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| Goursat E. — Functions of a complex variable. Part 1 of volume II |
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| Предметный указатель |
Abel 19 ftn. 170 76 180 244
Abelian integrals (see Integrals)
Abel’s theorem 244 102
Addition formulae 27 12
Addition formulae for elliptic functions 166 74 250 103
Adjoint curves 191 82
Affixe 4 ftn.
Algebraic equations (see Equations)
Algebraic functions (see Functions)
Algebraic singular points (see Singular points)
Analytic extension 196 83 199 84
Analytic extension, functions of two variables 231 97
Analytic functions 7 3 11 4
Analytic functions analytic extension of 231 97
Analytic functions of several variables 218 91
Analytic functions, analytic extension 196 83 199 84
Analytic functions, Cauchy’s theorems 225 94 227 95
Analytic functions, derivative of 9 3 42 9 77 33
Analytic functions, elements of 198 83
Analytic functions, Lagrange’s formula 250 104
Analytic functions, new definition of 199 84
Analytic functions, series of 86 39
Analytic functions, singularities of 232 97
Analytic functions, Taylor’s formula 222 92 226 94
Analytic functions, zeros of 88 40 Functions Integral Single-valued
Anchor ring 54 ex. 3
Appell 84 38 217
Associated circles of convergence 220 92
Associated integral functions 218 ex. 7
Bertrand 58 ex. 22
Bicircular quartics 193
Binomial formula 40 18
Birational transformations 192 82 252
Blumenthali 132 ftn.
Borel 130 ftn. 132 218
Bouquet (see Briot and Bouquet)
Branch point (see Critical points)
Branches of a function 15 6 29 13
Briot and Bouquet 126 ex. 27 195
Burman 126 ex. 26
Burman’s series 126 ex. 26
Cauchy 7 2 9 3 60 25 71 32 78 34 82 106 51 114 53 127 57 139 63 200
Cauchy — Taylor series 79 35
Cauchy’s integral 75 33
Cauchy’s integral, fundamental formula 76 33 227 95
Cauchy’s integral, fundamental theorem 233 98
Cauchy’s integral, integral theorems 75 33
Cauchy’s integral, method Mittag — Leffler’s theorem 139 63
Cauchy’s integral, theorem 66 28 71 74 32 75 33 78 34 216 90
Cauchy’s integral, theorem for double integrals 222 93 225 94
Caucliy — Laurent series 81 35
Change of variables, in integrals 62 26
Circle of convergence 18 8 202 84 209 87 212 88
Circle of convergence, associated circles of convergence 220 92
Circle of convergence, singular points on 202 84
Circular transformation 45 19 57
Class of an integral function 132 58
Clebsch 186 ftn.
Complex quantity 3 1
Complex variable 6 2
Complex variable, analytic function of a 9 3
Conformal maps (see Maps)
Conformal representation 42 19 45 20 48 20 52 23
Conformal transformations (see Transformations)
Conjugate imaginaries 4 1
Conjugate isothermal systems 54 24
connected region 11 4
Continuity, of functions 6 2
Continuity, of power series 7 2 56
Continuous functions (see Functions)
convergence of integrals 229
Convergence of series 7 2 88 39
Convergence, circle of (see Circle of convergence)
Convergence, uniform of infinite products 22 10 129 57
Cousin 232 ftn.
Critical points 15 6 29
Critical points, logarithmic 32 14 113 53
Cubics (see Curves)
Curves of deficiency one 172 77
Curves, adjoint 191 82
Curves, bicircular quartics 193 ex.
