Index

Abelian , 521, 526 A-. 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 , 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 . 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 , 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 , 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 ; , 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 , 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 , 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 ; , 524 transcendental , 491 Monotone increasing (decreasing ), 7 Number circle , 498 Monotone limits , 58 Number sphere , 262 Monotone , 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 ; , 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 , 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 , 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 , 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 , 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 ,