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Luke Y.L. — Special Functions and Their Approximations. Volume II

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Название: Special Functions and Their Approximations. Volume II

Автор: Luke Y.L.

Аннотация:

These volumes are designed to provide scientific workers with a self-contained and unified development for many of the mathematical functions which arise in applied problems, as well as the attendant mathematical theory for their approximations. These functions are often called the special functions of mathematical physics or more simply the special functions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1969

Количество страниц: 506

Добавлена в каталог: 17.11.2013

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Предметный указатель

$\psi$ -function      see "Hypergeometric function (confluent)"
$\tau$ -Method      II 66—91
Airy functions      I 205 217 see "Approximation" "Expansion"
Anger — Weber functions      I 218 II 39 54 55
Approximation      II 287—291
Approximation, Airy functions      I 217 see
Approximation, ber, bei      I 216 see
Approximation, Bessel functions, $2(v+1)I_{v+1}(z)/zI_{\nu}(z)$ , RFP      II 82 83
Approximation, Bessel functions, $e^{-z(1-r)}K_{n}(rz)/K^{'}_{n}(z)$ , RF      II 259—265
Approximation, Bessel functions, $H_{\nu}^{(1)}(x), BCRF      II 290 291
Approximation, Bessel functions, $H_{\nu}^{(1)}(z)=J_{\nu}(z) + iY_{\nu}(z)$       II 229—232 436—448
Approximation, Bessel functions, $J_{0}(z)$ , by approximate inversion of its Laplace transform      II 265
Approximation, Bessel functions, $J_{n}(w)$ , $w=(x^{2}-y^{2})^{1/2}$ , TR      II 220
Approximation, Bessel functions, $J_{n}(z)$ , TR      II 218 219 265
Approximation, Bessel functions, $J_{\nu}(x)$ , $Y_{\nu}(x)$ , $I_{\nu}(x)$ , $K_{\nu}(x)$ , BCP      II 290 291
Approximation, Bessel functions, $J_{\nu}(x)$ , $Y_{\nu}(x)$ , $I_{\nu}(x)$ , $K_{\nu}(x)$ , R      II 162 165
Approximation, Bessel functions, $J_{\nu}(x)$ , $Y_{\nu}(x)$ , zeros of, RF      II 228—233
Approximation, Bessel functions, $K^{'}_{\nu}(z)$ , zeros of, RF      II 228—233 259 264
Approximation, Bessel functions, $K^{'}_{\nu}(z)/K_{\nu}(z)$ , RFP      II 83 84
Approximation, Bessel functions, $K_{\nu}(z)$ , P      II 111—116
Approximation, Bessel functions, $K_{\nu}(z)$ , RF      II 111—116 229 230 436—448
Approximation, Bessel functions, $K_{\nu}(z)$ , TR      II 221—225
Approximation, Bessel functions, Airy functions      I 217
Approximation, Bessel functions, ber, bei, ker, kei, and derivatives, BCP      II 291 I 216
Approximation, best in sense of, Chebyshev      I 303—305 II 68
Approximation, best in sense of, least square      I 270
Approximation, best in sense of, mean square      I 268 305
Approximation, best in sense of, other      II 72 73
Approximation, beta function (incomplete), P, RF, RFP      II 178
Approximation, binomial function, $(1 + 1/z)^{-\sigma$ }$, RFP      II 179
Approximation, binomial function, $(1 + 1/z)^{\pm 1/2$ }$, Newton — Raphson process      II 183 184
Approximation, binomial function, $(1 + 1/z)^{\pm 1/2$ }$, Newton — Raphson process, P      II 387 388
Approximation, binomial function, $(1 + 1/z)^{\pm 1/2$ }$, Newton — Raphson process, RFP      II 182—185 269 395
Approximation, binomial function, $(1 + 1/z)^{\pm 1/2$ }$, Newton — Raphson process, sum of partial fractions      II 183 269
Approximation, binomial function, $(1 + 1/z)^{\pm 1/3}$ , RFP      II 397
