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Cambridge University Press 978-0-521-80937-5 - Complex analysis Kunihiko Kodaira Index More information

Index

0-chain, 191 boundary, 65, 73, 191, 261 1-chain, 191 boundary point, 261, 285 1-cycle, 316 branch, 178 1-form, 247 1-form, harmonic, 258, 295 canonical divisor, 386–388 1-form, holomorphic, 257, 258 Cauchy–Hadamard formula, 18 1-form, real, 247 Cauchy–Riemann equations, 10, 258 2-form, 248 Cauchy’s criterion, 4, 6 Cauchy’s integral formula, 36, 104 a-point, 119 Cauchy’s Theorem, 101, 103, 194, 396, 399 Abelian differential, 380, 386 Cauchy’s Theorem, homotopy variant, 190 Abelian differential of the first kind, 376, 379, cell, 74, 77 387, 390 cellular decomposition, 80, 400 Abelian differential of the second kind, 380 cellularly decomposable, 80 Abelian differential of the third kind, 380, center of a power series, 16 389, 390, 392 chain group, 0-dimensional, 191 Abelian differential, residue of an, 380 chain group, 1-dimensional, 191 Abelian differential, zero of the nth order, 380 chain, 0-, 191 Abelian integral, 380, 392 chain, 1- , 191, 360 Abel’s Theorem for meromorphic functions, circle of convergence, 18 391 closed differential form, 256, 318 absolute value, 1 closed region, 6 absolutely convergent, 4, 53 closed set, 261 absolutely convergent, uniformly, 43 closure, 261 absolutely convergent integral, 251 cocycle, 1-, 342 accumulation point, 272, 273 cohomologous, 342 analytic continuation, 153, 162 cohomology class, 342 analytic continuation, direct, 154 cohomology class, d-, 341 analytic continuation, result of, 164 cohomology group, first, 342 analytic function, 46, 154 cohomology group, first d-, 341, 375 analytic function, complete, 154 compact set, 262 analytic Jordan arc, 232 complete analytic function, 154 analytically continuable, 153, 162 complex coordinate, 137 analytically continuable, freely, 165, 178, 399 complex number, 1 arc, 26 complex plane, 1 argument, 23 complex plane, extended, 133 complex projective space, 1-dimensional , 138 Betti number, first, 343, 400 complex sphere, 133 biholomorphic, 127, 264, 267 conformal, 123 border, 352 conformal mapping, 267, 398, 399 bordered Riemann surface, 352 conformally equivalent, 214, 355 403

404

conjugate, 2 entire function, 58, 395 connect, 26 essential singularity, 57, 136 connected, 5, 207, 321 exact differential form, 256, 318 connected, simply, 172 extended complex plane, 133 connected set, 261 exterior derivative, 249, 270 connectivity, 333, 400 exterior of a Jordan curve, 321 continuous, 6, 7, 25 exterior product, 248 continuous, uniformly, 8 continuous mapping, 262 final point, 26 continuously differentiable, 55, 255 fractional linear function, 139 contractible, 171 fractional linear transformation, 141, 397 convergence, 4, 6 freely analytically continuable, 165, 178, 399 convergent, uniformly, 15, 43 fundamental group, 174 convex, 172 Fundamental Theorem of Algebra, 59 coordinate , 24,137 coordinate disk, 275 genus, 375, 400 coordinate transformation, 137 Green’s Theorem, 251, 255, 287, 399 cross, 70 cross ratio, 143 harmonic 1-form, 295 curve, 26, 271 harmonic 1-form of the first kind, 376 curve, analytic, 272 harmonic form, 258, 271 curve, smooth, 272 harmonic function, 259 cycle, 1-, 192, 360, 365 harmonic part of a form, 377 Hausdorff space, 261, 276, 400 holomorphic, 11 de Rham’s Theorem, 342 holomorphic 1-form, 257, 258 degree of a divisor, 380 holomorphic function , 256, 259, 267 derivative, 10 homeomorphism, 215, 262 differentiable, 9,10, 25 homogeneous coordinates, 138 differentiable, continuously, 25 homologous, 193, 316 differential coefficient, 9 homology, 372 differential form, 247 homology class, 360 differential form, exact, 256, 318 homology group, first, 360, 375 differential form of degree 1; see 1-form, homotopic, 166 differential form of degree 2; see 2-form, homotopy, 314, 360 differential form on the Riemann sphere, 268 homotopy class, 167 direct analytic continuation, 154 homotopy variant of Cauchy’s Theorem, 190 Dirichlet integral, 312 horizontal slit region, 329, 400 Dirichlet’s Principle, 299, 377 disk, 35 image, 117 disk, unit, 128 imaginary axis, 2 distance, 340 imaginary part, 2 divergence, 4 index of speciality, 385 divides, Jordan curve, 320 infinitesimal, 10 divisor of a meromorphic function, 379, 385, inhomogeneous coordinates, 139 391 initial point, 26 divisor of an Abelian differential, 381 inner product, 295 divisor, canonical, 386–388 integral, 32 divisor, linearly equivalent, 387 integral operator, 307 domain, 5, 6 interior, 261 dual form, 250, 258 interior of a Jordan curve, 321

