l)! bracketleftBigg nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbigg (2n + 1?2k)2l bracketrightBigg z2l, a82Q2n?1(z)a522n?2a160a49a102a160a62a141a56a53a235a58a131a81a58a87a55a216a117a58a99a118 a136 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. a51a131 limz→∞ 1z2n+1 = 0, a139 a113 lim R→∞ integraldisplay CR 1 z2n+1e i(2n+1?2k)zdz = 0; a236 a51a131 limz→∞z · 1z2n+1 = 0, a139 a113 lim R→∞ integraldisplay CR 1 z2n+1Q2n?1(z)dz = 0. a237a238a239 a121a142a123a150 lim R→∞ integraldisplay CR 1 z2n+1f(z)dz = 0. a240 a97a110a147a53 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, Wu Chong-shi a191a192a193a194 a195a196a197a198a199a200a201a202 ( a203) a20411a205 a139 a113 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. a241 a227a171δ → 0, R →∞a53a82a123integraldisplay ∞ ?∞ 1 x2n+1f(x)dx = pii (?)n (2n)! nsummationdisplay k=0 (?)k parenleftbigg2n+ 1 k parenrightbigg (2n + 1?2k)2n. a242a243 a68a69a53 a238a244a245 Q 2n?1(x)a49a235a58a131a81a58a53a142a123a150integraldisplay ∞ ?∞ 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. a165a246a142a122a234a163 integraldisplay ∞ ?∞ sin2n+1 x x2n+1 dx = pi (2n)! nsummationdisplay k=0 (?)k parenleftbigg2n + 1 k parenrightbiggparenleftbigg2n + 1 2 ?k parenrightbigg2n . Wu Chong-shi a230a231a232a233 a20412a205 a206a207a208a209a210a211a212 integraldisplay ∞ ?∞ sin2n x x2n dx, n = 1,2,···a213a214a215 a55a216Eulera217a56a53 a136 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 a51a71a53a127a128a129a130a77a78 contintegraldisplay C 1 z2nf(z)dz, a77a78a218a219C a100a220a221a53a222 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)a52a59a223a1152n?2a160a49a62a141a56a53a224z = 0a131a225a77a57a58f(z)/z2n a49a97a226a227a228a53a82z = 0a131 f(z) a492n?1a226a229a228a53 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. a98a52 Q2n?2(0) = n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg + (?) n 2 parenleftbigg2n n parenrightbigg = 0 parenleftBig a51a131sin2nvextendsinglevextendsinglex=0 = 0 parenrightBig Qprime2n?2(0) = i n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2m?2k) Wu Chong-shi a191a192a193a194 a195a196a197a198a199a200a201a202 ( a203) a20413a205 Qprimeprime2n?2(0) = ? n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2m?2k)2 = 0 parenleftbigg a51a131 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 a51a131 d2n?2 dx2n?2 sin 2nvextendsinglevextendsingle x=0 = 0 parenrightbigg a168a71a82a103a99a234 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, a82Q2n?2(z)a522n?3a160a49a247a160a62a141a56a53a235a58a131a248a68a58a87a55a216a117a58a99a118 a136 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. a51a131 limz→∞ 1z2n = 0, a139 a113 lim R→∞ integraldisplay CR 1 z2ne i(2n?2k)zdz = 0; a236 a51a131 limz→∞z · 1z2n = 0, a139 a113 lim R→∞ integraldisplay CR 1 z2ndz = 0; a159a51a131 limz→∞z · 1z2nQ2n?2(z) = 0, a139 a113 lim R→∞ integraldisplay CR 1 z2nQ2n?2(z)dz = 0. a132a179a69a78 a237a238a239 a121a142a123a150 lim R→∞ integraldisplay CR 1 z2nf(z)dz = 0. Wu Chong-shi a230a231a232a233 a20414a205 a240 a97a110a147a53 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, a139 a113 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. a241 a227a171δ → 0, R →∞a53a82a123integraldisplay ∞ ?∞ 1 x2nf(x)dx = (?) n pi (2n?1)! n?1summationdisplay k=0 (?)k parenleftbigg2n k parenrightbigg (2n?2k)2n?1. a242a243 a81a69a53 a238a244a245 Q 2n?2(x)a49a235a58a131a248a68a58a53a142a123a150integraldisplay ∞ ?∞ 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. a165a246a142a122a234a163 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.