312
Index
Embedding of a manifold into another,
241
Formula
– Hankel, 84, 85
– Mellin inversion, 105
– chart change, 221
– Green-Riemann, 194
– Hankel, 83–85
– Mellin inversion, 111
– multivariable chain rule, 149
– of change of variables in an integral,
204
– of change of variables in integrals,
201
– of duplication of the function Γ , 80
– Stirling’s for the gamma function,
118
– Stokes, 196, 201, 215, 269
Fourier transform
– complex, 94
– de (x − a)
−1 , 75
– of e
−x x
s−1 , 78
– de 1/ cosh πx, 125
– of a rational function, 69–74
Function
– algebraic, 280
– analytic, 8
– constant up to r, 35
– differentiable, 4
– elliptic, 55
– e(z) = exp(2πiz), 63
– gamma, 78–82, 84, 85
– holomorphic, 7, 276
– holomorphic at infinity, 50
– meromorphic, 47, 276
– of class C
1/2 , 3
– regulated, 3
– Riemann zeta, 108
Holomorphic
– analyticity, 40
– at infinity, 50
– differential, 277
– distribution, 39
– function, 7
Homotopy
– homotopic paths, 177, 288
– linear, 22, 177
– path homotopic, 19
Immersion, 155, 233
Index of a point with respect to a path,
32
Integral of a differential form
– along a path, 172
– and inverse image, 197, 215
– inverse image, 174
– over an oriented manifold, 266
Integral of a holomorphic function
– homotopy invariance, 23
– along a path, 10
– as a Stieltjes integral, 12
Irreducible (polynomial), 281
Jacobian, Jacobian matrix, 148
– and integration, 204
– of a holomorphic map, 53
– product formula, 150
Laurent (series), 40
Length of a path, 34
Lifting
– of a homotopy, 290
– of a homotopy, 288, 289, 291
– of a path, 288
Linear functionals, 12
Local Chart
– C
r -compatible, 220
Local chart
– holomorphic, 275
– in a manifold, 220
Local description
– in a Cartesian space, 156
Local uniformizer, 275
Logarithm (function), 26, 31
– uniform branch, 26
Manifold of class C
r
– immersed, 246
– orientable, 202, 264
– product, 225
Matrix exponential, 258
Mellin Transform
– inversion formula, 105, 111
Mellin transform, 103
Meromorphic
– differential, 277
– function, 47
– number of zeros and poles, 47
– on the Riemann sphere, 54, 55
Moving frame, 158
Multilinear (form or function), 134
Orientation
– canonical of R
n , 266
– of a Cartesian space, 266
– of a manifold, 264
Index
Embedding of a manifold into another,
241
Formula
– Hankel, 84, 85
– Mellin inversion, 105
– chart change, 221
– Green-Riemann, 194
– Hankel, 83–85
– Mellin inversion, 111
– multivariable chain rule, 149
– of change of variables in an integral,
204
– of change of variables in integrals,
201
– of duplication of the function Γ , 80
– Stirling’s for the gamma function,
118
– Stokes, 196, 201, 215, 269
Fourier transform
– complex, 94
– de (x − a)
−1 , 75
– of e
−x x
s−1 , 78
– de 1/ cosh πx, 125
– of a rational function, 69–74
Function
– algebraic, 280
– analytic, 8
– constant up to r, 35
– differentiable, 4
– elliptic, 55
– e(z) = exp(2πiz), 63
– gamma, 78–82, 84, 85
– holomorphic, 7, 276
– holomorphic at infinity, 50
– meromorphic, 47, 276
– of class C
1/2 , 3
– regulated, 3
– Riemann zeta, 108
Holomorphic
– analyticity, 40
– at infinity, 50
– differential, 277
– distribution, 39
– function, 7
Homotopy
– homotopic paths, 177, 288
– linear, 22, 177
– path homotopic, 19
Immersion, 155, 233
Index of a point with respect to a path,
32
Integral of a differential form
– along a path, 172
– and inverse image, 197, 215
– inverse image, 174
– over an oriented manifold, 266
Integral of a holomorphic function
– homotopy invariance, 23
– along a path, 10
– as a Stieltjes integral, 12
Irreducible (polynomial), 281
Jacobian, Jacobian matrix, 148
– and integration, 204
– of a holomorphic map, 53
– product formula, 150
Laurent (series), 40
Length of a path, 34
Lifting
– of a homotopy, 290
– of a homotopy, 288, 289, 291
– of a path, 288
Linear functionals, 12
Local Chart
– C
r -compatible, 220
Local chart
– holomorphic, 275
– in a manifold, 220
Local description
– in a Cartesian space, 156
Local uniformizer, 275
Logarithm (function), 26, 31
– uniform branch, 26
Manifold of class C
r
– immersed, 246
– orientable, 202, 264
– product, 225
Matrix exponential, 258
Mellin Transform
– inversion formula, 105, 111
Mellin transform, 103
Meromorphic
– differential, 277
– function, 47
– number of zeros and poles, 47
– on the Riemann sphere, 54, 55
Moving frame, 158
Multilinear (form or function), 134
Orientation
– canonical of R
n , 266
– of a Cartesian space, 266
– of a manifold, 264