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| Henrici P. — Applied and Computational Complex Analysis. I: Power Series, Integration, Conformal Mapping, Location of Zeros. |
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| Предметный указатель |
Function, analytic, scalar 89 139
Function, complex, of real variable 180
Function, complex-differentiable 325 328
Function, complex-valued 286
Function, continuous 73
Function, differentiable 143
Function, domain of definition of 72
Function, domain of values of 72
Function, doubly periodic 421
Function, elliptic 422
Function, even 93
Function, harmonic 332 336
Function, holomorphic 143 328
Function, homogeneous 646
Function, inverse 95
Function, inverse of exponential function 113 291 293
Function, inverse of Joukowski function 296
Function, inverse of Moebius transformation 302
Function, linear 288
Function, matrix-valued 122 138
Function, meromorphic 214 215 249 312 596 602 607 612 641 655
Function, odd 93
Function, one-to-one 95 291
Function, periodic 109 223 224 227 269 659
Function, quadratic 289
Function, range of 72
Function, rational 139 217 553 625 626
Function, scalar complex 139
Function, summatory 265
Function, trivial 95
Function, univalent 291
Function, value of 72
Functions, analytic, sequence of see “Weierstrass double series theorem”
Fundamental theorem of algebra 105 159 378
Fundamental theorem of algebra, proofs of 138 280 552
Galois, theorem of 497
Gamma function 153 186 412 655
Gauss-Lucas theorem 463 465 474 518
Gaussian elimination 554 627
Gegenbauer polynomials 43 121 137
Generating function 60 61 138 578—585 587 590
Generating function of Bernoulli numbers 13 111
Generating function of Bessel functions 221
Generating function of Jacobi polynomials 119
Generating function of Laguerre polynomials 119
Generating function of Legendre polynomials 60 247
Geometric series 207 210 225 460 580 597
Gershgorin bounds 663
Gershgorin disk theorem 636 639 640
Goursat's theorem 285 330
Grace, theorem of 472 473 474
Gradient, complex 334 346 349 544
Graeffe's method 500 552
Graph of real function 286
Grid, rectangular or hexagonal 503 505
Group 1 2 8 45
Group of Moebius transformations 302 307
Group, abelian 2 608
Group, non-abelian 8
Hadamard polynomials 625 626 627 630 633 634 642 663
Hadamard product 13 576
Hadamard, theorem of, on product of power series 576
Half-plane right 145
Hankel determinants 574 591 594 596 600 615 616 662 663
Hankel determinants, associated with rational function 603
Hankel determinants, asymptotic behavior of 602 612 646
Hankel determinants, recursive computation of 596
Hankel polynomials 622 625
Heine — Borel lemma 160 161 168
Helmholtz equation 342
Hermite polynomials 37 43
Hodograph plane 369
Homeomorphism 382
Homotopy 167 179
Horner algorithm 435 437 508 513 551 561 567 569
Horner scheme 437 438 556
Hurwitz, theorem of 283 284
Hyperbola 295 343
Hyperbola, mapping of interior of branch of 395
Hypergeometric series, classical 27 34 39 40 42 44 48 62 118 150 226 229 417 421
Hypergeometric series, confluent 27 29
Hypergeometric series, confluent, zeros of 662
Hypergeometric series, generalized 27 32—34 37 39 40 43 44 47 62 65 220 221
Hypergeometric series, generalized, differential equation satisfied by 29
Hypergeometric series, generalized, radius of convergence of 80
Hypothesis (H) 642 649
Identity element 1 8 11 45 303
Identity principle for analytic functions 88 147
Imaginary part 7
Imaginary unit 4
Inclusion disk 484 540
Inclusion radius 457 473 500
Inclusion radius, proper 457
Inclusion region, for poles of rational function 640
Inclusion set 514
Index critical 642 645 646 647 650 653
Influence 352
Instability, numerical 613
Integral of trigonometric functions 183
Integral of trigonometric functions, improper 254 255 262—264
Integral with respect to arc length 193
Integral with respect to parameter 183 184—186
Integral, along arc 188 189—194
Integral, along closed curve 195 200—206 233 243 245 246 247 249 252 257 265 272 277 350 351 353 366 628
Integral, domain 2 11 12 51 89
Integral, elliptic 417
Integral, improper 251
Integral, improper, of rational functions 253—254 263 264
Integral, improper, with fractional powers 257 264
Integral, improper, with logarithms 257 264
Integration of ordinary differential equations 662
Integration, numerical 234 517 590 621
Interior of Jordan curve 235 243 245—249 252 278 280 282 382 383
Interpolation by polynomials 572
Interpolation by polynomials, Lagrangian formula for 601
Interpolation by sums of exponentials 571 577 622 662
Interpolation by sums of exponentials, algorithmic solution of 631—632
Inverse in Banach algebra 72
Inverse of element in group 1 4 8 12 46
Inverse of Moebius transformation 303
Inversion  303 381 591
Involution 306
Isomorphism 15 46 302
Iterates of Moebius transformation 306
Iteration 523—552
Iteration function 533 542
Iteration function of arbitrary order 529
Iteration function of order three 531
Iteration function, Laguerre's 500 532 534
Iteration function, Newton's 529 531 532
Iteration function, order of 526
Iteration sequence 523 527 546
J.C.P. Miller formula 42 43 64 65 579
J.C.P. Miller formula, extension of 55
Jacobi polynomial 119
Jacobi's identity for Hankel determinants 595 611
Jacobi's identity for Hankel polynomials 622
Jacobi's transformation 63 65 228
Jacobian matrix of conformal map 339
Jordan block 128 130