Approximation, binomial function, $(1 + x)^{-1}$ , BCP      II 288
Approximation, binomial function, $x^{x}$ , $x^{1/x}$ , RF      II 180 181
Approximation, binomial function, $z^{1/2}$ , RFP      II 185 396
Approximation, binomial function, |x|, BCP      II 288
Approximation, binomial function, |x|, RFP      II 185 396
Approximation, cosine      see "Approximation sine
Approximation, cosine integral      see "Approximation sine
Approximation, Debye functions, RFP      II 273—281
Approximation, derivatives, repeated, R      II 165
Approximation, differential equation, solution of, first order Riccati, RFP      II 77—86
Approximation, differential equation, solution of, generalized second order Riccati, RFP      II 86—91
Approximation, differential equations, integral equations, and other functional equations, solution of, collocation, or selected points      II 69
Approximation, differential equations, integral equations, and other functional equations, solution of, P      II 66—69
Approximation, differential equations, integral equations, and other functional equations, solution of, RF      II 66—69 256—269
Approximation, Duffing's equation, RFP      II 89 90
Approximation, economized      II 66 105 106 111
Approximation, elliptic functions (Jacobian), sn(x,k), cn(x,k), RFP      II 90 91
Approximation, elliptic integrals, first two kinds, complete, BCP      II 291
Approximation, elliptic integrals, three kinds, complete and incomplete, RFP      II 85 86 269—273
Approximation, elliptic integrals, three kinds, complete, TR      II 220 221
Approximation, error functions, $e^{z^{2}}\int^{z}_{0} e^{-t^{2}} dt$ , zeros of, RFP      II 227 228
Approximation, error functions, $\int^{z}_{0} e^{\pm t^{2}} dt$ , RFP      II 422—429
Approximation, error functions, $\int^{\infty}_{z} e^{-t^{2}} dt$ , P and RF      II 106 107
Approximation, error functions, $\int^{\infty}_{z} e^{-t^{2}} dt$ , RFP      II 206 207 213 422 423 430—435
Approximation, error functions, $\int^{\infty}_{z} e^{-t^{2}} dt$ , TR      II 225 226
Approximation, error functions, repeated integrals, R      II 165
Approximation, exponential function, $10^{z}$ , BCP      II 288
Approximation, exponential function, $e^{z}$ , BCP      II 288
Approximation, exponential function, $e^{z}$ , P and RF      II 69—73
Approximation, exponential function, $e^{z}$ , RFP      II 73—75 83 191—193 256 399
Approximation, exponential function, $z(e^{z}-1)^{-1}$ , RFP      II 273 274
Approximation, exponential integral, $\int^{z}_{0} t^{-1}(1-e^{-t}) dt$ , P and RF      II 196—198 403 411—414
Approximation, exponential integral, $\int^{z}_{0} t^{-1}(1-e^{-t}) dt$ , RFP      II 197 402 405—408
Approximation, exponential integral, $\int^{\infty}_{z} t^{-1} e^{-t} dt$ , P and RF      II 106 107
Approximation, exponential integral, $\int^{\infty}_{z} t^{-1} e^{-t} dt$ , RFP      II 206 207 212 213 403 404 415—421
Approximation, exponential integral, ratio involving, RFP      II 265—267
Approximation, exponential integral, repeated integrals, R      II 165
Approximation, Fermi — Dirac integrals, BCP and BCRF      II 291
Approximation, Fresnel integrals, P      II 290
Approximation, Fresnel integrals, RFP      II 207 422—435
Approximation, Fresnel integrals, TR      II 226
Approximation, G-function, P and RF      II 118—132
Approximation, gamma function      see also "Approximation incomplete
Approximation, gamma function, $ln\Gamma(x)$ , BCRF      II 288
Approximation, gamma function, $\Gamma(x+2)$ , BCP      II 287
