Macrobert T.M. — Functions of a complex variable :: Электронная библиотека попечительского совета мехмата МГУ

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Macrobert T.M. — Functions of a complex variable

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Название: Functions of a complex variable

Автор: Macrobert T.M.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 5-th edition

Год издания: 1962

Количество страниц: 436

Добавлена в каталог: 25.11.2012

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Предметный указатель

-plane      10
-plane      2 10
Abel’s test for convergence of series      80
Abel’s theorem on continuity of series      125
Addition of complex numbers      1 3
Amplitude      2 4
Amplitude, of a function, variation of      11—17
Amplitude, principal value of      2 4
Argand diagram      2
Argument      2
Asymptotic expansions      136; see Bessel E-functions Euler Gamma Kummer Whittaker
Barnes Integrals      151 374
Bernoulli numbers      132 328 402
Bessel function      236
Bessel function $G_{n}(z)$       240
Bessel function $G_{n}(z)$ , addition theorem for      254
Bessel function $G_{n}(z)$ , asymptotic expansion of      273
Bessel function $G_{n}(z)$ , in terms of Bessel functions of first and second kinds      240
Bessel function $G_{n}(z)$ , recurrence formulae for      241
Bessel function of Becond kind      238 239
Bessel function of first kind      237
Bessel function of first kind, addition theorem for      254
Bessel function of first kind, as a contour integral      267 268 270
Bessel function of first kind, as a function of its order      239 320
Bessel function of first kind, asymptotic expansions of      273 274
Bessel function of first kind, products of      340
Bessel function of first kind, recurrence formulae for      239
Bessel function of first kind, zeros of      241 274
Bessel functions, modified      267—270 347 351
Bessel functions, modified, asymptotic expansions of      271 293
Bessel functions, relations between      241
Bessel functions, relations between, Lommel integrals, for      321
Bessel’s equation      236 266
Bessel’s transformed equation      267
beta function      144 145
Binomial theorem      90
Branch of function      13
Branch point      14 39
Branch point, of an integral      257
Cauchy’s integral theorem      61 54
Cauchy’s integral theorem, integral      321
Cauchy’s integral theorem, residue theorem      57
Circular functions      33 83 90
Coefficients, undetermined      96
Collineanty of points on cubic      197
complex numbers      1
Complex numbers, geometrical representation of      1
Complex numbers, operations with      1—5
Complex variable      7
Complex variable, function of a      7
Complex variable, path of variation of a      7
Confluent hypergeometrio function      348
Conformal representation      37 370
Congruent points      179 180
Conjugate numbers      1 2
connected region      30
Continuation, analytical      122 208 350
Continuation, analytical, of hypergeometrio function      153 156 249 297—301 370 371
Continuation, analytical, of integral of differential equation      213
Continuation, analytical, theorems on      123 124
Continuity      23 24
Continuity, Abel’s theorem on      125
Continuity, of series      92
Continuity, uniform      24—26
Convergence of infinite product      107 108
Convergence of infinite product, unconditional      107
Convergence of sequence      42
Convergence of sequence, uniform      42
Convergence of series      76
Convergence of series, absolute      76 78 358 359
Convergence of series, circle of      80
Convergence, of a double aeries      78
Convergence, of power series      80 82 95 358 359 360
