Index
Abelian group, 521, 526 A-set. SeeAnalytic set Absolutevalue, 190 Asymptoticallyequal. 479 Accumulation, point of, 196 Atlas , 231; of holomorphically related Adjoint differentialform, 157, 167 charts, 245 Adjoint operator, 403 Atomic theory, 415 Adjoint space, 397 Automorphism group, 510, 511 Algebra, 524; Boolean, 91, 92; Axiomatic method, in geometry, 507-508 fundamentaltheorem of, 195-196; homo logical, 519-520; normed, 516 BAIREclasses, 460; first, 460, 462, 463; Almost all, 479 of functions, 448 Almost continuous, 460 BAIREcondition, 464 Almost equal, 479 BAIREfunction, 464, 473; non-, 474 Almost everywhere, 70 BAIREspace, 464 Almost linear equation, 321, 323 BAIREsystem, of functions, 459, 460 Alternating differentialform, 185; BAIREtheorem, 448, 460, 462 differentialoperations for, 159-165; BANACH, S., 516 theory of, vi, 143 BANACHfixed point theorem, 423 Alternative theorem, 296, 413 BANACHspace, 338, 340, 393, 399, 432, Analysis, v, 1; axiomaticmethod in, 435,437, 516; adjoint, 400; 512-518; complex, vi ; functional, conjugate, 400; dual, 400; theory of, vi 391; harmonic, 518; and number BANACHtheorem, 446, 447 theory, 500-501 Band spectra, 418 Analytic function, definedby function BAYES theorem, 109 element, 242 BELTRAMIdifferential equation, 325 Analytic numbertheory, 480 BERNOULLI, DANIEL, 23 Analytic operation, 468 BERNOULLI, JACOB, 89, 360 Analytic set, 448, 458, 465, 468, 469; BERNOULLI, JOHANN, 23 linear, 466 BERNOULLIdistribution, 96 Angle-preservingtransformation, 194 BERNOULLIlaw, of large numbers, 116 a-points, of function, 225 BERNOULLInumbers, 360, 362 Approximatelyequal, 479 BERNOULLIpolynomials, 360, 362n Approximation. best, 496; BERTRAND-CEBYSEVtheorem, for Diophantine, 495, 497 primes, 483 A priori estimate, 339 BESSELdifferential equation, 313 Arc, 125; smooth, 127 BESSELinequality, 398 Arc-length, 128 Bicompactification, 257 ARTIN, E., 510 BIEBERBACH, L ., 274, 321 A Rz EL A theorem, 410, 435, 437 BIEBERBACHclosure, 274
529 530 INDEX
Bilateral , J32 CAUCHY condition , 2, 396 Bilinear form , bounded, 402 CAUCHY criterion , generalization of , ] 2 Binomial distribution , 96 CAUCHY filter , ] 8, ] 9 BOHR-LA!\:DAU theorem, 486, 487 CAUCHY integral , and power series , Bo LTZMA!\:N, L., 89 222 - 223 BOLZA!\:O theorem, 447 CAUCHY integral representation , 238 Boolean algebra, 91, 92 CAUCIIY integral theorem , 207 , 217 , BOREL, F. E. E., 499, 500 220 , 223 , 224 , 236 , 237 BOREL-CANTELLItheorem. J J9 C A U Clly - Ko V A LE V SKI theorem , 336 BORELmeasurable function , 93, 101, CAUCIIY principal value , 347n , 407 111, 474n CAUCHY remainder formula , 37 , 216 BORELset, 93, 110, 448, 458, 464, 465, CAUCHY- R I EM A1' N dif Terential equations , 469; projection of , 470 204 , 205 BORELsystem, 458 CAUCHY sequence . 396 . 424 . 425 Bour-;o, least upper and greatest lower, CAUCHY theorem , fundamental , 2 , 10 - 11 , 55n 16 , 18 , 19 Boundary condition , 308 CAYLEY , A ., 509 Boundary point , 125 CEBYSEV, 89, 497 Boundary value problem, 294, 314, CEBYSEVapproximation theorem, 497 336; for elliptic differential CEBYSEVfunction, 489 equations, 337; first , 295; second, CEBYSEVinequality, 102- 103 295n; third , 295n Cru YSE V theorem, 484 Boundcd, 402 Chain conductor , 353 Boundcd convergence, 104 Chain rule , 29, 203 , 205 ; generation Boundcd variation , function of, 65 of , 44 BOURBAKI, N ., v, vi, 67, 78n, 506, 521, Characteristic curve . 303 , 333 522, 525, 527 Characteristic equation , 287, 380 Branching case for integral equations, Characteristic function , 54n , 107 ; 440n, 442 continuity of , 105 ; definition of , 103 BRIGGS, 494 Characteristic initial value problem , 335 BROUWEI{ fixed point theorem, 303, Characteristic manifold , 309 , 311 - 321 , 432, 433 322 , 337 , 344 B-sct. See BOI{EL set Chart , 231 ; holomorphically related , 232 Chordal distance , 263 CAHEN, 501 CHOWiA , 494 Calculus, of alternating differentials , Circle , quadrature of , 491 154; of differential forms, 125; of Closed curve , 130 residues, 348; theorems of infinitesimal , Closed form , 161 136; of variatiolls , 349, 422 Closed hull , 64n CANTO!{, GEOI{G, 508, 513 Closed plane , 233 ; theory of CANTOI{ intersection theorem, 451n functions on , 229 - 238 CANTOI{ set, 447, 450 Closure , of complex plane , 194 ; CANTOI{ theory, of sets, 513 torus - like , 267 CAf{ATf I E O DO I{Y, C., 194 Codif Terentiation , 164 Cardinal number, 524 Coefficients , undetermined , 280 CARLSON. 487 COHEN independence theorem , 475 CARMICHAELconjecture, 503 Cohomology , 520 CARTAN, ELlE, 124 Combinatorial topology , 519 CARTAN, HENRI, v Compactification , 257 , 261 , 262 , 269 ; Cartesian coordinates, 430 axioms for , 265 ; concept of , 253 ; Cartesian space. 26. 42. 45: n- neighborhoods and , 264 dimensional, 49 Compactness , 233 CA-set, 47] , 472, 473 Compact support , 68 Caslls irredllcibili .s-, 191 Comparison function , 294 Category. 