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Abelian Group, 521, 526 Absolute Value, 190 Accumulation, Point Of 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 Convolution ConvolutiontheoremConvex , , 404 108 , 404n , 32 Coordinates , introduction of , 143 ; Convergent sequence , 1, 424 ; of Complex analysis , vi Content , 76 ; n-dimensional Constructibilityelementary , - questions of , 474 Convergence , v , 3, Continuum 10, 12, 427 ; hypothesisContinuous , 448 , 475 , 476 Continuous spectrummapping , , 24,413 455 Congruence class, 189 Congruence , 189 Conformal mappingConfiguration , 210 , 325 , Cone508 , 322 Composition , 514 ; outer , 524 ; of Continuous function , 26 ; power series Conoid , 322 Complex numberComplex , 188 ; fundamental manifold , 272 Complete set functionCompleteness , 63 , 11, 19Complete ; property differentialof , 2 , 202 INDEX Continuity , 23-Continued 28, 454 ; of additionfractions , 6 ; Continuation , 499 , Content analytic , 481 ; holomorphicaxiom , , 63 Consistency , 5, 55 ; amount of , 97 Connectedness , 26 Conjugate , 190 ; harmonic , 211 Complete (contentComplementary ), 76 set, 198 radius ofnormwise , 216 , ; 71nstrong ; in probability , 396 ; , theorem 115 ; functions , 33 weak , 105 of Peano , 278 ; uniform , 20 - 21 , 27 ; of , 131 ; polar , 177 ; position , 416 ; coordinate -wise , 25 ; maximal , 438 ; positive order of , 131 ; transformation Laplace operator in , 177 ; local , as , 34 subsets , 17 of , 192 ; relation laws to forelementary , 189 ; Gauss representation 217 ; meromorphic , 241 geometric , 57 ; Peano -Jordan , 62 bounded , 104 ; concept of , 1, 479 ; uniform of , 410 measure , 92 ; points of inverseof , , 4627 ; of ; multiplication , 7 ; geometry , 191 - 195 concept of , v ; definition of , 396 ; 185 ; momentum , 416 ; negative order of characteristic function , 105 ; '" CO '" g. .: 0 . ~~ N~' ' ~3 PJ' ' . 0 g~ . , : ~' ~ . ~I ~ i3I ,. ~. 3 ~ . ~ . .IoN.Io tiIN)( .- "3: 03 -0 g . 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