a0 a1 I a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16a17 a18a19a20a21a22a23a24 a25a26a27a28a29a30 a19 a31a32a33a34a35 a20a36a37 a38a39a40a41a42a43 a44a45a46a47a48(20 a49) 1. a50a51a52a51w = ln(1 + z)a53a54a55a52a51a27a56a57a58a53a59 (1) argz a36a60a61a62a63 (2) arg(1 + z)a36a60a61a62a63 (3) z a36a64a61a65a66a67a63 (4) 1 + z a36a64a61a65a66a67a68 2. a69a52a51f(z)a70a71a72a73a74a75G a76 a77a78a27C a79G a76 a80 a49a81a82a83a84 a85a27a86a87 a79Aa88Ba27a89a90a49 integraldisplay C f(z)dz (1)a91a92a93a94a95a96 a37 a27a97 a91a98a99a100a101a102 a37 a63 (2)a91a92a93a94a95a102 a37 a27a97 a91a98a99a100a101a96 a37 a63 (3)a91a92a93a94a95 a41 a98a99a100a101 a31 a96 a37 a63 (4)a91a92a93a94a95 a41 a98a99a100a101 a31 a102 a37 a68 3. a103a52a51f(z)a70z = aa87a77a78a27 f(a) = fprime(a) = ··· = f(n?1)(a) = 0, f(n)(a) negationslash= 0, a89 a52a51f prime(z)/f(z) a70z = aa87a80a104a51a79 (1) 1?na63 (2) n?1a63 (3) ?na63 (4) na68 4. z = ∞a53f(z) = 1sinz a80 (1)a18a105a106a99 a63 (2) a107 a62a108 a99 a63 (3)a109a110a99 a63 (4) a111 a112a113a108 a99 a68 5. Γ(z)Γ(1?z) = pisinpiz a80a114a115a74a75a79a59 (1)a116a117a118 a63 (2) a119a120a121 a122 0 < Rez < 1a63 (3)a123a124a117a118Rez > 0a63 (4)a125a124a117a118Rez < 1a68 a126 a45(10 a49) a127 a128a77a78 a52a51f(z)a70a129a130a131a132 a80 a51a55a79a133a134a51 a27a135 a134a136 v(x,y) = xx2 + y2 a27a137a138f(z)a68 2 a139 a45(20 a49)a140a52a51ln z ?1z + 1 a70z = ∞a80a141a75a76a142a143a79a144a145a51a27a146a147 ln z ?1z + 1 vextendsinglevextendsingle vextendsinglevextendsingle z=∞ = 0a68 a148 a45(40 a49)a149a150a151a152 a90 a49a59 (1) integraldisplay ∞ ?∞ dx (x2 + 1)(x2 ?2xcosθ + 1), 0 < θ < pi, a135θ negationslash= pi/2. (2) integraldisplay ∞ ?∞ sinx x(x2 + 4)dx. a153 a45(10 a49) a127 a128f(t) = 1 pi integraldisplay pi 0 cos(tcosθ)dθ a27a137a138a56a154a155a154a156a157a158a80a159a52a51 F(p)a68 a160a161a162a163a164a165a166a167a168 a44a45(20 a49) 1. (2) (4 a49) 2. (4) (4 a49) 3. (4) (4 a49) 4. (4) (4 a49) 5. (1) (4 a49) a126 a45(10 a49) a169a1701 a138a171u(x,y)a80a172a173 a49 du(x,y) = ?u?xdx + ?u?ydy = ?v?ydx? ?v?xdy = ? 2xy(x2 + y2)2 dx + x 2 ?y2 (x2 + y2)2dy, a90 a49 a27 u(x,y) = yx2 + y2 + C. a127 a128f(z) a70a129a130a131a132 a80 a51a55a79a133a134a51 a27a174a175a176 y = 0 a177 a27 u(x,y) = 0, a178a179a180a181a147a171 C = 0. a182a183a27 f(z) = y + ixx2 + y2 = iz ? zz? = i z. ?u ?x (2a49) ?u ?y (2a49) a138a171u(x,y) (2 a49) a90 a49a184a51C = 0 (2a49) a185a186a27a138a171f(z) (2 a49) a169a1702 a187a188 a185a186a189a171f(z): v(x,y) = xx2 + y2 = 12 z + z ? zz? = 12 bracketleftbigg 1 z? + 1 z bracketrightbigg 4 = 12i bracketleftbigg i z? + i z bracketrightbigg = 12i bracketleftbigg i z ? parenleftbiggi z parenrightbigg?bracketrightbigg = f(z)?f ?(z) 2i a178a179a180a181 a187a188a190a191 f(z) = iz. (a132a1925a193 a27a194 a193 2a49) a139 a45(20 a49) ln z?1z + 1 = ln 1? 