447, first , 451; second, 451 Comparison theorem , 298 Catenary, 422 Compatibility conditions , 522 CAUCHY, A . L ., 89 Complement , 91 ovlto term, 404 , theorem 404n , 108 , Convolution 32 , Convolution Convex oriae itouto o 13; 143 , of introduction , Coordinates Convergent sequence , 1, 424 ; of ; 424 1, , sequence Convergent Complex analysis , vi , analysis Complex otn 7 -iesoa eeetr - elementary 474 , n-dimensional ; of 76 , questions , Content Constructibility ovrec 3 1, 2 2 ; 427 12, 10, 476 3, , , 475 , v , 448 413 , , Convergence hypothesis spectrum Continuum 455 24, , Continuous mapping Continuous 189 class, Congruence 189 325 , , 210 , Congruence mapping 508 Conformal , 322 Configuration , of Cone ; 524 , outer ; 514 , Composition otnos ucin, 6; oe series power ; 26 , function Continuous ood, 322 , Conoid Complex number , 188 ; fundamental ; 188 , number 272 , Complex manifold Complex 63 , 2 , of function set property ; Complete 19 11, , 202 , Completeness differential Complete INDEX otniy,2-2,44;o adto 6; 6 , addition of ; 454 499 , 28, 23- , fractions Continuity , Continued holomorphic ; 481 , analytic , 63 , Continuation axiom Content 97 , of amount ; 55 5, , Consistency 26 , Connectedness Conjugate , 190 ; harmonic , 211 , harmonic ; 190 , Conjugate Complete (content ), 76 ), (content 198 Complete set, Complementary ais f, 1 srn 36; theorem ; 396 , strong ; ; 115 216 , , of probability in radius ; 71n , normwise ucin 33 , functions 105 , weak f en 28; nfr 2 1, 7; 27 , 21 - 20 , uniform ; 278 , Peano of f, 3 oa 17;psto 1 ; 416 , position ; 177 , polar ; 131 , of oriae-ie, 5; aia 48; 438 , maximal ; 25 , -wise coordinate positive order of , 131 ; transformation ; 131 , of order positive , local ; 177 , in operator Laplace 34 , as 17 , subsets f, 9 eain o elementary to relation ; 192 representation , of Gauss ; 189 , for laws 1 eoopi 241 , meromorphic ; 217 62 , -Jordan Peano ; 57 , geometric one 14; ocp o 1 7 ; 479 1, , of concept ; 104 , bounded 410 , ; 462 , ; of 7 , uniform points ; 92 , measure multiplication of of ; 7 , inverse of 195 - 191 , geometry ocp o v; eiiin f, 9 ; 396 , of definition ; v , of concept 8 oetm,46;ngtv order negative ; 416 , momentum ; 185 f hrceitc ucin, 0 ; 105 , function characteristic of
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Differential (continued ) Euclidean , 192 , 196 ; notion of , 71 meromorphic , 234 , 237 , 245 , 246 ; Distribution , binomial , 96 ; of a pole , 235 ; residue of , 236 , 246 ; differentiation of , 84 ; Gauss , 96 ; of second kind , 237 ; total , 129 , 145 , global behavior of , 81 ; infinitely 202 , 204 ; of zero , 235 divisible , 122 ; marginal , 97 ; Differential derivative , 204 multidimensional normal , 97 ; Differential equation , Beltrami , 325 ; normal , 96 ; partial derivative of , 85 ; Bessel, 313 ; Cauchy - Riemann , Poisson , 96, 122 ; probability , 92 ; 204 - 205 ; direction field of , 278 ; Schwartz , 53 , 74 , 78 , 82 , 84 , 516 ; elliptic , 322, 323, 326, 327, 337, 341 theory of , 80- 82 ; uniform , 497 eigenvalue problem for , 341 ; Distribution function , conditional , 111 ; elliptic -parabolic , 325 ; Euler , 287 , 422 ; cumulative , 94 ; one - dimensional , 94 ; exact , 284 ; explicit , 277, 290- 294 ; of random variable , 95 of first order , 287 ; homogeneous , Distributive lattice , 90 286 - 287 ;hyperbolic , 322, 323, 326, Distributive law , 467 ; dual , 90 343 ; hyperbolic -parabolic , 325 ; Divergence , 431 hyperbolic -parabolic -elliptic , 358 ; Domain , 261 ; connected , 199 ; of implicit , 277 , 282 ; Laplace , 211 ; definition , 392 ; of dependence , linear , 286 , of second order , 289 ; 309 - 311 ; fundamental , 392 ; nonhomogeneous , 286 ; nonlinear , multivalent , 242 ; non -schlicht , 242 ; 518 ; of nth order , 281 ; order of , 277 ; of operators , 524 ; perturbation , 312 ; ordinary , 306 , 424 ; parabolic , 322, simply connected , 199 ; star -shaped , 161 323, 326, 330 ; partial , of second Double points , 130 order , 306 ; perturbed , 302 ; DUff I NG equation , 434 quasilinear , 302, 321 ; quasilinear system , 326 ; Riccati , 290 ; Eigenfunction , 296 , 341, 410 , 411 , 414 , Schrodinger , 417 ; singular points 417 , 435 of , 282 ; solution of , 276 , 278 ; Eigenvalue , 296 , 348 , 410 , 411 , 412 , 417 , ultrahyperbolic , 322 ; uniqueness 435 theorem for , 291 Eigenvalue problem , 515 ; regular , 341 ; Differential form , adjoint , 157, 167, singular , 341 170 ; alternating , 154- 159, 185 ; El LE N BE R G , 271 complex -valued linear , 205 ; Elastic body , vibrations of , 307 conjugate , 207 ; in Euclidean plane , Electron , motion of , 417 176 ; in Euclidean three - dimensional Elementary differential , 154 space , 177 ; exact , 206 ; exterior Elementary function , 67, 214 ; potential multiplication of , 151 ; of first degree , theoretic method of , 249 145 ; Grassman ring of , 159 ; Elementary integral , 67 module of , 146 ; normal form of , 156 ; Elements de Mathematique , 506, 521 of second order , 150 , 237 Elliptic . See Differential equation Differential quotient , 22, 29, 30- 31 End point , 126 Diophantine approximation , 495 Energy integral method , 327, 331 , 332, Diophantine