Approximation, gamma function, logarithmic derivative, $\psi(x+1)$ , BCP      II 287
Approximation, hypergeometric function, confluent, the $\psi$ -function, $\psi(1;1+v,z)$       see "Approximation incomplete z)$"/>"
Approximation, hypergeometric function, confluent, the $\psi$ -function, $\psi(a,c;z)$ , P and RF      II 111—116
Approximation, hypergeometric function, confluent, the $\psi$ -function, $\psi(d,d-b;z)/\psi(d,d+1-b;z)$ , RFP      II 202
Approximation, hypergeometric function, confluent, the $_{1}F_{1}$ , $_{1}F_{1}(1;\nu +1;-z)$       see "Approximation incomplete z)$"/>"
Approximation, hypergeometric function, confluent, the $_{1}F_{1}$ , u'(z)/u(z), $u(z)=_{1}F_{1}(a,c;z)$ , RFP      II 80—83 196
Approximation, hypergeometric function, Gaussian, the $_{2}F_{1}$ , $_{2}F_{1}(1,\sigma;\rho + 1;-z)$       II 167—185
Approximation, hypergeometric function, Gaussian, the $_{2}F_{1}$ , $_{2}F_{1}(1,\sigma;\rho + 1;-z)$ , P and RF      II 167—169 175—176
Approximation, hypergeometric function, Gaussian, the $_{2}F_{1}$ , $_{2}F_{1}(1,\sigma;\rho + 1;-z)$ , RFP      II 84 169—173 176—178
Approximation, hypergeometric function, Gaussian, the $_{2}F_{1}$ , u'(z)/u(z), $u(z)=_{2}F_{1}(a,b;c;z)$ , RFP      II 173—175
Approximation, hypergeometric function, generalized, the $_{p}F_{q}$ , P and RF      II 92—132 150
Approximation, incomplete gamma function      II 186—213 see sine "Approximation error "Approximation exponential "Approximation Fresnel "Approximation hypergeometric "Approximation confluent"
Approximation, incomplete gamma function, $\gamma(\nu,z)$ , P and RF      II 186—188
Approximation, incomplete gamma function, $\gamma(\nu,z)$ , RFP      II 188—194
Approximation, integrals by use of orthogonal polynomials      II 243—245
Approximation, integrals, Simpson's rule and other Newton — Cotes formulas      II 218
Approximation, integrals, TR      II 214—226 see Laplace "Approximation sine
Approximation, ker, kei      I 216 see
Approximation, Laplace integral or transform, numerical integration      II 245
Approximation, Laplace integral or transform, P and RF      II 106—118
Approximation, Laplace transform, inversion of, numerical integration      II 251—255
Approximation, Laplace transform, inversion of, orthogonal polynomials      II 249—251
Approximation, Laplace transform, inversion of, RF      II 255—269
Approximation, Laplace transform, inversion of, TR      II 225
Approximation, Legendre functions, R      II 165
Approximation, logarithm, $ln{z+(z^{2}+1)^{1/2}}$ , RFP      II 393 394
Approximation, logarithm, ln(l+z), BCP      II 288
Approximation, logarithm, ln(l+z), P      II 385 386
Approximation, logarithm, ln(l+z), RFP      II 389 390
Approximation, logarithm, ln[(1+z)/(1-z)], RFP      II 391 392
Approximation, Lommel functions, P and RF      II 116—118
Approximation, Mellin — Barnes integral, TR      II 225
Approximation, Painleve's first transcendent, RFP      II 89
Approximation, Power series or ratio of two power series, RFP      II 76
Approximation, psi( $\psi$ )-function or logarithmic derivative of the gamma function      see "Approximation gamma
Approximation, Riccati differential equations, solutions of, RFP      II 77—91
Approximation, sine and cosine, integrals, $\int^{3\pi/2}_{0} t^{-\alpha}cos t dt$ , zeros of, RFP      II 228
Approximation, sine and cosine, integrals, $\int^{z}_{0} t^{-1}sin t dt$ , $\int^{z}_{0} t^{-1}(1-cos t) dt$ , RF      II 196—198 402—408 415—421