1z 1 + 1z = ∞summationdisplay n=1 (?)n?1 n bracketleftbiggparenleftbigg ?1z parenrightbiggn ? parenleftbigg1 z parenrightbiggnbracketrightbigg = ∞summationdisplay n=1 (?)n?1 n [(?) n ?1] parenleftbigg1 z parenrightbiggn = ?2 ∞summationdisplay n=0 1 2n + 1 parenleftbigg1 z parenrightbigg2n+1 , |z| > 1. a129a195a196 a171a197a85 (3 a49) a198 a55a49a199a200a201 (3a49) ln parenleftbigg 1? 1z parenrightbigg a142a143 (4 a49) ln parenleftbigg 1 + 1z parenrightbigg a142a143 (4 a49) a185a186 (4 a49) a202a203a204a205 (2 a49) a148 a45(1) (20 a49) a206a205a207 a79a132a208a209 a27a210a211 a71 a157a90 a49 contintegraldisplay 1 (z2 + 1)(z2 ?2zcosθ + 1)dz. a212a213a104 a51 a147a214a215 contintegraldisplay 1 (z2 + 1)(z2 ?2zcosθ + 1)dz = integraldisplay R ?R 1 (x2 + 1)(x2 ?2xcosθ + 1)dx + integraldisplay CR contintegraldisplay 1 (z2 + 1)(z2 ?2zcosθ + 1)dz = 2pii summationdisplay a216a217a218a219 res 1(z2 + 1)(z2 ?2zcosθ + 1) 5 = 2pii bracketleftbigg 1 (z2 + 1)(z2 ?2zcosθ + 1)a70z = i a80a104 a51 + 1(z2 + 1)(z2 ?2zcosθ + 1)a70z = eiθ a80a104a51 bracketrightbigg . a181a183 a149a150 a220 a90 a52a51a70z = ia80a104a51 = 12i(?2icosθ) = 14cosθ, a220 a90 a52a51a70z = eiθ a80a104a51 = 1(ei2θ + 1)2isinθ = 12cosθ(cosθ + isinθ)2isinθ = ?icosθ?isinθ4cosθsinθ = ?i 14sinθ ? 14cosθ. a221 a177 a27a58 a79 limz→∞z · 1(z2 + 1)(z2 ?2zcosθ + 1) = 0, a212a213a222 a209 a223a224a214a27a215 lim R→∞ 1 (z2 + 1)(z2 ?2zcosθ + 1)dz = 0. a225a226 a132a192a227a228 a27a206a229a230 R →∞a27a231 a190a191integraldisplay ∞ ?∞ 1 (x2 + 1)(x2 ?2xcosθ + 1)dx = pi 2sinθ. a90 a49 a205a207 (2 a49) a220 a90 a52a51 (2a49) a104 a51 a147a214 (3 a49) a232a87 i a233 a104 a51a149a150 (4a49) a232a87 eiθ a233 a104 a51a149a150 (4a49) a222 a209 a223a224a214 (2 a49) a227a228 (3a49) a148 a45(2) (20 a49) a206a205a207a234a2358.7a27a210a211 a71 a157a90 a49 contintegraldisplay eiz z(z2 + 4)dz. a212a213a104 a51 a147a214a215 contintegraldisplay eiz z(z2 + 4)dz = integraldisplay ?δ ?R eix x(x2 + 4)dx + integraldisplay Cδ eiz z(z2 + 4)dz = integraldisplay R δ eix x(x2 + 4)dx + integraldisplay CR eiz z(z2 + 4)dz 6 = 2piie ?2 ?8 = ?pii4 e?2. a58 a79 limz→∞ 1z(z2 + 4) = 0, a212a213Jordana224a214a27a215 lim R→∞ integraldisplay CR eiz z(z2 + 4)dz = 0. a236a58 a79 limz→0z · e iz z(z2 + 4) = 1 4, a212a213a237 a209 a223a224a214a27a215 lim δ→0 integraldisplay Cδ eiz z(z2 + 4)dz = ? pii 4 . a225a226 a132a192a227a228 a27a206a229a230 R →∞, δ → 0a27a231 a190a191integraldisplay ∞ ?∞ eix x(x2 + 4)dx = pii 4 parenleftbig1?e?2parenrightbig. a238a239 a134a136 a27a231 a190a191a240a241a227a228 integraldisplay ∞ ?∞ sinx x(x2 + 4)dx = pi 4 parenleftbig1?e?2parenrightbig. a90 a49 a205a207 (2 a49) a220 a90 a52a51 (2a49) a104 a51 a147a214 (3 a49) a232a87z = 2i a233 a104 a51a149a150 (4a49) a237 a209 a223a224a214a27 a149a150 (4a49) a222 a209 a223a27 Jordana224a214 (2 a49) a227a228 (3a49) a153 a45(10 a49) F(p) = 1pi integraldisplay pi 0 p p2 + cos2θdθ = 12pi integraldisplay 2pi 0 p p2 + cos2θdθ = 12pi contintegraldisplay |z|=1 p p2 + 14 (z + z?1)2 dz iz = 4p2pii contintegraldisplay |z|=1 z z4 + 2(2p2 + 1)z2 + 1dz. a196 a157a158ζ = z2 a27a14a15z a242a243a132 a80|z| = 1 a44a244 a157 a79ζ a242a243a132 a80|ζ| = 1 a245 a244 a27a182a183 F(p) = 4p2pii contintegraldisplay |ζ|=1 1 ζ2 + 2(2p2 + 1)ζ + 1dζ = 4p× 1ζ2 + 2(2p2 + 1)ζ + 1a70 a198a246 a209a76 a80a104 a51a88. 