Curves, conjugate isothermal systems 54 24
Curves, deficiency of 172 77 191 82 252
Curves, double points 184 80 191 82
Curves, loxodromic 53 ex. 1
Curves, parametric representation of curves of deficiency one 187 81 191 82 193
Curves, parametric representation of plane cubics 180 78 184 80 187 81
Curves, points of inflection 186 80
Curves, quartics 187 81
Curves, unicursal 191 82
Cuts 208 87
Cycles 244 101
Cyclic periods 244 101
Cyclic system of roots 238 99
D'Alembert 104
D'Alembert’s theorem 104
Darboux 64 27
Darboux’s formula, law of the mean 64 27
De Moivre 6
De Moivre’s Formula 6 1
Deficiency (see Curves deficiency
Definite integrals 60 25 72 31 97
Definite integrals, differentiation of 77 33 227 95
Definite integrals, evaluation of 96 45
Definite integrals, Fresnel’s 100 46
Definite integrals, law of the mean 64 27
Definite integrals, Note (see also Integrals)
Definite integrals, periods of 112 53 114
Derivative of analytic functions 9 3 42 19 77 33
Derivative of integrals 77 33 227 95
Derivative of power series 19 8
Derivative of series of analytic functions 88 39
Dominant function 56 ex. 7 81 35 227 94
Dominant series 21 9 157 69
Double integrals 222 93
Double integrals, Cauchy’s theorems 222 93 225 94
Double points 184 80 191 82
Double series 21
Double series, circles of convergence 220 92
Double series, Taylor’s formula 222 92 226 94
Doubly periodic functions 145 65 149 67
Eisenstein 252 ex. 3
Elements of analytic functions 198 83
Elliptic functions 145 65 150 68
Elliptic functions, addition formulae 166 74
Elliptic functions, algebraic relation between elliptic functions with the same periods 153 68
Elliptic functions, application to cubics 180 78 184 80
Elliptic functions, application to curves of deficiency one 187 81 191 82
Elliptic functions, application to quartics 187 81
Elliptic functions, even and odd 154 68
Elliptic functions, expansions for 154 69
Elliptic functions, general expression for 163 73
Elliptic functions, Hermite’s formula 165 73 168 75 195
Elliptic functions, integration of 168 75
Elliptic functions, invariants of 158 70 172 77 182 79
Elliptic functions, order of 150 68
Elliptic functions, p(u) 154 69
Elliptic functions, p(u) defined by invariants 182 79
Elliptic functions, periods of 152 68 172 77 184 79
Elliptic functions, poles of 150 68 154 68
Elliptic functions, relation between p(u) and p'(u) 158 70
Elliptic functions, residues of 151 68
Elliptic functions, zeros of 152 68 154 68 159 70
Elliptic integrals of the first kind 120 56 174 78 250 103
Elliptic integrals, periods of 56
Elliptic integrals, the inverse function 174 78
Elliptic transformation 57 ex. 15
Equations 5 98
Equations, algebraic 240 100
Equations, cyclic system of roots of 288 99 241 100
| Equations, D’Alembert’s theorem 104
Equations, Kepler’s 109 ex. 126 ex.
Essentially singular point 91 42
Evaluation of definite integrals (see Definite integrals)
Even functions 153
Expansions in infinite products 194 exs. 2
Expansions in series of ctn x 143 64
Expansions in series of elliptic functions 154 69
Expansions in series of periodic functions 145 65
Expansions in series of roots of an equation 238 99
Exponential function 23
FOURIER 170 76
Fuchsian transformation 57 ex. 15
Functions of a complex variable 6 2
Functions, algebraic 233 98 240 100
Functions, analytic (see Analytic functions and Analytic functions of several variables)
Functions, analytic except for poles 90 41 101 48 136 61
Functions, branches of 15 6 29 13
Functions, class of integral 132 58
Functions, continuous 6 2
Functions, defined by differential equations 208 86
Functions, dominant 56 ex. 7 81 35 227
Functions, doubly periodic 145 65 149 67
Functions, elementary transcendental 18 8
Functions, elliptic (see Elliptic functions)
Functions, even and odd 153
Gamma function 100 47 229 96
Gauss 125 ex. 21
Gauss’s sums 125 ex. 21
General linear transformation 44 ex. 2
Geographic maps (see Maps)
Gourier 126 ex. 28
Goursat 208 ftn. 216 ftn.
Goursat’s theorem 69 29
hadamard 206 ftn. 212 88 218
hermite 106 51 109 73 168 75 215
Hermite’s formula 215 90
Hermite’s formula for elliptic 165 73 168 75 195
Hermite’s formula for elliptic of rational functions 38 15 113 53
Hermite’s formula for elliptic of series 86 39
Hermite’s formula for elliptic, hyperelliptic 116 55 247 103
Hermite’s formula for elliptic, law of the mean (Weierstrass, Darbonx) 64 27
Hermite’s formula for elliptic, line 61 25 62 26 74 32 224 93
Hermite’s formula for elliptic, uniform convergence of 229 96
Holomorphic functions 11 ftn.