Convergence, radius of      80
Convergence, ratio tests for      77
Convergence, uniform      92 347 348
Coordinates, polar      2 29 294
Coordinates, polar, rectangular      2
Cross-cut      30
Cylindrical functions      236
Derivative      26 28
Derivative, of function of a function      30
Derivative, of holomorphic function      28 70
Derivative, of inverse function      30
Derivative, partial      31 70
Determinant of fundamental system      216 223
Determinant of fundamental system, index of      224
Differential equation, homogeneous linear      209
Differential equation, homogeneous linear, coefficients of      210
Differential equation, homogeneous linear, construction of      216
Differential equation, homogeneous linear, domain of ordinary point of      210
Differential equation, homogeneous linear, dominant equation      210
Differential equation, homogeneous linear, Frobenius’ method of solution      225
Differential equation, homogeneous linear, fundamental equation      220
Differential equation, homogeneous linear, fundamental system      215 257
Differential equation, homogeneous linear, indicial equation      225
Differential equation, homogeneous linear, integrals of      210
Differential equation, homogeneous linear, of the first order      210
Differential equation, homogeneous linear, of the second order      210
Differential equation, homogeneous linear, ordinary point of      210
Differential equation, homogeneous linear, singularity of      210
Differential equation, homogeneous linear, solutions of      210
Differentiation      26 28 29
Differentiation, of series      93
Differentiation, under integral sign      44 69 138
Discontinuity, removable      23
Division of complex numbers      1 4
Dixon’s theorem      362
Domain of a point      38 210
Dougall's theorems      371 372
E-functions      348 352 393—397 404 15
E-functions, asymptotic expansions of      351 358
E-functions, Barnes integral for      374
Elements of a function      208
Elliptic function      180
Elliptic function, order of      181 182
Elliptic function, poles of      180—183
Elliptic function, zeros of      182; see Jacobian and Weierstrassian functions
Elliptic integrals      169
Elliptic integrals, reduction of      170—173
Elliptic integrals, transformation of      170—174 295
Equation, roots of      16 69 363
Euler’s constant      135
Euler’s constant, asymptotic expansion of      134
Euler’s definition of gamma function      141
Euler’s numbers      328 402
Evaluation of $\pi$       329
Expansion of functions in series of fractions      103 105 115 116 326
Expansion, Lagrange’s      119
Exponential function      32 90
Ferrers’ function      307
Fourier series      86
Fourier — Beesel integral theorem      321 383
Fourier’s integral theorem      319 323 340 341
Fresnel integrals      62
Frobenius’ method of solving linear differential equations      225
Frobenius’ method of solving linear differential equations, indicial equation      225
Frobenius’ method of solving linear differential equations, solutions free from logarithms      228
Frobenius’ method of solving linear differential equations, uniform convergence of series with regard to index      227
Fuchsian type, equations of      243
Fuchsian type, equations of, sum of indices a constant      243 244
Function, analytic      29 208
Function, analytic, conjugate      31
Function, analytic, continuous      23
Function, analytic, dominant      210
Function, analytic, doubly-periodic      179
Function, analytic, elements of a      208
Function, analytic, even      33 97
Function, analytic, geometrical representation of a      7 10
Function, analytic, holomorphic      29 52 93 294