7 1 ζ2 + 2(2p2 + 1)ζ + 1 a80a232a87 a53 ζ = ?2(2p 2 + 1)±radicalbig4(2p2 + 1)2 ?4 2 = ?(2p2 + 1)±2p radicalbig p2 + 1, a182a183 1 ζ2 + 2(2p2 + 1)ζ + 1 a70 a198a246 a209a76 a247a215 a44a248 a232a87a27 ζ = ?(2p2 + 1) + 2p radicalbig p2 + 1, a249a135 a53 a44a250 a229a87a27a104 a51a79 1 2ζ + 2(2p2 + 1) vextendsinglevextendsingle vextendsinglevextendsingle ζ=?(2p2+1)+2p √ p2+1 = 14pradicalbigp2 + 1. a240a241 a231 a190a191 F(p) = 1radicalbigp2 + 1. cosωta80Laplacea157a158 (2a49) a157 a79a251 a198a246 a209|z| = 1a80a90a49 (2a49) a196 a157a158ζ = z2 (1 a49) a232a87 a233 a104 a51a149a150 (2a49) a227 a228 (2a49) a114a115a252a253 a59 (Rep > 0) (1a49) 8 a0 a1 II a5 a254 a255 a0 a1 a2 a6 a7 a3 a4 a5 a6 a7 a8 a160 a161 a9 a10 a11 Pl(x) = lsummationdisplay n=0 1 (n!)2 (l + n)! (l?n)! parenleftbiggx?1 2 parenrightbiggn (2l + 1)xPl(x) = (l + 1)Pl+1(x) + lPl?1(x) Pprimel+1(x) = xPprimel(x) + (l + 1)Pl(x) Pprimel+1(x) = (2l + 1)Pl(x) + Pprimel?1(x) Pl(x) = Pprimel+1(x)?2xPprimel(x) + Pprimel?1(x) Jν(x) = ∞summationdisplay k=0 (?)k k!Γ(k + ν + 1) parenleftBigx 2 parenrightBig2k+ν Laplacea157a158a11 a57 a52a51f(t) a159a52a51F(p) 1 1p erfc α2√t 1pe?α√p 1√ piα sin2 √αt 1 p√pe ?α/p 1√ pit cos2 √αt 1√ pe ?α/p 1√ pite ?α2/4t 1√ pe ?α√p 1√ pite ?2α√t 1√ pe ?α2/perfc α√ p a12a13 a59 a16a17 a18a19a14a15a22a23 a33a34a25a16a26a27 a28a29a30 a19 a31a17a33a34a35 a20a36a37 a38a39a40a68 a44a45(20 a49)a18 a171 a151a152a19a20a21a55a22 a48 a80a77 a59 (1) ?? ? ?? yprimeprime(x) + λy(x) = 0, y(0) = 0, y(l) = 0; (2) ? ?? ?? yprimeprime(x) + λy(x) = 0, yprime(0) = 0, yprime(l) = 0; (3) ?? ? ?? yprimeprime(x) + λy(x) = 0, y(x) = y(x + 2pi) = 0; 10 (4) ?? ? ?? d dx bracketleftbigg (1?x2)dy(x)dx bracketrightbigg + λy(x) = 0, y(±1)a215a23. a126 a45(25 a49)a138a77a151a152 a147a77 a22 a48 a59? ??? ??? ?? ??? ??? ?? ?ua ?xt?y? 2 ?u0 ?xx?y=, 0 < x < l, t > 0, ?u ?x vextendsinglevextendsingle vextendsinglevextendsingle x=0 = 0, ?u?x vextendsinglevextendsingle vextendsinglevextendsingle x=l = 0, t > 0, uvextendsinglevextendsinglet=0 = Acos2 pixl , ?u?t vextendsinglevextendsingle vextendsingle t=0 = 0, 0 < x < l, a56a24a a88Aa25a79a127 a128 a184a51 a68 a139 a45(20 a49)a138a77a26a76 a80a147a77 a22 a48 a59? ?? ?? ?2u = ?4pir cosθ, r < a, uvextendsinglevextendsingler=a = 0. a148 a45(15 a49)a149a150 a90 a49a59 integraldisplay ∞ 0 e?αx2 J0(x)xdx, a56a24α > 0a27J 0(x)a79a27a250 Bessela52a51 a68 a153 a45(20 a49)a28Laplacea157a158a169a170a138a208a29a23a30a31a32a33a22a48a80 Greena52a51a59 ? ??? ??? ? ??? ??? ? ?G(x,t;xprime,tprime) ?t ?