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| Bliss G.A. — Algebraic Functions |
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| Предметный указатель |
Abel iii iv 94 119
Abelian integrals, classification of 64 95
Abelian integrals, definition 64 93
Abelian integrals, for algebraic equations of the third degree in y 208ff (see also Elementary integrals)
Abelian integrals, for elliptic and hyperelliptic curves 191ff
Abelian integrals, inversion of 169ff
Abelian integrals, normal integrals 110
Abelian integrals, periods 96ff 105
Abelian integrals, singularities 94
Abel’s theorem 119ff
Abel’s Theorem, elementary applications of 121
Abel’s Theorem, for elementary integrals 129
Abel’s Theorem, for integrals of the first kind 116
Adjoined polynomials 153
Algebraic function, branch of 32
Algebraic function, continuations of 25
Algebraic function, defined by equation of the third or fourth degree in y 194ff
Algebraic function, definition of 24
Algebraic function, discriminant 24
Algebraic function, expansions 29 35
Algebraic function, genus of 58 90 147
Algebraic function, having genus  or  139
Algebraic function, hyperelliptic and elliptic cases 141 143
Algebraic function, integral algebraic function 195
Analytic functions 1
Arcs, regular 2
Arcs, regular arc on a Riemann surface 80
Arcs, simply closed 2
Barnum 123
Bases, complementary bases 59ff 200
Bases, for a divisor 48
Bases, for all rational functions 46
Bases, for special examples 196ff
Bases, for the divisor  and an equation of the third degree in y 196ff
Bases, normal at  52
Bases, normal for the divisor and an equation of the third degree in y 202
Bases, transformation of 51
Bauer v 197 211 213
Bauer, his methods for equations of the third degree in y 194ff
Bauer, theorem of 204
Birational transformations 133ff
Birational transformations, for curves of the third degree in y 203
Birational transformations, in the function- theoretic plane 167
Birational transformations, projective 144
Birational transformations, reducing singularities to double points in the projective plane 163
Birational transformations, reducing singularities to multiple points with distinct tangents 156
Branch cycle 32 56
Branch cycle, in the function-theoretic plane 166
Branch cycle, linear branch in the projective plane 160
Branch cycle, of a rational function 63
Branch cycle, primitive branch 32 159 166
Branch of an algebraic function 32
Brill iii
Canonical system of cuts 90
Canonical system of cuts, for elliptic or hyperelliptic curves 190
Cauchy iv 2 7 84
Cauchy’s integral formula 8
Cauchy’s Theorem 3
Cauchy’s theorem, consequences of 5 7
Cauchy’s theorem, on a Riemann surface 84 92
Clebsch iii
Column order of a rational function at 52
Complementary bases 59ff 200
Connected regions 5 26
Connectivity of surfaces 82 87
Connectivity of surfaces, of a closed surface 88
Connectivity of surfaces, of a Riemann surface 86ff
Continuations for an algebraic function 25
Cremona iv 156
Cremona transformations 133
Cremona transformations, reducing singularities to multiple points with dis-tinct tangents 156
Curves, in the function-theoretic plane with double points only 165
Curves, in the projective plane with double points only 159
Curves, of elliptic and hyperelliptic types 189
Curves, of genus not greater than 2 139
Curves, of the fourth degree in x and y 210
Curves, of the third degree in y 194ff
Cusps 41
Cuts 82
Cuts, canonical systems 90
Cuts, cross and loop cuts 82
Cycles for an algebraic function 32
Cycles for an algebraic function, branch cycles 56
Dedekind iii iv
Determinant  48
Discriminant of a polynomial 18
Divisor 47
Divisor, basis for a divisor 48
Divisor, ideal norm of 55
Divisor, multiples of 51 54
Divisor, of a rational function 62
Divisor, of branch cycles 56 63
Divisor, order of a divisor 62
Divisor, special divisor 71
Elementary integrals 64 102
Elementary integrals, for curves of elliptic or hyperelliptic types 191-2
Elementary integrals, for equations of the third degree in y 205 209
Elementary integrals, of first kind 65 98 151
Elementary integrals, of second kind 66 102
Elementary integrals, of third kind 66 102
Elementary integrals, period relations 106
Elliptic curves 139
Elliptic functions, analogies with rational functions on a Riemann surface 174ff
Elliptic functions, further properties 177
Elliptic integrals 94
Elliptic integrals, addition formulas 123
examples 183ff
Examples, algebraic functions defined by equations of the third or fourth degree in y 194ff 211 213
Examples, elliptic and hyperelliptic curves 189
