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Goursat E. Functions of a complex variable. Part 1 of volume II
Goursat E.  Functions of a complex variable. Part 1 of volume II









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: Functions of a complex variable. Part 1 of volume II

: Goursat E.

: en

: /

:

ed2k: ed2k stats

: 1916

: 269

: 31.10.2012

: | | ID
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Abel      19 ftn. 170 76 180 244
Abelian integrals      (see Integrals)
Abels 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, Cauchys theorems      225 94 227 95
Analytic functions, derivative of      9 3 42 9 77 33
Analytic functions, elements of      198 83
Analytic functions, Lagranges formula      250 104
Analytic functions, new definition of      199 84
Analytic functions, series of      86 39
Analytic functions, singularities of      232 97
Analytic functions, Taylors 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
Burmans 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
Cauchys integral      75 33
Cauchys integral, fundamental formula      76 33 227 95
Cauchys integral, fundamental theorem      233 98
Cauchys integral, integral theorems      75 33
Cauchys integral, method Mittag Lefflers theorem      139 63
Cauchys integral, theorem      66 28 71 74 32 75 33 78 34 216 90
Cauchys 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'Alemberts theorem      104
Darboux      64 27
Darbouxs formula, law of the mean      64 27
De Moivre      6
De Moivres 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, Fresnels      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, Cauchys theorems      222 93 225 94
Double points      184 80 191 82
Double series      21
Double series, circles of convergence      220 92
Double series, Taylors 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, Hermites 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, DAlemberts theorem      104
Equations, Keplers      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
Gausss sums      125 ex. 21
General linear transformation      44 ex. 2
Geographic maps      (see Maps)
Gourier      126 ex. 28
Goursat      208 ftn. 216 ftn.
Goursats theorem      69 29
hadamard      206 ftn. 212 88 218
hermite      106 51 109 73 168 75 215
Hermites formula      215 90
Hermites formula for elliptic      165 73 168 75 195
Hermites formula for elliptic of rational functions      38 15 113 53
Hermites formula for elliptic of series      86 39
Hermites formula for elliptic, hyperelliptic      116 55 247 103
Hermites formula for elliptic, law of the mean (Weierstrass, Darbonx)      64 27
Hermites formula for elliptic, line      61 25 62 26 74 32 224 93
Hermites 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, Veierstrasss theorem      283 98 inverse and
Independent periods, Jacobis 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
Jacobis form      125
Jacobis theorem      147 66
Jensen      104 50
Jensens formula      104 50
Kepler      109 ex. 126
Keplers equation      109 ex. 126
Lagrangem.      106 51 126
Lagranges formula      106 51 126
Laplace      10 3 54 24 55
Laplaces equation      10 3 54 24 55
Laplaces form      125
laurent      75 33 81 37 91 42 94 43 126
Laurents series      75 33 81 37 146 65
Law of the mean for integrals      64 27
legendre      106
Legendres 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
Liouvilles 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
Mercators 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, Cauchys method      139 63
Monodromic functions      17 ftn.
Monogenic functions      9 ftn.
Morera      78 34
Moreras 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, Weierstrasss      127 57
1 2
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