κ ?G(x,t;xprime,tprime)δ ?xx?y= (x?x prime)δ(t?tprime), 0 < x < ∞, t > 0, G(x,t;xprime,tprime)vextendsinglevextendsinglex=0 = 0, t > 0, G(x,t;xprime,tprime)vextendsinglevextendsinglet=0 = 0, 0 < x < ∞, a56a240 < xprime < ∞,tprime > 0a68 a20 a137 a48 a28 a34 a202 a35 11 a12a13a36a37a38a39a40a41a42 a43a44(20 a45) (1) λ n = parenleftBignpi l parenrightBig2 (2a45) yn(x) = sin npil x (2a45) n = 1,2,3,··· (1a45) (2) λ n = parenleftbigg2n + 1 l pi parenrightbigg2 (2a45) yn(x) = sin 2n + 1l pix (2a45) n = 0,1,2,3,··· (1a45) (3) λ n = n2 (2a45) yn(x) = sinnx, cosnx (2a45) n = 0,1,2,3,··· (1a45) (4) λ l = l(l + 1) (2a45) yl(x) = Pl(x) (2a45) l = 0,1,2,3,··· (1a45) 12 a46 a44(25 a45) u(x,t) = C0t + D0 + ∞summationdisplay n=1 bracketleftBig Cn sin npil at+ Dn cos npil at bracketrightBig sin npil x u vextendsinglevextendsingle vextendsingle t=0 = D0 + ∞summationdisplay n=1 Dn sin npil x = u0cos2pil x = u02 bracketleftbigg 1 + cos 2pil x bracketrightbigg ? D0 = u02 , Dn = u02 δn2 ?u ?t vextendsinglevextendsingle vextendsingle t=0 = C0 + ∞summationdisplay n=1 Cnnl pia·sin nl pix ? C0 = Cn = 0 u(x,t) = u02 + u02 cos 2pil x cos 2pil at a45a47a48a49 X(x)a50a51a52 (2a45) a53a54a55a56 (2 a45) T(t)a50a51a52 (2a45) a57a58a59a60a61 λ n (2a45) Xn(x) (2a45) na50a62 a59 (1 a45) a63 a64 T n(t),T0(t) (2a45) a43a65 a64 (2 a45) a66a67a68 C 0 (2a45) Cn (2a45) D0,D2 (2a45) Dn (nnegationslash= 0,2) (2a45) a64 a69 (2 a45) 13 a70 a44(20 a45)a71a72a73a74a75 a60a61 a50 a57a58a76a68a77a78 u(r,θ) = ∞summationdisplay l=0 Rl(r)Pl(cosθ) 1 r2 d dr parenleftbigg r2dRldr parenrightbigg ? l(l + 1)r2 Rl = ?4pirδl1 Rl(0)a79 a54 R l(a) = 0 a80l = 1 a81a82R1(r) = A1r + B1r?2 ? 25pir3 a82 R1(0)a79 a54 ? B 1 = 0 R1(a) = 0 ? A1 = 25pia2 R1(r) = 25piparenleftbiga2 ?r2parenrightbig a80l negationslash= 1 a81a82Rl(r) = Alrl + Blr?l?1 a82 Rl(0)a79 a54 ? B l = 0 Rl(a) = 0 ? Al = 0 Rl(r) = 0, l negationslash= 1 a83a84 u(r,θ) = 25piparenleftbiga2 ?r2parenrightbigrcosθ a68a85a86a87a88a89 (2 a45) a64a61 a51a90 a88a89 (2 a45) a91a92a93φ a94a95 (2a45) a57a58a76a68 (2 a45) Rl(r)a96a97a50a98a45a51a52 (2a45) Rl(r)a96a97a50 a53a54a55a56 (2 a45) l = 1a81a50 a64 (3 a45) l negationslash= 1a81a50 a64 (3 a45) a64 a69 (2 a45) 14 a99 a44(15 a45)a51a90 a43 a100a101Bessel a76a68 a50a102 a68a86a103a69a104a105a106 a45 integraldisplay ∞ 0 e?αx2J0(x)xdx = ∞summationdisplay n=0 (?)n n!n! parenleftbigg1 2 parenrightbigg2n integraldisplay ∞ 0 e?αx2 x2n+1 dx = ∞summationdisplay n=0 (?)n n!n! parenleftbigg1 2 parenrightbigg2n 1 2 integraldisplay ∞ 0 e?αt tn dt = ∞summationdisplay n=0 (?)n n!n! parenleftbigg1 2 parenrightbigg2n+1 n! αn+1 = 12α ∞summationdisplay n=0 (?)n n! parenleftbigg 1 4α parenrightbiggn = 12αe?1/4α a51a90a50 a88a89a107a108 (3 a45) a109a110a106 a45 (2a45) a111a112Γa76a68a113a114a106 a45 (5a45) a115 a116 (5 a45) a51a90 a46 a117a118 a49 a106 a45a90a119a120 I(b) = integraldisplay ∞ 0 e?