Hyperbolic transformations 57 ex. 15
Hyperelliptic integrals 116 55 247 103
Hyperelliptic integrals, periods of 116 55
Imaginaries, conjugate 4 1
Imaginary quantity 3 1
Implicit functions, Veierstrass’s theorem 283 98 inverse and
Independent periods, Jacobi’s theorem 147 66
Index of a quotient 103 49
Infinite number of singular points 184 60
Infinite number of zeros 26 11 98 42 128 57
Infinite products 22 10 129 57 194
Infinite products, uniform convergence of 22 10 129 37
Infinite series (see Series)
infinity (see Point at infinity)
Inflection, point of 186 80
Integral functions 21 8 127 57
Integral functions with an infinite number of zeros 127 57
Integral functions, associated 218 ex. 7
Integral functions, class of 182 58
Integral functions, periodic 147 05
Integral functions, transcendental 21 ftn. 92 42 136 61 230
Integral transcendental functions 21 ftn. 92 42 136 61 230
Integrals, Abelian 193 82 248 101
Invariants (integrals) 57 ex. 15
Invariants (integrals) of elliptic functions 158 70 172 77 182 79
Inverse functions (see Functions inverse implicit)
inversion 45 19 57
Irratioual functions 18 (see also Functions)
Irreducible polynomial 240 100
Isolated singular points 89 40 182 59
Isolated singular points, essentially singular 91 42
Isothermal curves 54 24
jacobi 125 ex. 18 147 66 154 69 170 76 180 78
Jacobi’s form 125
Jacobi’s theorem 147 66
Jensen 104 50
Jensen’s formula 104 50
Kepler 109 ex. 126
Kepler’s equation 109 ex. 126
Lagrangem. 106 51 126
Lagrange’s formula 106 51 126
Laplace 10 3 54 24 55
Laplace’s equation 10 3 54 24 55
Laplace’s form 125
laurent 75 33 81 37 91 42 94 43 126
Laurent’s series 75 33 81 37 146 65
Law of the mean for integrals 64 27
legendre 106
Legendre’s polynomials 108
Limit point 90 41
Line integrals 61 25 62 26 74 32 224 93
Linear transformation 59
Lines, singular (see Natural boundaries and
Liouville 81 36 150 67
Liouville’s theorem 81 36 150 67
Logarithmic critical points 82 14 113 53
Logarithms 28 13 118 53
Logarithms, natural or Napierian 28 13
Loops 112 53 115 54 244 101
Loxodromic curves 53 ex. 1
Maclaurin 83 ex.
Maps, conformal 42 19 45 20 48 20 52 23
Maps, geographic 52 23
Mercator’s projection 52 ex. 1
Meromorphic functions 90 ftn.
Mittag — Leffler 127 57 61 139 63
Mittag — Leffler's theorem 127 57 134 61 139 63
Mittag — Leffler's theorem, Cauchy’s method 139 63
Monodromic functions 17 ftn.
Monogenic functions 9 ftn.
Morera 78 34
Morera’s theorem 78 34
Multiform functions 17 ftn.
Multiple-valued functions 17
Napierian logarithms 28 13
Natural boundary 201 84 208 87 211 88
Natural logarithms 28 13
Neighborhood 88 40
Neighborhood of the point at infinity 109 52
Odd functions 154 68
Order of elliptic functions 150 68
Order of poles 89 40
Order of zeros 88 40
Ordinary point 88 40
Painlevg 85 38
Parabolic transformation 57 ex. 15
Parallelogram of periods 150 67
Parametric representation (see Curves)
Periodic functions 145 65
Periodic functions, doubly 145 65 149 67
Periodic integral functions 147 65
Picardi 21
Polar periods (see Periods polar)
Poles 88 40 90 41 133 59
Poles at infinity 110 52
Poles of elliptic functions 150 68 154 68
Poles, infinite number of 135 61 137 62
Poles, order of 89 40
Polynomials, irreducible 240 100
Power series 18 8 196 83
Power series, continuity of 7 2 56
Power series, derivative of 19 8
Power series, dominating 21 9
Power series, representing an analytic function 20 (see also Analytic extension Circle and
Primary functions, Weierstrass’s 127 57
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