equation , 495 334 DIRAC delta - function , 79 Energy opera tor , 417 DIRAC measure , 83 ; derivative of , 86 Entire function , 481 Directed set , 12 , 14 Entourage , 18 Direction filter , 15 Estimate , a priori , 339 DIRICHLET , P . G . L . , 23 , 497 Epsilon -language , 479 DIRICHLET approximation theorem , 497 Epsilon -neighborhood , 6 DIRICHLET discontinuous factor , 104 Equal in the limit , 479 DIRICHLET function , 214 , 448 , 459 Equation (see also differential DIRICHLET series , 489 equation ), almost linear , 321 , 323 ; DIRICHLET theorem , 479 ; on prime branching , 443 ; characteristic , 287 , numbers , 488 380 ; potential , 306, 314 Discontinuity , points of , 463 Equivalence class , of number triples , Discriminant manifold , 282 259 ; of parallel lines , 259 Distance , 11 ; definition of , 338 ; ERATOSTHENES , sieve of , 483 INDEX 533
ERDOS , 485 , 489 Field , 510 ; of sets , 63 , 76 Ergodic theorems , 1 J9 Filter , v , 2 , 12 - 16 , 525 ; Cartan , 513 ; ERLANGER Programm , 509, 517 Cauchy , 18 ; direction , 15; finer , 15 ; ESENIN , 475 Frechet , 14 ; image , 15 ; system of , 14 ESTERMANN , 491 Filter basis , 15 Euclidean algorithm , 383 First (second ) mean value property , 318 Euclidean distance , 192 , 196 Fixed point , of mapping , 423 Euclidean geometry , 507 ; points at Fixed point theorem , 430 ; nonlinear , infinity in , 255 434 - 437 Euclidean motion , 194 Flow of gases , 336 Euclidean plane , 192 ; differential forms Forced oscillation , 302 in , 176 ; function -theoretic closure of , Foundations of mathematics , 509 262 - 263 FOURIER coefficients , 398 , 399 , 401 Euclidean space , four - dimensional , 274 ; FOURIER integral , 518 subsets of , 24 - 25 FOURIER integral theorem , 347 Euclidean three - dimensional space , FOURIER series , 23 , 518 differential forms in , 177 FOURIER transform , 403 EUCLID - HILBERT system of axioms . 253 Fractions , continued , 499 EU CL I D' S theory of primes , 480 FRAt-;KL'S problem , 334 EULER , L . , 489 . 493 . 494 FRECHET differential , 419 , 421 , 423 , 439 EULER constant , 364 , 365 F RE C HE Tfilter , 14 EULER differential equation , 287 , 422 FREDHOLM , MOn , 515 EULER equation , 421 , 422 FREDHOLM theorems , 413 EULER function , 489 , 495 FRIEDRICHS , K . 0 ., 313 EULER functional equation , 367 FUETER, Rudolph , 274 EULER gamma -function , 355 Function , of a-points , 225 ; comparison , EULER identity , 479 294 ; complex -differentiable , 204 ; EULER integral formula , 374 , 388 complex -valued , 200 ; connected with EULER integral representation , gamma - measures , 82 ; continuous , 26 ; function , 371 - 374 definition of , 23 ; differentiable , 39 ; EULER multiplier , 285 Dirichlet , 214 , 448 , 459 ; domain - EULER - RIEMA !'-:N zeta - function , 481 continuous , 201 ; domain of , 200 ; EULER summation formula , 363 , 364 elementary , 67, 214 ; entire , 481 ; Events , 90 ; certain , 91 ; possible , 91 exponential , 22, 38- 42 ; extension Exact differential equation , 284 of , 448 ; general thcory of , 251 ; Exact differential form , 206 harmonic , 211 , 318 ; holomorphic , Existence , questions of , 336- 348, 474 207 , 232 , 240 , 246 , 250 , 257 , 271 ; Existence condition , 308 implicit , 473 ; influence , 410 ; Existence problem , 327, 328 inverse , 231 n ; kernel , 349 ; Jim -closed Expansion theorem , 343, 414 set of , 459 ; locally constant , 454 ; Expectation , 99- 103 ; conditional , 108, many -valued , 238 ; as mapping , 215 ; 112 - 113 measurable , 93 ; meromorphic , Exponential function , 22, 38- 42 228 , 232 , 240 , 243 , 246 , 250 ; Extended matrix , 51 minorant (majorant ), 328 ; Extension , methods of , 73 - 74 ; monotone , 463 ; negative part of , theorem of , 95 55n ; nowhere differentiable , 446 ; Extension for measures , theorem of , 95 perturbation of , 296 ; positive Exterior multiplication , 155 ; of definite , 304 ; positive part of , 55n ; differential forms , 151 real -analytic , 215 ; real - differentiable , 202 ; reciprocal , 231n ; FECHNER , 89 regular , 207 ; restrictions on , 448 ; FELLER , W . , 121 scalar , 26 ; separated , 297 ; set, 63, FERMAT numbers , 488 94 ; sine , 41 ; singularity of , 315 , 319 ; FERMAT theorem , 252 stem , 284 ; step , 54 - 56 , 99 ; upper - FIBONACCI (Leonardo of Pisa), 351 (lower -) semicontinuous , 463 ; value FIBONACCI numbers , 352 , 383 of , 55n ; variance of , 107 ; variation FIBONACCI sequence , 352 , 353 , 383 of , 454n , 456 ; wave , 416 534 INDEX
Functional , 57 ; bounded , 396 ; GREEN formulas , 182 , 314 , 316 continuation of , 58 ; extension of , 58 ; G RE ENS function , 329 , 410 , 434 ; linear , 56 , 392 , 393 , 397 ; existence of , 317 positive , 58 ; Riemann extension of , 68 Group , 313 , 321, 509, 519 ; Abelian , Function element . 241 521 , 526 ; of automorphisms , 510 , Function theory , in the large , 125 ; 511 ; Lie , 511 ; with operators , 511 ; in the small . 125 . 206 . 290 with parameter , 526 ; topological , Functional analysis , 391 4 - 5, 517 , 518 Functional dependence , 51 G -set , universal , 476 Functional equation , 353, 367 Guinea -pig problem , 351- 352