αx2J0(bx)xdx a121 Iprime(b) = ? integraldisplay ∞ 0 e?αx2J1(bx)x2 dx = 12αe?αx2 xJ1(bx) vextendsinglevextendsingle vextendsingle ∞ 0 ? b2α integraldisplay ∞ 0 e?αx2J0(bx)xdx = ? b2αI(b) a83a84 I(b) = Aexp braceleftbigg ?b 2 4α bracerightbigg , a122a123A = I(0) = 1/2α a119a124 integraldisplay ∞ 0 e?αx2J0(x)xdx = I(1) = 12α exp braceleftbigg ? 14α bracerightbigg . a51a90a50 a88a89a107a108 (3 a45) a125a126I(b) a50a98a45a51a52 (4a45) a64 a98a45a51a52 (4a45) a111a112a63a127a59a66a106 a45a128 a68 (2 a45) a115a126a83a115a106 a45 (2a45) 15 a129 a44(20 a45)a130 G(x,t;xprime,tprime) equaldotleftright G(x,p;xprime,tprime), a121a66a64a60a61 a48a131 ? ?? ?? pG(x,p;xprime,tprime)?κd2d xG(x,p;xprime,tprime) = e?ptprimeδ(x?xprime), G(x,p;xprime,tprime)vextendsinglevextendsinglex=0 = 0, G(x,p;xprime,tprime)vextendsinglevextendsinglex→∞ → 0. a64a132a133 G(x,p;xprime,tprime) = ? ?? ?? Asinh radicalbiggp κx, x < x prime, B exp braceleftbigg ? radicalbiggp κx bracerightbigg , x > xprime. a134a135a105a55a56 G(x,p;xprime,tprime) vextendsinglevextendsingle vextendsingle x=xprime+0 x=xprime?0 = 0, ?κ ddxG(x,p;xprime,tprime) vextendsinglevextendsingle vextendsingle x=xprime+0 x=xprime?0 = e?ptprime a136 B exp braceleftbigg ? radicalbiggp κx prime bracerightbigg ?Asinh radicalbiggp κx prime = 0, Bexp braceleftbigg ? radicalbiggp κx prime bracerightbigg + Acosh radicalbiggp κx prime = 1√ κp e ?ptprime, a137a84a66a126 a128 a68 A = 1√κpe?ptprime exp braceleftbigg ? radicalbiggp κx prime bracerightbigg , B = 1√κpe?ptprime sinh radicalbiggp κx prime. a138a139 a82 G(x,p;xprime,tprime) = ? ?? ?? 1√ κpe ?ptprime exp braceleftbigg ? radicalbiggp κx prime bracerightbigg sinh radicalbiggp κx, x < x prime, 1√ κpe ?ptprime exp braceleftbigg ? radicalbiggp κx bracerightbigg sinh radicalbiggp κx prime. x > xprime, a140a86 a82 a136a133a141a142 G(x,t;xprime,tprime) = 12radicalbigκpi(t?tprime) braceleftbigg exp bracketleftbigg ?(x?x prime)2 4κ(t?tprime) bracketrightbigg ?exp bracketleftbigg ?(x+ x prime)2 4κ(t?tprime) bracketrightbiggbracerightbigg η(t?tprime). Laplacea48a143a82a128a98a45a51a52 a66a64a60a61 (4 a45) a64 a51a52a144 x < xprime (2a45) x > xprime (2a45) a135a105a55a56 a144a145 a69 (4 a45) a146a147 (2 a45) a141 a142 a144 η(t?tprime) (2a45) a140a86 a82 a133a148a149a146a147 (4 a45) 16 a0 a1 III a5 a254 a255 a0 a1 a150 (B) a6 a7 a151 a152a153a154a155a156a157a158a159a160a161a162a162a163 a44a45 a16430 a49a165 a186a166 a48 1. a167a168a52a51 x x2 + y2 ?i y x2 + y2 a70a169a170 a181a33 a164 a249a138a171a56a33 a51a165 a45 a70a169a170 a77a78 2. a149a150 a90 a49 contintegraldisplay |z|=2 cosz z3 dz 3. a18a171a52a51f(z) = cos zz ?1 a70a56a182a215a171a115a232a87a164a172a173∞a87a165a170a80a104a51 4. a18 a171 a52a51f(z) = lnzsinpiz a70 a56a182a215a171a115a232a87 a164 a172a173∞a87 a165a170 a80a104 a51 5. a18 a171a169a174zd 2w dz2 + dw dz ?w = 0a70 a56a182a215 a129 a89a232a87 a170 a80a175a176 a126 a45 a16425 a49a165 a138a77 a151a152 a147a77 a22 a48 ?2u ?t2 ?a 2?2u ?x2 = 0, 0 < x < l,t > 0 ?u ?x vextendsinglevextendsingle vextendsinglevextendsingle x=0 = ?u?x vextendsinglevextendsingle vextendsinglevextendsingle x=l = 0, t ≥ 0 uvextendsinglevextendsinglet=0 = x, ?u?