Examples, for expansions 183
Examples, for Riemann surfaces 186
Expansion theorems 17 25 29 32
Expansion theorems, at singular points of special types 40
Expansion theorems, Laurent expansion 9
Expansion theorems, Newton’s polygon method 35
Fields iii
| Function-theoretic plane 165
Functions, algebraic 24
Functions, analytic 1
Functions, holomorphic on a plane 1
Functions, inverse 171ff
Functions, on a Riemann surface 80
Functions, rational 12ff 43ff 70
Functions, special rational 71
Fundamental systems of integrals, construction of 107
Fundamental systems of integrals, for curves of elliptic or hyperelliptic type 193-4
Fundamental systems of integrals, of the first and second kinds 103
Fundamental systems of integrals, of the first kind 98
Fundamental theorem of algebra 15
Genus, invariant property 62
Genus, of a Riemann surface 88
Genus, of an algebraic function 58 90 147
Genus, of an equation of the third degree in y 207
Genus, of elliptic or hyperelliptic curves 191
Gordan iii
Halphen 155
Hensel iii iv v 156
Holomorphic functions, on a Riemann surface 80
Holomorphic functions, on a z-plane 1
Holomorphic functions, together forming an algebraic function 28
Hyperelliptic curves 139 189
Ideal norm of a divisor 55
Integral algebraic function 195
Integrals, associated with an algebraic function 64 93 Elliptic
Integrals, of a function f(z) 2
Inverse function, when 171
Inverse function, when 172
Inversion of Abelian integrals 169ff
Irreducible algebraic equations 78
jacobi 176
Klein 155
kronecker iii v 155-6
Landsberg iii iv v 156
Laurent expansion 9
Laurent expansion, principal part of 13 15
Leading coefficients of a rational function 74
Logarithm of a rational function 114
Multiple points with distinct tangents 40
Multiples of a divisor 47 48 51 54 72
Newton 35
Newton’s polygons 36
Noether iii 155
Noether, Emmy iii
Norm, ideal norm of a divisor 55
Norm, of a rational function 44
Normal basis for a divisor 52
Normal form of equation of third degree in y 194
Normal integrals 110
Ordinary point 9 11
Ordinary point, ordinary place 80
Periods, of an Abelian integral 96ff 101
Periods, period relations 100 105-6
Periods, primitive periods of an elliptic function 179
Picard 155
Place on a Riemann surface 34 77
Place on a Riemann surface, for an equation of the third degree in y 207
Place on a Riemann surface, singular places 80
Place on a Riemann surface, special set of places 71
Primitive branch 32
Principal parts 15
Projective intersections of two curves 150
Projective plane 159
Projective transformations 144ff
Rational function, analogies with elliptic functions 174ff
Rational function, conjugate values 44
Rational function, expressed in terms of integrals 113
Rational function, in the plane 12ff
Rational function, integral rational function 196
Rational function, on a Riemann surface 43ff
Rational function, special rational function 71 73
Rational function, with prescribed poles 14-15 70 72
Reducibility of polynomials 21ff
Reduction of singularities 155ff
Reduction of singularities, to double points in the function-theoretic plane 167
Reduction of singularities, to double points in the projective plane 163
Reduction of singularities, to multiple points with distinct tangents 156
references 23 42 74 92 118 132 154 168 182 213 214
Region 26
Region, connected 5
Region, simply connected 5 26
Regular arc 2
Regular arc, on a Riemann surface 80
Residues 7 12 44
Residues, of a function on a Riemann surface 81
Residues, theorems concerning 8 11 14 45
Resultant of two polynomials 18
Riemann iii iv 67 75
Riemann surface T 75ff
Riemann surface T, construction of T, T’, and T’’ 75 92 96
Riemann surface T, examples 186ff
Riemann surface T, for an irreducible algebraic equation 78
Riemann surface T, genus of 88 147
Riemann surface T, places on 77
Riemann — Roch theorem 67ff
Roch iv 67
Severi iii
Simart 155
Singular places, of Abelian integrals 94
Singular places, of an algebraic function 34
Singular places, of functions on a Riemann surface 80
Singular points 9 11
Singular points, for an algebraic function 25
Singular points, of special types 40
Singularities, reduction of 155ff (see also Reduction of singularities)
Special places and rational functions 71 73
Special places and rational functions, special divisor 71
Surface, connected or simply connected 82
Surface, connectivity 87
Trace of a rational function 44
Transformations, projective 144ff (see also Birational transformations Cremona
Unicursal curves 139ff
Weber iii iv
Weierstrass iii 176-7
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