Functional matrix , 45 ; rank of , 51 . Fundamental solution , 315 , 316 , 317 , 356 HADAMARD , J ., 313 , 314 , 321 Fundamental theorem of algebra , 195- 196 HADAMARD method , of descent , 310 Fundamental theorem of CAUCHY , 10 HARDY , G . H . , 498 Fundamental vibration , 411 Harmonic analysis , 518 Harmonic functions , 211 , 318 GALOIS , E ., 510 HAUSDORFF axioms , 264 , 472 GALOIS group , 510 H A US DO Rf F, separation axiom , 4 ; GALOIS theory , 518 space , 2 , 4 , 245n , 264 , 266 ; Gamma -function , vi , 355 , 365 - 379 , connected , 245n ; connected compact , 500- 501 ; Euler functional equation 272 ; example of , 6 ; locally Euclidean , for , 367 ; Euler integral representation 262 of , 371- 374 ; Stirling formula for , 371 Heat conduction , equation of , 307 , 319 GAUSS , K . F ., 89 HEAYISIDE function lI , 80 , 85 GAUSS complex plane , 255 HEILBRONN . 491 GAUSS distribution , 96 HEINE - Bo RE L theorem , 141 , 198 , 233n , GAUSS integral theorem , 176, 181, 334 256 , 433 , 513 GAuss -KRONECKER integral , 429 , 431 - 432 HELLY , theorem of , 105 - 106 GAUSS plane , 226 , 354, 355 ; line at HELMERT - PEARSON , distribution of , 108 infinity in , 253 HENSEL , L ., 501 , 519 GAUSS representation , of complex HERMITE , C ., 494 numbers , 192 I -IERMITE theorem , 494 GAUSS theorem , 430 HESSE normal form , 192 G ELF ON D, A . 0 ., 517 Hierarchy of structures , 521 GELLERSTEDTproblem , 335 HILBERT , D ., 491 , 507 , 508 , 515 , 517 , 523 General solution , 292 HILBERT axioms , 253 , 259 Geodesic lines , 422 HILBERT sequence space , 395 , 397 Geometry , absolute , 508 ; axiomatic HILBERT space , 393 , 394 , 399 , 400 , 401 , method in , 507 - 508 ; Euclidean , 507 ; 405 , 412 , 515, 516 ; complete , 397 ; Riemannian , 170 theory of , vi GLEASON , A ., 517 H I LB ER T' S fifth problem . 517 Global , meaning of , 439 H I LB ER T' S seventh problem , 494 Global behavior , of distributions , 81 HINCIN , ] 21 Global properties , 453 HINCIN theorem , of continued fractions , Globaluniformizingparameter , 250 499 GNEDENKO , B . W ., 121 HOHEISEL , 487 GODEL , 471 HOLDER continuous function , 337 GODEL independence theorem , 475 HOLDER theorem , 354 , 375 - 379 GOLDBACH conjecture , 489 , 490 Holomorphic differential , 234 GOURSAT , E . J . B ., 207 Holomorphic function , 207 ; in Graded ring , 159 complex plane , 215- 226 ; differential Gradient , 129 of , 208 ; local derivative , 234 ; GRASSMAN , II ., 124 sequence of , 223 GRASSMAN algebra . 159 : of alternating Holomorphic part , 229 differential forms , 154 - 159 ; exterior , Holomorphy , vi 525 Ho PF sigma -process , 275 GRASSMAN ring , of differential forms , 159 Hull of a set , 450 INDEX 535
HURWITZ polynomial , 300 Iteration, 428, Picard-Lindelof method, Hyperbolic (See also Differential 279 equation ), motion , 194 ; Riemann surface , 250 JENTZSCHtheorem, 435 Hypercomplex system , 525 JORDANarc, 201 Hypertranscendental number , 493 JORDANcontent, 62, 63, 66, 76 JORDANcurve, 200, 201 Ideal , 512 JoRDAN-measurable, 62 Identity theorem . 223 . 228 . 247
Incidence theorem , 259 KELLOGG theorem , 339
Inclusion . 428 . 521 KEPLER , J . , 254
Indcfinite integral , 87n ; of holomorphic Kernel , 436 ; open , 64n ; resolvent , 440
function , 208 Kernel function , 349 , 438
Independence , 98- 99 Kernel matrix , 401 , 403 , 407
Independent SCHWARTZ distribution , 82 Kinetic energy , 416
Indeterminacy , limits of , 8- 10 KIRCHHOFF law , 353
Index set , 2 KLEIN , FELIX , 507 , 509
Inertia , index of , 322 KoKsMA , J . K . , 497
Infimum (greatest lower bound ), 55n KOLMOGOROV , A . N . , 89 , 118 , 121
Infinitely divisible distribution , 122 KOLMOGOROV inequality , 114 - 122
Infinitesimal increment , 44 , 45 KONIG , H . , 86 , 87
Infinity , line at , 253 ; plane at , 273 ; KRULL , 511 , 512
point at , 252 , 253- 259, 260 ; KUREPA , 475 , 485 neighborhoods of , 230
Influence , range of , 309 , 311 , 322 LAGRANGE , J . L . , 491
INGHAM , 486 LAGRANGE remainder formula , 37
Initial conditions , 292 LAME theorem , 383
Initial data , continuous dependence on , LANDAU , E . , 484 , 487 , 491
308 LANDAU symbol , 479 , 482
Initial point , 126 LAPLACE differential equation , 211
Initial value problem , 343 LAPLACE operator , 164 , 177
Initial values , propagation of , 313 LAPLACE transform , 346 , 386 - 389 , 404 ,
Inner composition , 514 405
Inner - mathematical , 509 Lattices , structure of , 522 ; theory of , 512
Integrability condition , 284 LAURENT series , 229 , 236
Integrable form , 161 Law of large numbers , " empirical , "
Integral , 53 ; curvilinear , 127, 178, 235 ; 115 ; strong , 116 - 119 ; weak , 115
definite , 57 ; step , 57 ; surface , 131 , LEBESGUE , H . L . , 72 , 470
137 , 179 ; volume , 179 LEBESGUE convergence theorem , 72
Integral curve , 278 , 290 , 291 LEBESGUE extension , of functional , 68
Integral equation , 292 , 341, 514 ; LEBESGUE integrability , of functions , 87
algebraic , 421 , 435 ; nonlinear , 437- 444 ; LEBESGUE - integrable , 83 , 399