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = 0, 0 ≤ x ≤ l a56a24a a53a127 a128a80a177x a88ta178a29a179 a80 a184a51a180 a139 a45 a16415 a49a165 a138a77a221a181a26a182 a76 a80a147a77 a22 a48 ?2u = 0, a < r < b, 0 < θ < pi, 0 < ? < 2pi uvextendsinglevextendsingler=a = 0, u|r=b = cos2 θ, 0 ≤ θ ≤ pi, 0 ≤ ? ≤ 2pi a148 a45 a16415 a49a165 a138a77 a209a76 a147a77 a22 a48 ?u ?t ?κ? 2u = 0, 0 < r < a,0 < ? < 2pi,t > 0 uvextendsinglevextendsingler=0a215a23, uvextendsinglevextendsingler=a = sin?, 0 ≤ ? ≤ 2pi,t ≥ 0 uvextendsinglevextendsinglet=0 = 0, 0 ≤ r ≤ a,0 ≤ ? ≤ 2pi a56a24κ a53a127 a128a80a177ra27? a88ta178a29a179 a80 a184a51a180 a153 a45 a16415 a49a165 a138 a44a183 a208a29 a23 Helmholtza169a174a184a185a80 Green a52a51G(x, xprime). a186a187a188 ? ?? ?? d2G(x,xprime) dx2 + k 2G(x,xprime) = ?δ(x?xprime) 0 < x,xprime < ∞ Gvextendsinglevextendsinglex=0 = 0, Ga70x →∞a247a215a171a189a190a164a206a177a191a58a192a79 e?iωt a165 a56a24k a53a127 a128a80a177x a88xprime a178a29a179 a80 a184a51a180 18 a193a194a195a196a197a198a199 a44a45 a16430 a49a165 1. xx2 + y2 ?i yx2 + y2 = 1z a70a200z = 0a201 a80a202a203a80a172 a242a243 a164 a172a173∞a87 a165a132 a181a33 a45 a77a78a27 parenleftbigg1 z parenrightbiggprime = ? 1z2 a181a33 a74a75 (2a49) a77a78 a74a75 (2a49) a33 a51 a9a10 (2 a49) 2. contintegraldisplay |z|=2 cosz z3 dz = ?pii (6a49) 3. z = 1a53f(z) = cos zz ?1 a80a20a204 a232a87a27 resf(1) = ?sin1a63 (3 a49) z = ∞a53f(z) = cos zz ?1 a80a77a78a87a27 resf(∞) = sin1 (3a49) 4. z = 0a88z = ∞a53lnz a80a199 a87a27 a251a205a129a134a131a206 z = 0a191z = ∞a196 a197a85a27 a146a147lnz vextendsinglevextendsingle z=1 = 0 a27a89 a196 a197a85 a45 a146a147a198 a55a49a199 (1a49) z = 1a53 a181a207a232a87a27 resf(1) = 0a27 (2 a49) z = k, k = ?1, ±2, ±3, ··· a53 a44a250 a229a87a27 (1 a49) resf(k) = ? ??? ?? ??? ?? (?1)k pi lnk, k = 2, 3, 4, ··· i k = ?1 (?1)k pi ln|k|?(?1) ki, k = ?2, ?3, ?4, ··· (1a49) (1a49) a146a147lnzvextendsinglevextendsingle z=1 = 2npii, n = ±1, ±2, ±3, ··· a27a89 z = k, k = ±1, ±2, ±3, ··· a178a53 a44a250 a229a87a27 resf(k) = ?? ??? ??? ? ??? ??? ?? ?2ni k = 1 (?1)k pi lnk + (?1) k2ni, k = 2, 3, 4, ··· ?(2n?1)i k = ?1 (?1)k pi ln|k|+ (?1) k(2n?1)i, k = ?2, ?3, ?4, ··· 5. z = 0a53 a169a174zd 2w dz2 + dw dz ?w = 0 a80a208 a44 a129 a89a232a87a27 (3 a49) a169a174zd 2w dz2 + dw dz ?w = 0a70z = 0 a87a80a175a176 a79ρ1 = ρ2 = 0 (3a49) a126 a45 a16425 a49a165 a209 a44a210 a77 a79 u(x, t) = C0t + D0 + ∞summationdisplay n=1 parenleftBig Cn sin npil at + Dn cos npil at parenrightBig cos npil x, uvextendsinglevextendsinglet=0 = D0 + ∞summationdisplay n=1 Dn cos npil atcos npil x = x, 19 ?u ?t vextendsinglevextendsingle vextendsinglevextendsingle t=0 = C0 + ∞summationdisplay n=1 Cnnpil acos npil x = 0, =? ? ??? ??? ? ??? ???? Cn = 0 n = 0, 1, 2, ··· D0 = 1l integraldisplay l 0 xdx = l2 Dn = 2l integraldisplay l 0 xcos npil xdx = 2ln2pi2 [(?1)n ?1] n = 1, 2, 3, ··· a211a57 a22 a48 a80a77 a79 u(x, t) = l2 ? 