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Bibliography Bibliography 1. E. Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964 (translated by M. Butler from the German original, Einfubrung in Die Theorie Der Gamma Function, B. G. Teubner, 1931) 2. F. Bowman, Introduction to Elliptic Functions with Applications, Dover, New York, 1961 3. N. Bourbaki, Function D'une Variable Reel/e (Theorie Elementaire), Hermann, Editeur Des Sciences et Des Arts, Paris, 1949 4. T. J. I. Bromwich, Infinite Series (Second Edition Revised), Macmillan & Co. Limited, London, 1942 5. Burrill and Knudsen, Real Variables, Holt, Rinehart and Winston, New York, 1969 6. G. Chrystal, Textbook of Algebra, Vol. II, Dover Edition 7. Courant and John, Introduction to Calculus and Analysis, Interscience Divi­ sion, John Wiley and Sons, New York, 1965 8. Phillip Franklin, Treatise on Advanced Calculus, John Wiley and Sons, New York, 1940 9. Avmer Friedman, Advanced Calculus, Holt, Rinehart and Winston, New York, 1971 10. Watson Fulks, Advanced Calculus (Third Edition), John Wiley and Sons, New York, 1978 11. Eimar Hille, Analysis, Vol. I, Blaisdell Publishing Co., 1964 Ila. E. W. Hobson, The Theory of Functions of a Real Variable, Vol. II, Dover Edition, New York, 1957 12. Konrad Knopf, Theory and Application of Infinite Series, Blackie and Sons, London, 1951 13. Konrad Knopf, Infinite Sequences and Series, Dover Edition, 1956 14. D. Mitronovic, Analytic Inequalities, Springer-Verlag, New York 15. Poly and Azego, Aufgabe Der Analysis (2 vols.) Dover 16. John Olmsted, Advanced Calculus, Appleton, Century, Crofts, New York, 1956 16a. James Pierpont, Theory of Functions of Real Variables, Vol. II, Ginn and Company, Boston, New York, Chicago, London, 1912 17. John F. Randolph, Basic Real and Abstract Analysis, Academic, NY, London, 1968 18. Walter Rudin, Principles of Mathematical Analysis (Second Edition), McGraw­ Hill, New York, 1964 Index Abel's limit theorem 596 bounded Abel's partial sum formula 589 function 94 absolute value 39 set of real numbers 41 accumulation point 209, 211 boundedness 24 extended 210 bounded 24 one-sided 210 bounded from above 24 addition formulas 747 bounded from below 24 analytic functions lower bound 24 series of 580 upper bound 24 analyticity of a function 548 at a point 548 Archimedean-ordered field 145, 148 Cantor set 706 Archimedean property 26 Cartesian product 42 arithmetic-geometric inequality 389 Cauchy criterion for limits of functions asymptotically equivalent functions 238 392 Cauchy-Schwarz inequality 87 asx~a 392 Cauchy's condensation test 497 for infinite series 497 Cauchy sequence 144 Bernoulli numbers 565 chain rule 303 recurrence relation for 566 closed set 268 Bernoulli's inequality 56 bounded 270 Bernstein polynomial 526 cluster point 266 Beta function 729 complete integral of the second kind big 0 163 761 asx~d 391 Completeness Axiom 26 binary operation 6 order-completeness 38 binomial series 409 complex number 427 binomial theorem 59 arguJDentofa 439 Bolzano-Weierstrass theorem 268 conjugate of 434 for sets 268 direction of a 438 sequences 268 modulus of a 434 766 Index complex number (cont.) differentiable polar form of a 439 from the left 293 composite of 68 from the right 293 correspondences 68 function 296 function 68 in the extended sense 298 concave function 345 of order n 307 strictly 345 differentiable function 290 continuity 240 differential 643 in the large 250 discontinuity 248 local 250 infinite 249 one-sided 247 jump 249 sequential 243 point of 248 continuous 240 removable 248 at Xo 240, 241 distance 40 from the left 247 domain of discourse from the right 247 function 242 piecewise 700 e 107 uniformly 285 elliptic functions 744 continuous but nowhere differentiable addition formulas 747 function 521 delta amplitude 744 convergent Jacobian 744 sequence 100 modular cosine 744 senes 152 modular sine 743 elliptic integrals 742 convex function 342 strictly 345 complete 742 Euclid's theorem 28 convex set of real numbers 202 Euler-Mascheroni constant 332 correspondence 44 Euler numbers 579 inverse of 69 a Euler's formula 447 one-to-one 69 Euler's second integral 717 