of second kind , 410 ; theory of , vi LEBESGUE integrable theorem , 73
Integral representation , 220 ; LEBESGUE integral , v , 53 , 66 , 68 , 101 ,
of functional , 64 453 ; definite , 73 , 87n
Integral ring , 224 LEBESGUE measure , 53 , 66
Integrating factor , 285 LEBESGUE measure zero , 60
Integration , elementary theory of , 53- 66 ; LEBESGUE - $ TIELTJES integral , 68 , 73 , 78
Lebesgue theory of , 86 ; LEBESG UE theory , 53 ; classical , 78 ; of
methods of , 277- 290 ; Stieltjes , 65 integration , 86 ; of measure and
Integration and measure , theory of , 77 integration , 508
Interior points , 125 Left reciprocal , of matrix , 406
Interval , 54 LEIBNIZ , 23 , 507 Inverse , continuity of , 7 Leonardoof Pisa. SeeFIBONACCI Isolated , 198 , 449 LEVI, B., 72 Isomorphism, 92 LEVY, P., 121 Isoperimetricproblem, 422 LEWY, H., 313 536 INDEX
Liber abbaci, 351 Majorizing sequence , 8 LIE, SOPHUS, 517 Manifolds , characteristic , 309 , 311 , 321 , LIE groups, 511, 517 322, 337, 344 ; complex , 272 ; LIE ring , 511 differentiable , 184 - 186 , 513 ; Limes inferior (lower limit ), 8. discriminant , 282 ; orientable , 140 , Limes superior (upper limit ), 8. 185 ; Riemannian , 124 - 125 , 184 - 186 ; Limit , calculation with , 5- 6; concept two - dimensional , 231 , 245n of, 2, 3, 512; monotone, 58; theorems Mapping , 23, 26, 41, 42 ; angle - of, 114-122; uniqueness of, 4 preserving , 210 ; biholomorphic , 258 ; Limit number, 196 complete , 424 ; conformal , 210 , 325 ; Limit point , 11, 196, 198 continuous , 24 , 455 ; contractive , Limits of indeterminacy, 8- 10 423 , 424 ; defined by holomorphic LINDEBERG-FELLER, theorem of, 120 functions , 209 ; equiform , 194 ; LINDELOFconjecture, 487 extension of , 449 ; fixed point of , 423 ; LINDEMANN-WEIERSTRASStheorem, 495 function as, 215 ; homogeneous , 201 ; Linear difference equation, 353, 354; homogeneous linear , 45 ; linear , 516 ; existence theorems for , 388 locally univalent , 209 ; open , 455 ; Linear differential equations, 286; of projection , 239 ; scale-preserving , 209 ; second order, 289 schlicht , 209 ; segment -preserving , Linear form , 57 209 ; sense-preserving conformal , 211 ; Linear functionals , 392, 393; sense-reversing conformal , 211 integration of, 56; and operators, 393 Marginal distribution function , 97 Linear independence, of functions , 388 Marginal probability distribution , 97 Linear operator, 203, 392, 393; Mathematical logic , 509 completely continuous , 408 Matrix , left reciprocal , 406 Line at infinity , 259 , 260 Maximum , of two functions , 54n Line -element , 278 , 325 ; regular , 282 ; Maximum -minimum principle , 327, 328 singular , 282 Maximum principle , for functions of a L -integrable . See LEBESGUEintegrable complex variable , 226 L -integral . See LEBESGUEintegral MAXWELL , J . C ., 89 LIOUVILLE , J ., 221 , 296 , 496 MAXWELL equations , 307 LIOUVILLE theorem , 221 . 233 , 492 Mean of k , 102 LIPSCHITZ condition , 279 , 281 , 291 , Mean density , 497 292 , 425 , 426 Mean - value theorem , 30 LIPSCHITZ constant , 279 Measurability , of functions , 66 LITTLEWOOD , 498 Measurable function , 93 LJAPUNOV criterion , for stability , 301 Measurable set , 63 LJAPUNOV Theorem , 120 , 122 Measurable space , 93 Local , meaning of , 439 Measure , 53, 67 ; theory of , 77 Local compactness , 453 Measure and integration , theory of , 77 Local coordinates , 185 Measure /If , density of , 83 Local density , 453 Mechanical system , 416 Local parameter , 231 , 234 Medium , of function , 74 Local properties , 453 MELLIN , 501 Local theory , 439 Membrane , vibrations of , 307 Local uniformizing parameter , 244 Meromorphic continuation , 241 Locally compact topological space , 514 Meromorphic differential , 234 Locally countable , 453 Meromorphic function , algebraic Logarithms , natural , 38 equation for , 248 ; algebraic function Logarithm function , 22, 40 field of , 248 ; in complex plane , 226 - 229 LOOMAN - MENCHOFF theorem , 208 Meromorphy , 228 LoWER limit , 8 Metric spaces , 11 - 12 , 423 , 513 LUZIN , N ., 466 MIN , 486 Minimax , 411 MACLANE , 271 Minimum , of two functions , 54n Majorant (minorant ) functions , 328 Minorant (majorant ) functions , 328 Majorant series, 441 Minorizing sequence , 8 INDEX 537
Mirror image , 193 Nonlinear differential equation , 518 MITROYIC , D ., 494 Nonorientability , 270 Mixed structure , 522 Norm , 69 , 190 , 396 ,\1-neighborhood , 449 Normal form , 323 ; of differential form , MC)BIUS strip , 262 156 Model , 508 Normalization , 398 Modification , concept of , 275 ; Normed al .l:!;ebra , 516 meromorphic , 275 ; proper Normcd space , 69n continuous , 275 ; theory of , vi , 253 North pole , 263 Module , 519 ; of sets , 452 NOVIKOV independence theorem , 471 , 475 Modulus problem , 250n NOVIKOV separation theorem , 472 Molecule , 418 Number , cardinal , 514 ; complex , Moment , 102 ; " central ," 102 ; first , 102 188- 191 ; hypertranscendental , 493 ; Momentum coordinates , 416 natural , 524 ; revolution , 429 , 432 ; Monoid , 524 transcendental , 