4lpi2 ∞summationdisplay n=0 1 (2n + 1)2 cos 2n + 1 l piatcos 2n + 1 l pix λ = 0a50a185a80a212a77 (4a49) a20a21a52a51 (4a49) Tn (4a49) a213a246a214 (2 a49) a213a215a216 (2 a49) Cn = 0 (1a49) D0 (3a49) Dn (3a49) a77a10 (2 a49) a139 a45 a209 a217 a28 a26a218a176a219a68a178a169a174a183 a233a220 a252a253a128 ua177? a29a179 a27a211a169a174 a79 1 r2 ? ?r parenleftbigg r2?u?r parenrightbigg + 1r2 sinθ ??θ parenleftbigg sinθ?u?θ parenrightbigg = 0 a44a210 a77 a79 u(r, θ) = ∞summationdisplay l=0 parenleftbigA lrl + Blr?l?1 parenrightbigP l(cosθ) a221a222 a220 a23a252a253a27 uvextendsinglevextendsingler=a = ∞summationdisplay l=0 parenleftbigA lal + Bla?l?1 parenrightbigP l(cosθ) = 0 uvextendsinglevextendsingler=b = ∞summationdisplay l=0 parenleftbigA lbl + Blb?l?1 parenrightbigP l(cosθ) = cos2 θ a180 a190 Alal + Bla?l?1 = 0 Albl + Blb?l?1 = 2l + 12 integraldisplay pi 0 cos2 θPl(cosθ)sinθdθ = 2l + 12 parenleftbigg1 3δl,0 + 2 15δl,2 parenrightbigg 20 =? ? ??? ??? ??? ??? ??? ??? A0 = b3(b?a) B0 = ? ab3(b?a) A2 = 2b 3 3(b5 ?a5) B2 = ? 2a 5b3 3(b5 ?a5) a211a57 a22 a48 a80a77 a79 u(r, θ) = b3(b?a) parenleftBig 1? ar parenrightBig + 2a 2b3 3(b5 ?a5) parenleftbiggr2 a2 ? a3 r3 parenrightbigg P2(cosθ) ua177?a29a179 (1a49) a26a218a176a219 a151 Laplacea169a174 (2a49) a44a210 a77 a59a20a21a52a51 (2a49) Rl(r) (1a49) a223a169 (2 a49) Al a177Bl a80a169a174a224 (1a49) Al a177Bl a80a169a174a225 (1a49) A0 (1a49) B0 (1a49) A2 (1a49) B2 (1a49) a77a10 (1 a49) a148 a45 a209 a69u(r, ?, t) = v(r, t)sin?a27a89v(r, t)a187a188 a147a77 a22 a48 ?v ?t ?κ bracketleftbigg1 r ? ?r parenleftbigg r?v?r parenrightbigg ? vr2 bracketrightbigg = 0, 0 < r < a,t > 0 vvextendsinglevextendsingler=0a215a23, vvextendsinglevextendsingler=a = 1, t > 0 vvextendsinglevextendsinglet=0 = 0, 0 < r < a a69v(r, t) = ra + w(r, t)a27a89w(r, t)a187a188 a147a77 a22 a48 ?w ?t ?κ bracketleftbigg1 r ? ?r parenleftbigg r?w?r parenrightbigg ? wr2 bracketrightbigg = 0, 0 < r < a,t > 0 wvextendsinglevextendsingler=0a215a23, wvextendsinglevextendsingler=a = 0, t > 0 wvextendsinglevextendsinglet=0 = ?ra, 0 < r < a a20a21a55a22 a48 1 r d dr parenleftbigg rdR(r)dr parenrightbigg + parenleftbigg λ? 1r2 parenrightbigg R(r) = 0 21 R(0)a215a23, R(a) = 0 a80a77 a79 λi = parenleftBigμ1i a parenrightBig2 , Ri(r) = J1 parenleftBigμ1i a r parenrightBig , i = 1, 2, 3, ··· a56a24μ 1i a53J1(x) a80a226i a248 a129a27 a87 a180 a58a179w(r, t)a80 a44a210 a77 a79 w(r, t) = ∞summationdisplay i=1 CiJ1 parenleftBigμ1i a r parenrightBig e?