Function, analytic, initial value of      10
Function, analytic, integral      88
Function, analytic, integrals of      48
Function, analytic, inverse      30
Function, analytic, limit of      22
Function, meromorphic      39 40 89 160
Function, multiform or multiple-valued      7 161 209
Function, odd      33 97
Function, of a complex variable      7
Function, of a function      24 30 49
Function, of two complex variables      69 137
Function, of two real variables      26
Function, periodic      32 86
Function, periodic, of the second kind      187
Function, periodic, of the third kind      189
Function, rational      89
Function, rational integral      88
Function, region of existence of      7
Function, regular      29
Function, simply-periodic      86
Function, single-valued      7 209
Function, transcendental integral      88
Function, uniform      7 209
Function, uniform, classification of      88
Fundamental equation      220
Fundamental system of integrals      210 257
Fundamental system of integrals, associated with fundamental equation      220
Fundamental system of integrals, in neighbourhood of singularity      219
Fundamental theorem of algebra      68 69
Gamma function      67 75 109 139 141
Gamma function, asymptotic expansion of      146 374
Gamma function, duplication formula for      145
Gamma function, Euler’s definition of      141
Gamma function, expression as a contour integral      143
Gamma function, Gauss’s definition of      141
Gamma function, the derived function $\Psi(z)$       141
Gauss’s differential equation      228 258
Gauss’s differential equation, function $\Pi(z)$       141
Gauss’s differential equation, sum      117
Gauss’s differential equation, theorem      144 275
Generalised hypergeometric functions      343 358
Generalised hypergeometric functions, expressed as multiple series      363
Geometrical representation      see Complex numbers Functions
Green’s Theorem      45
Gregory’s aeries      84
Harmonic functions      31
Hyperbolic functions      33 90
Hypergeometric equation      228 258
Hypergeometric equation, relations between integrals of      249 298
Hypergeometric equation, the twenty-four integrals of      247 297
Hypergeometric function      77 151 229 246 247 297 298
Hypergeometric function, analytical continuation of      153 156 249 297—301
Hypergeometric function, as a contour integral      259
Hypergeometric function, asymptotic expansion of      301
Hypergeometric function, confluent      348
Hypergeometric function, generalised      343 358
Hypergeometric series      77 78 144
Hypergeometric series, convergence of      77
I( $\rho$ ) notation      1
identities      83
Image of point      9
Indented contour      65
Indicial equation      225
Indicial equation, fundamental system      227
Infinity, point at      9
Infinity, point at, continuity at      23
Infinity, point at, integral at      51 137 139
Infinity, point at, integrals of differential equations at      212 213 224
Infinity, point at, loop about      162 168
Infinity, point at, residue at      58 96
Infinity, point at, singularity at      38 39
Integrals of differential equation      210
Integrals of differential equation, analytical continuation of      213
Integrals of differential equation, at infinity      212 213
Integrals of differential equation, existence of      210
Integrals of differential equation, existence of, fundamental system of      215
Integrals of differential equation, existence of, in form of infinite series      213
Integrals of differential equation, existence of, initial values of      210
Integrals of differential equation, existence of, lineally independent      215
Integrals of differential equations in form of definite integrals      255 266
Integrals of differential equations in form of definite integrals, branch points of      257
Integrals of differential equations in form of definite integrals, fundamental system of      257
Integrals of differential equations near a singularity      219
Integrals of differential equations near a singularity, at infinity      224
Integrals of differential equations near a singularity, fundamental system of      219
Integrals of differential equations near a singularity, index of      222
Integrals of differential equations near a singularity, regular      222
Integrals, contour      59 97 113
Integrals, contour, convergent      136
Integrals, contour, curvilinear      42
Integrals, contour, definite      48
Integrals, contour, double      69 138
Integrals, contour, elliptic      169
Integrals, contour, evaluation of definite      59 97 113
Integrals, contour, finite moduli of definite      50
Integrals, contour, Fresnel      62
Integrals, contour, indefinite      53
Integrals, contour, independent of paths      52
Integrals, contour, limiting values of definite      60 63 113 115
Integrals, contour, of holomorphic functions      50—52
Integrals, contour, of meromorphic functions      160
Integrals, contour, of multiform functions      161
Integrals, contour, principal values of      65 323
Integrals, contour, uniformly convergent      137
Integrals, contour, with infinite paths      51 137 139
Integrand, infinite      136 139
Integration, change of order of      69 138
Integration, change of order of, of series      93
Integration, change of order of, partial      53
Integration, change of order of, under integral sign      69 138
Invariants      see Weierstrass
Inverse points      9
Inverse sine function      163
Inverse tangent function      34 84 86
Jacobian elliptic functions      167 182 198
Jacobian elliptic functions, addition theorems for      202
Jacobian elliptic functions, complementary modulus of      166 167 200
Jacobian elliptic functions, derivatives of      200
Jacobian elliptic functions, differential equations of quarter periods of      176 231
Jacobian elliptic functions, duplication formulae for      204
Jacobian elliptic functions, Legendre’s relation for      175
Jacobian elliptic functions, moduli of      167 200
Jacobian elliptic functions, orders of      182 202
Jacobian elliptic functions, periods of      167 201 202
Jacobian elliptic functions, poles of      200 202
Jacobian elliptic functions, relation to Weierstrassian functions      201
Jacobian elliptic functions, relations between      200
Jacobian elliptic functions, relations between periods of      202
Jacobian elliptic functions, residues at poles of      202 205
Jacobian elliptic functions, transition from Weierstrassian function to      198
Jacobian elliptic functions, zeros of      200
Jacobi’s imaginary transformation      205
Kummer’s function      346
Kummer’s function, asymptotic expansion of      350
Lagrange’s expansion      119 125
Landen’s transformation      174
Laplace’s equation      31
Laurent’s series      84 95
Laurent’s series, absolute convergence of      85
Legendre functions      234
Legendre functions, addition theorems for      314—318 333
Legendre functions, of the first kind      214 235 236
Legendre functions, of the second kind      235 236
Legendre functions, recurrence formulae for      236 252 289 290 310
Legendre polynomials      99 214 235
Legendre polynomials, expression in series      102 103 121
Legendre polynomials, in definite integral forms      100—102
Legendre polynomials, integrals involving      122
Legendre polynomials, recurrence formulae for      102 124 129
Legendre polynomials, Rodrigues’ formula for      120

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