Approximation, sine and cosine, integrals, $\int^{z}_{0} t^{-1}sin t dt$ , $\int^{z}_{0} t^{-1}(1-cos t) dt$ , RFP      II 197 403 404 409 410
Approximation, sine and cosine, integrals, $\int^{\infty}_{z} t^{-1}sin t dt$ , $\int^{\infty}_{x} t^{-1}cos t dt$ , RF      II 207 403 404 415—421
Approximation, sine and cosine, inverse, arc sin z, RFP      II 393 394
Approximation, sine and cosine, sin z, cos z, BCP      II 288
Approximation, sine and cosine, sin z, cos z, RF      II 74 193 399—401
Approximation, Struve functions, P and RF      II 116—118
Approximation, Struve functions, RF      II 449—452
Approximation, tangent, inverse, arc tan z, BCP      II 288
Approximation, tangent, inverse, arc tan z, P      II 385 386
Approximation, tangent, inverse, arc tan z, RFP      II 391 392
Approximation, tangent, tan z, BCP      II 288
Approximation, tangent, tan z, RF      II 399—401
Approximation, tangent, tan z, RFP      II 83
Approximation, Whittaker function      see "Hypergeometric function confluent"
Associated Bessel function      I 219
Asymptotic expansion      I 1—7 see asymptotic"
Asymptotic expansion, elementary properties      I 3 4
Asymptotic expansion, expansion definition      I 2
Asymptotic expansion, Watson's lemma      I 4—7
Basic series      I 291 II 283
bcp      see "Best polynomial in the Chebyshev sense"
BCRF      see "Best rational function in the Chebyshev sense"
Ber, bei functions      I 216 II 291 347—351 see "Approximation" "Expansion"
Bernoulli and generalized Bernoulli polynomials, definition and elementary properties      I 18—22
Bernoulli and generalized Bernoulli polynomials, Fourier series for $B_{n}(x)$       I 23
Bernoulli and generalized Bernoulli polynomials, integrals of      I 22 23
Bernoulli and generalized Bernoulli polynomials, recurrence formulas      I 20 22 35
Bernoulli and generalized Bernoulli polynomials, table of $B^{(2\rho)}_{2n}(\rho)$       I 34
Bernoulli and generalized Bernoulli polynomials, table of $B^{(a)}_{n}(x)$       I 19
Bernoulli and generalized Bernoulli polynomials, table of $B_{n}(x)$       I 20
Bernoulli numbers, definition      I 19 20
Bernoulli numbers, expansion of cot z, tan z, csc z, ln cos z in series involving      I 23

Bernoulli numbers, table of      I 20
Bessel functions, Airy functions, connection with      I 217
Bessel functions, approximation      see "Approximation"
Bessel functions, associated      I 219
Bessel functions, asymptotic expansions      I 203—205 215
Bessel functions, ber, bei functions, connection with      I 216
Bessel functions, computation by use of recurrence formulas      II 162
Bessel functions, confluent hypergeometric function, connection with      I 120 135 213
Bessel functions, definitions, connecting relations, power series      I 39 40 212 213
Bessel functions, derivatives with respect to order      II 52 53
Bessel functions, derivatives with respect to order at half an odd integer      I 216
Bessel functions, difference-differential properties      I 48 214 II 232
Bessel functions, expansions, expansions in series of      see "Expansion"
Bessel functions, G-functions, connection with      I 226—234
Bessel functions, integrals involving      I 115 165 177 219 220 287 289 II 33 34 39 53 54 see
Bessel functions, Jacobi polynomial, connection with      I 52
Bessel functions, ker, kei functions, connection with      I 216
Bessel functions, products      I 216 228—230 232—234 II 50—53
Bessel functions, Wronskians      I 214 215
Bessel functions, zeros of      I 205 II 228—234