491 Monotone increasing (decreasing ), 7 Number circle , 498 Monotone limits , 58 Number sphere , 262 Monotone sequences , 7 - 8 ; Number theory , 499 , 500 fundamental theorem on , 7 Number triples , equivalence class of , 259 ; Mo N T GO ME R Y, 517 singular equivalence class of , 260 MOOI { E- SMITII convergence , 526 Numerical sequence , limit of , 197 MOO ({ E- SMITH sequences , v , 2 , 12 - 13 , 14 , 15 Open kernel , 64n Motion , 42 ; of electron , 417 ; Open set, 18, 197 hyperbolic , 194 Operator , 392 ; adjoint , 403 ; completely Mountain , 43 continuous , 341 , 342 , 413 ; domain (mu * -)almost everywhere , 70 of , 524 ; in a domain , 415 ; energy , mu * -equivalence , 70 417 ; groups with , 511 ; integral (mu *-) integrable function , 71 representation of . 415 ; linear , 400 ; mu -integral , 71 nabia , 180 ; self -adjoint , 408 , Multidimensional normal distribution , 97 413 - 418 ; solution , 346 ; star , 159 ; Multiple integrals , vi ; laws for , 125 symmetric , 341 ; unbounded , 415 Multiplication , alternating , 151, 153 ; Operator equations , 423 , 434 - 437 continuity of , 7 ; exterior , 151, 155 Ordered pair , 17 Multiplicity , of zero , 195 Ordered set , 521 (mu -)measurable , function of , 74 Ordinary differential equations , vi (mu -)summable function , 71 Ordinary point , 239 mu -upper integral , 69 Orientability , 268 (mu *-)zero function , 70 Orientable , 185 , 268 ; manifold , 140 , 185 ; surface -segment , 137 Nabia operator , 180 Orientation , 126 ; induced , 132 , 140 ; Natural number , 524 opposite , 126 n - dimensional space , functions in , 42 - 52 Oriented boundary , 126 Neighborhood , 3, 20, 429 ; and Oriented Riemann surface , 246 compactification , 264- 266 ; concept Oriented surface -segment , 134 of , 196 ; coordinate , 184 ; deleted , 225 ; Orthogonality , 398 elementary , 196 ; system of , 263 Orthonormal system , 398 , 412 , 442 ; Neighborhood filter , 14 complete , 398 Neighborhood set, 198 Orthonormality , 168, 186 NEUMANN series , 428 Oscillation , forced , 302 ; free , 302 ; NEWTON , Sir ISAAC , 507 theorem of , 298 NOETHER , EMMY , 510 , 512 OSGOOD space , 273 Nonanalytic segment , of surface , 325 Outer composition , 524 Non - Bo RE L set , 458 Non - Euclidean motion , 194 p -adic numbers , 519 Nonhomogeneous differential equation , PAPPUS and PASCAL theorem , 254 , 255 , 286 270 538 INDEX
Parabolic curves , 325 Position coordinates , 416 Parallel lines , 252 ; equivalence class of , Potential energy , 416 259 Potential equation , 306 , 314 Parameter , 126 : global uniformizing , Potential - theoretic method , for 250 ; local uniformizing , 244 , 271 ; elementary functions , 249 variation of , 289 Power , 446 Parameter planes , 244 Power series , 34 - 35 , 215 , 229 ; and Parametric representation , of arc , 126 Cauchy integrals , 222 ; derivatives PARSEVAL equation , 404 of , 216 Partial derivative , 43 ; of distribution , 85 Preservation of domains , theorem on ,226 Partially ordered , 55n Prime ideal , 517 Partition , 54 ; of integral , 13 Prime number theorem , 482 ; PASCAL and PAPPUS theorem , 254 , 255 , 270 consequences of , 484 - 485 ; Paths , 234 ; bounding system of , 217 ; elementary proofs of , 489 ; second , 487 , simple , 219 ; simple closed , 220 488 PCA -set , 471 Primes, Bertrand -Cebysev theorem, PCPCA -set , 471 483 ; Bohr - Landau theorem , 486 ; PEA NO, G ., 508 ; axioms of , 518 ; Cudakov theorem , 485 ; Dirichlet convergence theorem of , 278 theorem , 488 ; Euclid theorem , 480 ; PEA NO-JORDAN content , 62 Selberg theorem , 486 PEA NO theorem , 290 Principal axes, 322 Periodic solution , 302 Principal part , 229 Perpendicular , 193 Principe du recollement des morceaux , 81 PERRON , 0 ., 326 , 328 , 336 , 338 Probability , 499 ; concept of , 89- 93 ; Perturbation , 296 , 312 conditional , 108 , 1 ] 0 , 111 ; Perturbed differential equation , 302 converg (~nce in , 115 ; discrete , 96 ; Pfaffian form , 145 , 156 , 160 , 168 ; events and , 90 ; limit theorems in , 119 ; orthonormal basis of , 170 , 186 objective , 89 ; subjective , 89 ; Phase plane , 303 theory of , vi , 90 Phi -function , 358 Probability density , 95 Pi , 351n , 491 Probability distribution , 92 PICARD - LINDELOF method of iteration , Probability measure , 92 279 Probability space , 93 Piecewise smooth curves , 130 Product rule , 29 Piecewise smooth surfaces , 130 Projection mapping , 239 , 243 ; PLANCK quantum of action , 417 stereo graphic , 194, 230 , 262 Plane , closing of , 266- 271 ; Euclidean , Projective plane . 259 ; properties of , 176, 192, 259 , 262- 263 ; infinitely 259 - 262 distant , 273 ; nonorientable , iparameter Projective sets, 471 , 244 ; projective , 259- 262 Property , descriptive , 447 ; metric , 447 ; Plane triadic set , 477 strong maximum (minimum ), 318 ; POINCARE lemma , first , 160 , 164 ; topological , 447 second , 161 , 164 , 176 Point , 3 ; of accumulation , 196 ; Quadratic form , 321, 326 branch , 210 , 239 ; continuity at , 454 ; Quadrature , 284 ; of circle , 491 ; of continuity , 461 ; double , 130 ; end , elementary problem of , 277 126 ; hyperinfinitely distant , 274 ; Quantum mechanics , 416 ideal , 227 ; at infinity , 252 , 259 , 260 , Quarternion , 511 neighborhoods of , 230 , use fulness of , Quaternion function , right -regular , 274 253 - 259 ; initial , 126 ; isolated , 198 ; Quasilinear , 302 , 321 , 326 limit . 11 . 196 . 198 : O -Den set of . 197 : ordinary , 239 ; oriented , 126 ; singular , Radius , of convergence , 216 282 RAoo theorem , 275 Point spectrum , 413 RAOO - BEHNKE - STEIN - CARTAN theorem , POISSON , S . D ., 89 257 POISSON distribution , 96 , 122 Radon measure , 74 , 82 , 83 , 84 ; Pole , 235 ; of a differential , 235 derivative of , 86 INDEX 539