405

interior point, 261 partition of unity, 278, 400 inverse image, 117 path of integration, 32 isolated singularity, 55 period of a form, 374 Picard’s Theorem, 241 Jordan arc, 26, 62, 272 piecewise smooth, 64, 181 Jordan arc, analytic, 232 planar Riemann surface, 319, 320 Jordan curve, 27, 73, 320, 395 point at infinity, 133 Jordan Curve Theorem, 68, 73, 321 Poisson’s formula, 397, 399 pole, 56, 136 Lambert series, 395 power series, 16 Laplacian, 259, 271 primitive function, 32 Laurent expansion, 55 principal part, 55 Laurent series, 53 principal value, , 331 Lebesgue measurable, 312 principle of reflection , 230, 233 limit, 6 line integral , 250, 274 radius of convergence, 18 linear (fractional) function, 139 range, 5 linear (fractional) transformation, 141, 397 real axis, 2 linearly equivalent divisors, 387 real part, 2 Liouville’s Theorem, 59 reflection, 227 local complex coordinates, 137, 264 reflection, principle of, 230, 233 local complex coordinates, system of, 264, region, 6, 261, 265 267 region without holes, 208 locally finite open covering, 276 removable singularity, 55 logarithmic function, 155 residue, 106 residue of an Abelian differential, 380 maximum principle, 51, 376 Residue Theorem, 107 Mean Value Theorem, 51 Residue Theorem for Abelian differentials, meromorphic function, 379, 385, 386, 381 391–393 restriction, 14 metric space, 340 result of analytic continuation, 164 modular function, 240 Riemann Mapping Theorem, 200, 332 Monodromy Theorem, 168, 315 Riemann sphere, 133, 263, 265, 397 Montel’s Theorem, 202 Riemann surface, 264, 265, 274 multiple point, 290 Riemann surface, bordered, 352 multiplicity, 56 Riemann–Roch Theorem, 382, 386 multiply connected, 333 schlichtartig; see planar Riemann surface, natural boundary, 160 Schwarz–Christoffel formula, 246 neighborhood, 261 Schwarz’s lemma, 130 norm, 296 simply connected, 172 north pole, 132 simply connected Riemann surface, 329 single-valued function, 267 open covering, 262 singularity, essential, 57, 136 open covering, finite, 262 singularity, isolated, 55 open mapping, 121 smooth, 26, 29, 60, 62 order, 56, 136 south pole , 132 orientation, 62 stereographic projection, 133, 264, 398 subcovering, 262 parameter, 62 support, 278 parameter representation, 62 system of open sets, 260

406

tangent, 28 uniformly convergent, 15, 43 tangent vector, 28 unit disk, 128 Taylor series, 22 univalent; see also single-valued, 127 topological space, 260 touch, 30 Weierstrass’ Theorem, 57 transversal intersection, 70 Weyl’s Lemma, 313 triangle inequality, 3 Weyl’s method of orthogonal projection, 312 uniformly absolutely convergent, 43 uniformly continuous, 8 zero of the nth order, 380