Bessel polynomials      I 274 II 194 253
Best approximation      see "Approximation"
Beta function, complete      I 15 16 18
Beta function, incomplete      I 210 II 178
Beta transform      I 58 59 170—176 286
BF      see "Bessel function or
Binomial coefficient      I 9 35
Binomial function      I 38 40 49 209 210 II 179—185 see binomial "Expansion powers
Catalan's constant      II 293
Chebyshev polynomials of the first kind      I 308—312 see "Approximation"
Chebyshev polynomials of the first kind, connection with Pade approximation for square root      II 183
Chebyshev polynomials of the first kind, definition and basic properties      I 273 296 297 300 301
Chebyshev polynomials of the first kind, difference-differential properties      I 297—299 301 302 316 321 324
Chebyshev polynomials of the first kind, evaluation of series of, by use of backward recurrence formula      I 325—329
Chebyshev polynomials of the first kind, expansions in series of (based on orthogonality property with respect to integration)      see also "Expansion"
Chebyshev polynomials of the first kind, expansions in series of (based on orthogonality property with respect to integration), asymptotic estimate of coefficients      I 293—296
Chebyshev polynomials of the first kind, expansions in series of (based on orthogonality property with respect to integration), evaluation of coefficients      I 286—293
Chebyshev polynomials of the first kind, expansions in series of, differential and integral properties      I 314—325
Chebyshev polynomials of the first kind, expressed in powers of x      II 294 296 297
Chebyshev polynomials of the first kind, integrals involving      I 293—295 299 300—302 316—325
Chebyshev polynomials of the first kind, Jacobi polynomial, connection with      I 273 296
Chebyshev polynomials of the first kind, minimax and mean square properties      I 303—307
Chebyshev polynomials of the first kind, orthogonality property with respect to integration      I 273 299 301
Chebyshev polynomials of the first kind, orthogonality property with respect to summation      I 307 310 311
Chebyshev polynomials of the first kind, powers of x expressed in terms of      II 295 298
Chebyshev polynomials of the first kind, solution of differential and integral equations by expansions in series of      II 66—69 234—241
Chebyshev polynomials of the second kind      I 312—314 see
Chebyshev polynomials of the second kind, connection with Pade approximation for square root      II 183
Chebyshev polynomials of the second kind, definition and basic properties      I 273 296—300
Chebyshev polynomials of the second kind, integrals involving      I 299 300
Chebyshev polynomials of the second kind, Jacobi polynomial, connection with      I 273 296
Chebyshev polynomials of the second kind, orthogonality property with respect to integration      I 273 299
Chebyshev polynomials of the second kind, orthogonality property with respect to summation      I 312 313
Chebyshev, best approximation in sense of      I 303
Christoffel — Darboux formulas      I 272
Collocation      II 69
Computation, by use of recurrence formulas      I 317—319 325—329 II 26—28 159—166 282—285
Confluence principle and theorems      I 48—57
Constants, mathematical, table of      II 292 293
Continued fractions      see "Approximation RFP"
Continued fractions, convergence      II 80
Cosecant      I 23 210
Cosine      see "Sine and cosine"
Cosine integral      see "Sine and cosine integrals"
Cotangent      see "Tangent"