κ(μ1i/a)2t a221a222a213a252a253 wvextendsinglevextendsinglet=0 = ∞summationdisplay i=1 CiJ1 parenleftBigμ1i a r parenrightBig = ?ra a178a179 a190 Ci = integraldisplay a 0 ?raJ1 parenleftBigμ1i a r parenrightBig rdr integraldisplay a 0 J21 parenleftBigμ1i a r parenrightBig rdr = ? 1a parenleftbigg a μ1i parenrightbigg3 x2J2(x) vextendsinglevextendsingle vextendsinglevextendsingle vextendsingle x=μ1i x=0 a2 2 J prime12 (μ1i) = 2μ 1iJ0(μ1i) a182a183a57 a22 a48 a80a77 a79 u(r, ?, t) = ra sin? + 2 ∞summationdisplay i=1 1 μ1iJ0(μ1i)J1 parenleftBigμ1i a r parenrightBig e?κ(μ1i/a)2t sin?. a242a243 a229a218a176 a151? 2 a80a11a227a10 (2 a49) a228a229a185 a52a51 (2a49) w(r, t)a80a147a77a22a48 (2a49) a20a21a55 a177 a20a21a52a51 (2a49) Ti(t) (1a49) Ci a80a11a227a10 (2a49) a223a169 (1 a49) a49 a192 (2 a49) a77a10 (1 a49) a153 a45 a209 a59a230a231a232 x negationslash= xprime a177a233 a169a174a234 d2G(x,xprime) dx2 + k 2G(x,xprime) = 0 22 G(x, xprime) = ? ?? ?? Asinkx + B coskx x < xprime Ceikx + De?ikx x > xprime (2a235) (2a235) G(x, xprime)vextendsinglevextendsinglex=0 = 0 =? B = 0 (2a235) Ga236x →∞a247a215a171a189a190a164a206a177a191a237 a192a234 e?iωt a165 =? D = 0 (2a235) G(x, xprime)vextendsinglevextendsinglex=xprime? = G(x, xprime)vextendsinglevextendsinglex=xprime+ =? Asinkxprime = Ceikxprime (2a235) dG(x, xprime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=xprime+ ? dG(x, x prime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=xprime? = ?1 =? ikCeikxprime ?Akcoskxprime = ?1 (2a235) =? ? ?? ?? A = 1keikxprime x < xprime C = 1k sinkxprime x > xprime G(x, xprime) = ? ?? ?? 1 ke ikxprime sinkx x < xprime 1 k sinkx primeeikx x > xprime (1a235) (3a235) (3a235) a230a231a238 a239xa196Laplacea240a241a233a242 G(x, xprime) equaldotleftright g(p, xprime) a243 dG(x, xprime) dx equaldotleftright pg(p, x prime) (2 a235) d2G(x, xprime) dx2 equaldotleftright p 2g(p, xprime)? dG(x, xprime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=0 (2a235) a244a245a246a247a174 a240 a234 p2g(p, xprime)? dG(x, x prime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=0 + k2g(p, xprime) = ?e?pxprime (2a235) g(p, xprime) = 1p2 + k2 braceleftbiggdG(x, xprime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=0 ?e?pxprime bracerightbigg (2a235) a248a249a250 G(x, xprime) = dG(x, x prime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=0 1 k sinkxη(x)? 1 k sink(x?x prime)η(x?xprime) (2 a235) Ga236x →∞a251a252a253 a189a190 a164 a254a255 a191a237 a192a234 e?iωt a165 =? dG(x, x prime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=0 1 k sinkx? 1 k sink(x?x prime) ∝ eikx (2 a235) =? dG(x, x prime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=0 ?coskxprime = isinkxprime (1a235) =? dG(x, x prime) dx vextendsinglevextendsingle vextendsinglevextendsingle x=0 = coskxprime + isinkxprime = eikxprime (1a235) a244a245 G(x, xprime) = ?? ? ?? 1 ke ikxprime sinkx x < xprime 1 k sinkx primeeikx x > xprime (1a235) 23