Random variable , 93 ; distribution of , 94 ; 257 ; of parabolic type , 250 ; simply independent , 117, 121 ; connected , 246 , 249 ; in the small , multidimensional , 95 ; Poisson - 246 ; theory of functions on , 243 ; distributed , 108 ; probability type of , 250 ; univalent (schlicht ), 239 distribution of , 112 ; uniqueness RIEMANN zeta -function , 481 , 485 theorem for , 104 RIESZ , F . , 65 Range (of a function ), 200 RIESZ space , 56 Range of influence , 309 , 311 , 322 Right reciprocal , of matrix , 406 Real -analytic functions , 215 Ring , 511 ; graded , 159 ; topological , 4 Reciprocal , 190 ; left , of matrix , 406 ; Rotation , 135 ; surface of , 422 right , of matrix , 406 Rotation -dilation , of Euclidean plane , Recursion , process of , 353 189 Recursive law , 351 Rotation group , two - parameter , 269 Recursive sequence , 351 - 353 ROTH theorem , 492 Reflection , 194 Regular solution , 282 Scalar functions , 26 Relation , transitive , 12 Scalar prodllct , 178, 192 Relaxation , 302 SCHAUDER , 339 REMAK , 328 SCHAUDER fixed point theorem , 432 , Removable -singularity theorem , 274 435 , 437 Residue , 224 ; calculus of , 348 Schlicht , 239 Residue class ring , 512 SCHMIDT , E . , 442 , 515 Resolvent kernel , 440 SCHRODINGERdifferential equation , 417 Restriction , of a function , 448 SCHUR , I ., 300 , 301 Revolution number , 429 , 432 SCHWARTZ , L ., 53 , 78 , 79 , 82 , 86 , 87 RICCATI differential equation , 290 SCHW ARTZ distribution , 53 , 74 , 78 , 84 , RIEMANN , G . F . B . , 501 , 507 516 ; independent , 82 RIEMANN conjecture , 486 , 487 SCHWARZ , H . A ., 426 ; alternation RIEMANN extension , of functional , 68 method of , 426 RIEMANN filter basis , 61 SCHWARZ inequality , 342, 395, 408 Riemannian geometry , 170 SCHWARZ reflection principle , 314 Riemannian manifolds , 125 , 126 ; SCHWARZ theorem , 427 differential forms on , 184 - 186 ; SEBASTIAO e SILVA , J ., 82 , 87 theory of , 124 Second category , 447 Riemannian metric , 185 SELBERG , 489 RIEMANN integrable , 56, 58, 59, 60, 64 ; SELBERG theorem , 486 improperly , 59n Semicontinuity (upper and lower ), 464 RIEMANN integral , v , 13, 53, 61, 62, Semigroup , 321 , 346 , 347 133 , 527 ; content associated with , Semi norm , 69 ; defined by abstract 62 - 64 ; definite , 59 , 64 ; indefinite , 59n ; measures , 68 ; in the wider sense , 69n rules for , 84 ; sum - definition of , 60 - 61 Sense-preserving (and sense- RIEMANN lemma , 226 , 247 reversing ) conformal mapping , 211 RIEMANN mapping theorem , 317 Separability , 472 RIEMANN - MELLIN inversion formula , 405 Separated functions , 297 RIEMANN sphere , 230 , 263 , 271 ; Separated variables , 285 Cartesian product of , 273 Separation axiom , 472 RIEMANN-STIELTJES integral , 101 Separation of variables , 346 RIEMANN sum , limiting value of , 61 Sequence , 2- 7 ; convergence of , 196 ; RIEMANN surface , 200 , 238 - 241 , 263 , Fibonacci , 352 , 353 ; majorizing , 8 ; 271 , 513 ; abstract , 245 ; of algebraic minorizing , 8 ; monotone decreasing , 7 ; functions , 248 ; compact , 246 ; monotone increasing , 7 ; of compact meromorphic function , Moore - Smith , 12 - 13 247 ; concrete , 245 ; defined by Sequence of points , limit of , 197 function element , 242 ; of hyperbolic Set, analytic , 448 , 458n , 465 , 466 , 468 , type , 250 ; in the large , 246 ; 469 ; Borel , 458 ; closed , 198 , 261 , 450 ; meromorphic separability of , 248 ; compact , 198 ; complementary , 198 ; oriented , 246 ; and points at infinity , coverable , 199 ; dense , 451 ; 540 INDEX
Set (continued ) Star-shapeddomain, 161 dense - in - themselves , 449 , 450 ; State, 313 everywhere -dense , 451 ; Fa , 456 ; F a6, STEINITZ, 510 470 ; field of , 63, 76 ; G6, 456 ; hull of , Stemfunction, 284 450 ; larger , 446 ; Jim-closed , 459 ; Step function , 54 - 56 , 99 ; integration module of , 452 ; nowhere - dense , 451 ; of , 56 open , 261 , 450 , 456 ; ordered , 521 ; Step integral , 57 perfect , 450 , 451 ; plane triadic , 477 ; Stereo graphic projection , 194 , 230 , 262 projective , 448 ; richer , 446 ; STIEL TJES integral , 64 - 66 , 105 ; sequentially compact , 432 ; ternary generalization of , 65 ; integration Cantor , 447 , 450 ; theory of , 508 ; function of , 65 uncountable analytic , 451 ; well - STIEL TJES sum , 65 ordered , 470 ; zero , lOOn STIRLING formula , 368 - 371 , 501 Set