Coulomb wave functions      I 135 212
cp      see "Chebyshev polynomials of the first kind"
Cylinder function      I 204 213
D operator      I 24—26
Darboux, method of generating functions      I 254—259
Debye functions      II 273—281
Delta ( $\delta$ ) operator      I 24—26
Difference equations (or recurrence formulas), use of in computation      I 317—319 325—329 II 159—166 282—285
Differential, integral and functional equations      see "Approximation" "Expansion"
Dilogarithm      II 310
Dixon's theorem      I 104
Duffing's equation      II 89 90
E-function      I 148
Economized approximation      II 66 105 106 111
Elliptic functions      II 90 91
Elliptic integrals      I 211 II 91 see "Expansion"
Error functions      I 135 223 224 II 195 201 see "Expansion"
Euler — Mascheroni constant, $\gamma$       I 9 II 293
Evaluation of expansions in series of functions where functions satisfy a linear finite difference equation      I 325—329
expansion      II 287—291
Expansion, $\psi$ -function, CP      II 25 28 238—240
Expansion, $\psi$ -function, G-functions      II 25
Expansion, Airy functions      I 217 see
Expansion, Anger — Weber functions, BF      II 39 54 55
Expansion, Anger — Weber functions, CP      II 39
Expansion, ber, bei      I 216 see
Expansion, Bessel functions, $H^{(1)}_{\nu}(z)$ , $H^{(2)}_{\nu}(z)$ CP      II 27
Expansion, Bessel functions, $J_{n}(z)$ , $Y_{n}(z)$ , $I_{n}(z)$ , $K_{n}(z)$ , BF      II 37 49 50 52 53
Expansion, Bessel functions, $J_{n}(z)$ , $Y_{n}(z)$ , $I_{n}(z)$ , $K_{n}(z)$ , CP      II 37 38 290 331—333 338—341 347—351
Expansion, Bessel functions, $J_{n}(z)$ , $Y_{n}(z)$ , $I_{n}(z)$ , $K_{n}(z)$ , GP      II 38
Expansion, Bessel functions, $J_{\mu}(z)$ , $J_{\nu}(z)$ , BF      II 50—53
Expansion, Bessel functions, $J_{\nu}(z)$ , $I_{\nu}(z)$ , BF      II 49 50
Expansion, Bessel functions, $J_{\nu}(z)$ , $I_{\nu}(z)$ , CP      II 35 37 234—238 290 352—356 359—367
Expansion, Bessel functions, $J_{\nu}(z)$ , $I_{\nu}(z)$ , GP      II 37
Expansion, Bessel functions, $J_{\nu}(z)$ , $Y_{\nu}(z)$ , zeros of, CP      II 233 234
Expansion, Bessel functions, $Y_{\nu}(z)$ , $K_{\nu}(z)$ , BF      II 49
Expansion, Bessel functions, $Y_{\nu}(z)$ , $K_{\nu}(z)$ , CP      II 240 290 352—356 359—367
Expansion, Bessel functions, $\partial J_{\nu}(z)/\partial \nu$ , $\partial I_{\nu}(z)/\partial \nu$ , BF      II 52 53
Expansion, Bessel functions, Airy functions      I 217 see
Expansion, Bessel functions, ber, bei, ker, kei, and derivatives, CP      II 347—351 I 216
Expansion, Bessel functions, integrals involving, $\int^{x}_{0} J_{0}(t) dt$ , $\int^{x}_{0} t^{-1}[1-J_{0}(t)] dt, BF      II 39 54
Expansion, Bessel functions, integrals involving, $\int^{x}_{0} J_{0}(t) dt$ , $\int^{x}_{0} t^{-1}[1-J_{0}(t)] dt, CP      II 39 334
Expansion, Bessel functions, integrals involving, $\int^{z}_{0} t^{\mu} J_{\nu}(t) dt$ , $\int^{z}_{0} e^{-t} t^{\mu} I_{\nu}(t) dt$ , BF      II 33 34 53 54
Expansion, Bessel functions, integrals involving, $\int^{\infty}_{z} t^{\mu} K_{\nu}(t) dt$ , CP      II 240—242
Expansion, Bessel functions, integrals involving, $\int_{0}^{x} I_{0}(t) dt$ , $\int_{0}^{x} t^{-1}[I_{0}(t)-1] dt$ , $\int_{0}^{x} K_{0}(t) dt$ , $\int_{x}^{\infty} t^{-m} K_{0}(t) dt$ , m=0,1, CP      II 242 342—345
Expansion, Bessel functions, integrals involving, $\int_{0}^{x} I_{\nu}(t) dt$ , $\int_{x}^{\infty} K_{\nu}(t) dt$ , $\nu=\pm\frac{1}{3}$ , CP      II 368 369