Jordan canonical form 127 130 305 306
Jordan curve 235 243 245 246 266 269 272 278 279 282 330 331 366 383 485 515 520 628
Jordan curve theorem 235 285
Jordan curve, positively oriented 235
Jordan region 382 383 384 391
Jordan's lemma 255 256
Joukowski flow 367
Joukowski hypothesis 365 370
Joukowski map 294 299 315 345 353 359 367 394 395 415
Kahan's theorem 534
| Kakeya-Enestroem theorem 284 462 497
Kapteyn series 165
Kepler's equation 119 120 130 226 230
Kummer's first identity 29
L-R algorithm 641
Lagrange — Burmann expansion 58 65 97 100 101 104 138 226 249 370 529
Lagrange — Burmann expansion, convergent 100 102 104 119 120 138
Laguerre polynomial 119 618
Laguerre polynomial, generating function of 119
Laguerre's theorem on polynomial zeros 466 473 480
Laguerre's theorem, generalization of 478 534
Laplace's expansion 592 593 607
Laurent series 209 224 226 369 370 381 559
Laurent series, formal 51 52 54—56 59 627 628 630
Laurent series, multiplication of 219
Laurent series, uniqueness of 209 219
Laurent's theorem 209
Legendre function 226
Legendre polynomial 42 60 164 284
Legendre polynomial, generating function of 60 247
Lehmer covering 512
Lehmer's method 492
Leibniz rule for differentiation of product 144
Lemniscate 299
Length of rectifiable curve 188
Limes inferior 76
Limes superior 76
Limit, of function 73
Liouville's theorem 159 218 400 525 659
Logarithm 112 147 179 203 232 233 257 294
Logarithm of matrix 131 138
Logarithm, analytic branch of 203
Logarithm, principal value of 114
Logarithm, simple branch of 114
Logarithmic series 49 50 115
MAP 377
Map, conformal 329
Map, topological 382
Mapping of interior of ellipse 394
Mapping of interior of parabola 392
Mapping of regular n-gon 411
Mapping of triangle 407
Mapping of two-gon 406
Mapping theorem for disks 321
Mapping theorem for simply connected regions 378
Matrix, diagonal 306 630
Matrix, equivalent 299
Matrix, exponential 122
Matrix, non-singular 16 72 303
Matrix, partitioned 653
Matrix, semi-circulant 14
Matrix, similar 123
Matrix, triangular 171
Matrix, tridiagonal 635 653
Matrix, unitary 313 314
Matrix, upper triangular 14 45 47 314
Maximum for scalar analytic functions 104
Maximum, principle of 86 332 355 379 541
Method of descent 541
Metric, chordal 311
Mittag — Leffler theorem 655
Model problem 344 346 352 357 358 364 368 407 409
Modulus 5
Modulus of doubly connected region 386 391
Moebius transformation 300 348 376 384 389 418 419 463 465 474 491 524 662
Moebius transformation, composition of Schwarz — Christoffel map with 404
Moebius transformation, entire 316
Moebius transformation, group of 302
Moebius transformation, standard representation of 304 314
Moivre's formula 9
Moment of inertia polar 341
Monodromy theorem 167
Monodromy theorem for simply connected regions 168
Montel's theorem, on polynomial zeros 459
Neighborhood 73
Neumann series (of Bessel functions) 164
Newton correction 454 467 468 477 478 544
Newton's formula 479
Newton's method 121 528 532 533 547 651
Newton's method for polynomials 534—541
Newton's method with wrong sign 534
Nilpotent 79
Non-unit, in integral domain of formal power series 12 35 38 40 41 43 45 51
Norm 68 173
Norm, column sum 71
Norm, Euclidean 68
Norm, induced 71
Norm, of vector 173
Norm, row sum 71
Norm, spectral 71
Normality of formal power series 605
Null function 464
Nyquist diagram 279 280
Obstacle 380
Orthogonality of Hadamard polynomials 628
Osgood — Caratheodory theorem 383 384 397
Pade table 663
parabola 290 392
Partial fraction decomposition 218 565 631 662
Partial fraction decomposition of rational function 553 557
Partial fraction decomposition real 556
Partial fraction expansion 658
Partial fraction expansion of cotagens function 660
Partial fraction expansion of meromorphic function 655
Partial fractions 159 553—661
Path 165
Period 109
Permutation 591 598 645
Picard's theorem 216
Pivotal points, system of 187
Plana summation formula 271 274 285
Plane, cut along negative real axis 145
Plane, model 337
Plane, physical 337
Pochhammer symbol 24 25—30 32—34 39 40 43 47 48 51 53 54 57 62 94 117 121 148 151 155—157 163—165 174 254 417 633
Point at infinity 300
Poisson equation 371
Polar representation, of curve 236
Pole 214
Pole, order of 214
Poles, of equal modulus 641
Polygon, mapping of exterior of 406 413
Polygon, mapping of interior of 396
Polygon, unbounded 401
Polynomial 10 85 105 139 159 198 264 281 283
Polynomial zeros, determination of 137 497—550 617—651
Polynomial, characteristic 635
Polynomial, convergence-generating 655 656
Polynomial, deflated 513
Polynomial, reciprocal 492
Polynomial, self-reciprocal 496
Polynomial, stable 488
Polynomial, Taylor coefficients of 437
Polynomial, Taylor expansion of 433
Polynomial, trigonometric 284
Polynomials, algebraic theory of 433
Polynomials, analytic theory of 551
Power function, general 115
Power function, principal value of 116
Power function, simple branch of 116 117
Power series, differentiation of 18
Power series, formal 9 576 578 585 604 609
Power series, hypernormal 606 607 615 617 620 621
Power series, normal 605 610 611 613 628—630 638 642
Power series, radius of convergence of 77 151 156 656
Power series, reciprocal of 12 615
Power series, reversion of 47 48 49 55—61
Power series, scalar 84
Power series, ultimately normal 605
Power sum 454 481
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