function , 63 , 94 Stochastic process , 89 Sheaf theory , 520 ST OK ES' theorem , 153 , 171 - 175 , 176 , S I D LO YS K Il, 493 181 ; special , 135 Sieve of ERATOSTHENES , 483 STOLTZ , 0 ., 203 Sigma -additivity , 76, 77 STONE , M . H ., 68n , 92 Sigma -algebra , 92 STONE isomorphism theorem , 92 Sigma -field , 76, 77, 92n String , infinitely long , 308 ; Sigma -finite measure , 77 vibrations of , 307 , 309 Sigma -module , 452 Strong maximum ( minimum ) property , Sigma -operations , 466 , 468 318 Simple path , 219 Structure , 507 ; algebraic , 521 ; Simply connected curve , 131 complex -analytic , 245 ; hierarchy of , Sine function , 41 521 ; monomorphic , 50S ; theory of , Singularity function , 315 , 319 vi ; topological , 521 ; uniform , v , 2 , Singular solution , 282 17 , 18 , 71n , 513 SKEWES number , 488 Subsets , composition of , 17 Skew field , 511 Sum , of two functions , 54n Skew -symmetric tensor , 148 Summation method , for difference " Smallest complete measure ," 77 equations , 355 G - measurable , lIOn Support , of function , 68 Smooth segment , 131 Supremum ( least upper bound ) , 55n Solutions , a priori estimates for , 329 ; Surface , nonorientable , 262 ; piecewise fundamental , 315 , 316 , 318 , 356 ; smooth , 130 ; Riemann ( see RIEMANN fundamental system of , 288 ; general , surface ) ; of rotation , 422 380, 381 ; linearly independent Surface -element , 210 , 245n ; vector , 135 fundamental , 514 ; periodic , 302 ; Surface integral , 131 , 137 , 179 singular , 282 ; special , 380 ; stable , Surface -segment , oriented , 134 , 137 298 ; trivial , 295 SUSLIN , 465 , 470 , 475 Solution operator , 316 , 346 Synchronization , 302 Space , 507 ; adjoint , 397 ; complex - projective , 273 ; locally compact , Tangent vector , 22 , 43 , 128 ; outwardly 521 - 522 ; metric , 11- 12 , 423 , 513 ; directed , 132 semicompact , 265 ; sequentially TAYLOR formula , 36 ; remainder term in , compact , 265 ; topological , 1, 3, 230 , 22 513 ; uniform , 16 - 21 TAYLOR series , 22 , 37 , 38 , 215 Spectrallines , 417 Tensor , contravariant , 148 ; covariant , Spectrum , 415 ; of operator , 413 148 ; skew - symmetric , 148 , 149 Spherical waves , 313 Tensor product , 525 Square -integrable , 399 Termwise differentiation , 35 Stability , 298 ; Ljapunov criterion for , 301 Ternary CANTOR set , 447 , 450 Stable solution , 298 Theorem , ergodic , 119 Standing waves , 417 TIHONOV , 320 Starlike , 334 TITCHMARSH , 487 Star operator , 159 TOEPLITZ theory of linear equations , 515 INDEX 541
Topological field , 4 , 6 56 , 58 , 59 ; line - element , ] 28 ; Topological group , 2 , 4 - 5 , 517 , 518 product , ] 78 ; space , 55n , 80 ; surface - Topological ring , 4 element , ] 35 Topological space , ] , 3 , 230 , 513 Velocity , 43
Topological structure , 52 ] Vibration , of beam , 4 ] 0 ; fundamental , Topological vector space , 516 41 ] ; of string , 309 , 4 ] 0
Topology , 273 , 453 ; combinatorial , VIETA rule . 382
519 ; countable , 264 ; fundamental VINOGRADOV , 484 , 491 , 498 ; symbol , theorem of , 270 ; uniform , 28 482 ; theorem , 479 Torus , 267 VOLTERRA integral equation , 426 Torus - line , 270 Volume - integral , 179 Total additivity , 76 VON KOCH , 515
Total derivative , 22
Total differential , 44 , ] 29 , ] 45 , 160 , WARING problem , 490 202 ; of second order , 152 Wave equation , 307 - 313 Total variation , of function , 65n Wave function , 416
Transcendental number , 491 Wave , cylinder , 313 ; spherical , 313 ; Transformation , angle - preserving , 1 ~ 4 ; standing , 417 of coordinates , ] 43 , 165 - 171 ; of Weak convergence , 119 , 396
differentials , 167 ; circle - preserving ~ Wedge . 333
194 ; linear - fractional , 192 , 194 ; WEIERSTRASS approximation theorem , Moblus circle - preserving , ] 92 ; of 28 , 207 , 215 , 366 ; elliptic function , 495 parameter , 140 - ] 43 ; sense - WE I ERS T R Ass - Ba Lz A No theorem , 256 ,
reversing , ] 94 ; transitive group of , 269 261 . 264
Triadic set ( plane ) , 477 WENNEBERG , 487 Triangle , 135 WEYL , H . , 498 Triangle inequality , 1 ] , 17 , 190 WIARDA . G . . 428
Triangulable , 136 Wronskian determinant , 287 , 288 , 389 , TRICOMI problem , 335 514
Trivial solution , 295
TURAN , 487 Zero , 235 ; of differential , 235 ; Type problem , 250 multiplicity of , 195 Zero set . lOOn Uniform space , 16 - 21 Zeta-function , 48] Uniform structure , v , 2 , 17 , 18 , 71n , 513 ZIPPIN, 517 Uniform topology , 18 ZORN lemma, 523
Uniformization , 250 , 473
Uniformly distributed , 497 ; modulo I , 498
Uniformly elliptic , 329
Uniqueness , condition , 308 ; problem , 327
Uniqueness theorem , 104 , 291 , 331 ,
332 , 334 ; for differential equations ,
291 ; for random variable , 104
Upper limit , 8
URYSON , 466
VAN DER W AERDEN , 512 , 525
Variables , separated , 285 ; separation of , 346
Variance , 102 , 107
Variation , of a function , positive and
negative , 66n
Variation , calculus of , 349 ; of function ,
454n , 456 ; of parameter , 289
Vector , contravariant , 144 ; covariant ,
144 ; field , 129 ; function , 26 ; lattice ,