Expansion, Bessel functions, integrals involving, $\int_{0}^{x} J_{0}(t) dt$ , $\int_{0}^{x} Y_{0}(t) dt$ , $\int_{x}^{\infty} t^{-m} J_{0}(t) dt$ , $\int_{x}^{\infty} t^{-m} Y_{0}(t) dt$ , m=0,1, CP      II 334—336
Expansion, Bessel functions, integrals involving, $\int_{0}^{x} J_{\nu}(t) dt$ , $\int_{x}^{\infty} J_{\nu}(t) dt$ , $\nu=\pm\frac{1}{3}$ , CP      II 357 358
Expansion, Bessel functions, integrals involving, $\int_{0}^{x} u^{-1} \int_{0}^{u} J_{0}(t) dtdu$ , $\int_{0}^{x} u^{-1} \int_{0}^{u} t^{-1}[1-J_{0}(t)] dtdu$ , CP      II 337
Expansion, Bessel functions, integrals involving, $\int_{0}^{x} u^{-1} \int_{0}^{u} J_{0}(t) dtdu$ , $\int_{0}^{x} u^{-1} \int_{0}^{u} t^{-1}[1-J_{0}(t)] dtdu$ , CP, the same integrals with J replaced by I, CP      II 346
Expansion, binomial function      see "Expansion powers
Expansion, Chebyshev polynomial of first kind, powers of x      II 294 296 297
Expansion, Chebyshev polynomial of first kind, powers of x, derivatives and integrals of, CP      I 297—299
Expansion, cosine      see "Expansion sine
Expansion, cosine integral      see "Expansion sine
Expansion, cotangent      see "Expansion tangent"
Expansion, differential equations, integral equations, other functional equations, solutions of, CP      II 234—242
Expansion, dilogarithm, $\int_{0}^{x} t^{-1} ln(1+t) dt$ , CP      II 310
Expansion, elliptic integrals, $_{2}F_{1}$ 's      II 63 64
Expansion, elliptic integrals, first two kinds (complete and incomplete), sin $2n\phi$       II 35 376—382
Expansion, elliptic integrals, first two kinds (complete), CP      II 291 383 384
Expansion, elliptic integrals, Legendre polynomials      II 36
Expansion, error functions, $\int_{0}^{z} e^{\pm t^{2}} dt$ , BF      II 42 43 57 58
Expansion, error functions, $\int_{0}^{z} e^{\pm t^{2}} dt$ , CP      II 42 43 57 289 290 323 324
Expansion, error functions, $\int_{x}^{\infty} e^{-t^{2}} dt$ , CP      II 289 290 324
Expansion, error functions, repeated integrals, CP      II 290
Expansion, exponential function, $a^{\pm x}$ , $2^{\pm x}$ , $10^{\pm x}$ , BF      II 32
Expansion, exponential function, $a^{\pm x}$ , $2^{\pm x}$ , $10^{\pm x}$ , CP      II 32 315
Expansion, exponential function, $e^{-z}z^{\lambda}$ , BF      II 45
Expansion, exponential function, $e^{-z}z^{\lambda}$ , JP      I 285
Expansion, exponential function, $e^{z}$ , BF      II 32 45 46
Expansion, exponential function, $e^{z}$ , CP      II 32 288 313 314 363
Expansion, exponential function, $e^{z}$ , GP      I 287 II 32
Expansion, exponential function, $e^{z}$ , JP      I 285
Expansion, exponential function, $z^{-1}(e^{z}-1)$ , BF      II 33 34
Expansion, exponential function, $z^{-1}(e^{z}-1)$ , CP      II 34
Expansion, exponential integral, $\int_{0}^{x} e^{-y^{\alpha}} dy$ , $\int_{x}^{\infty} e^{-y^{\alpha}} dy$       II 289
Expansion, exponential integral, $\int_{0}^{x} t^{-1} (1-e^{\pm t}) dt$ , BF      II 41 56
Expansion, exponential integral, $\int_{0}^{x} t^{-1} (1-e^{\pm t}) dt$ , CP      II 41 289 321
Expansion, exponential integral, $\int_{x}^{\infty} t^{-1} e^{-t} dt$ , P.V. $\int_{-\infty}^{x} t^{-1} e^{t} dt$ , CP      II 289 322
Expansion, Fresnel integrals, $\int_{0}^{x} t^{-1/2} e^{it} dt, BF      II 43 58
Expansion, Fresnel integrals, $\int_{0}^{x} t^{-1/2} e^{it} dt, CP      II 43 290 328
Expansion, Fresnel integrals, $\int_{x}^{\infty} t^{-1/2} e^{it} dt$ , CP      II 27 290 329 330
Expansion, G-function, $_{p}F_{q}$ 's      I 139 142 145—147
Expansion, G-function, CP      II 27 28

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