single-valued 45 exponent 75 rational 75 exponents Darboux's theorem 333 real 196 Darboux sum 603 lower and upper 603 decimal 159 factorial of degree 57 infinite 159 factor theorem 449 nonterminating 159 fluctuation of a function 623 representation 159 function 43 terminating 159 absolute value 46 truncation 159 analytic at Xo 548 De Moivre's theorem 445 Beta 730 derivation of a function bounded 94 second 307 codomain 43 derivative continuous 242 from the left 293 derivative of a 291 from the right 293 differentiable 296 nth-order 307 Dirichlet's 244 derivative of a function 290 distance to the nearest integer 205 at a 290 domain of 44 notation for 291 even 205 derived set 209 greatest integer 67 Index 767 function (cont.) infinite product 465 identity 47 absolutely convergent 474 inverse of a 70 convergent 466 limit of 212 divergent 466 mapping 43 nth partial product of an 465 monotonically decreasing 66 rearrangement of an 475 monotonically increasing 66 sine and cosine as 478 notation for 639 infinite series odd 205 absolutely converging 157 one-to-one 65 inflection point 358 onto 61 integer 17 primitive of 638 negative integer 21 range of 44 nonnegative 17 rational 203 integrable 606 real-valued 45 Darboux 606 restriction of 47 improperly 711 signum 46 Riemann 631 strictly monotonic 66, 75 integral 605 transformation 45 convergent 711 uniformly continuous 285 divergent 711 fundamental theorem of the calculus improper 711 649 indefinite 651 lower Darboux 606 Riemann 631 Gamma Function 419 upper Darboux 605 as an infinite product 468 integral of a function 639 continuity of 421 integrals functional equation for the 423 first mean-value theorem for 690 integral representation of 726 second mean-value theorem for 690 Gauss' multiplication formula 492 integration by parts 666 geometric-arithmetic inequality 84 Riemann integral counterpart of greatest common divisor 30 669 interior point 270 greatest lower bound 25 from the left 293 from the right 293 intermediate-value property 253 Holder's inequality 85 strong 257 hyperbolic functions 204 intermediate-value theorem 253 inverses 281 interval 11 one 454 bounded 24 hypergeometric series 499, 504 closed 12 finite 12 half-closed 12 improper integral 711 half open 12 absolutely convergent 721 infinite 12 Cauchy criterion for convergence of left endpoint of II 719 open 11 comparison tests for convergence of right endpoint of II 715 unbounded 24 conditionally convergent 721 irrational number 36 inequality of Jensen 390 isolated point 241 infimum 25 of a function 94 of a set of real numbers 25 Jacobi's zeta function 763 768 Index Kummer's test 173 natural logarithm 258 neighborhood 207 deleted 207 Lagrange interpolation formula 313 t:-neighborhood 207 least upper bound 25 from the left 208 Legendre polynomials 336 from the right 208 L'Hopital's rule of +00 208 for functions 379, 381 of -00 208 L'Hopital's theorem 139 one-sided 209 for sequences 139 noncountable set 702 limit nth root 63 of a function 212 of a complex number 452 of a sequence 100 of unity 452 one-sided 224, 227 ofy>O 63 limit inferior criteria for 110 open of a real sequence 98 set 271 limit superior open set 271 criteria for 110 of a real sequence 98 ordered field 24 Archimedean-ordered 145 little 0 163 Cauchy-complete 145, 148 as X--M 393 complete 38 log convexity 376 order-complete 38, 148 ordered n-tuple 50 ordered pair 42 mathematical induction 18 order t 387 complete induction 21 principle of 18 maximum 24 'IT 256 local 413 partial fractions 672 of a real-valued function 318 integration by 672 of a set of real numbers 24 partially ordered set 92 mean 78 linearly ordered set 92 arithmetic 78 ordering 92 arithmetic--geometric 139 partition geometric 78 norm of 602 harmonic 78 refinement of a 604 mean-value theorem 321 partition of an interval 601 Cauchy's 378 periodic function 206 consequences 322 period of the a 206 extended 399 pointwise convergence 505 for integrals 689 of a series of functions 513 with one-sided derivatives 319 sequence of functions of a 505 measure zero 700 polynomial 64 Mertens's theorem 177 Bernstein 526 minimum 24 root of mUltiplicity k of 311 local 413 zero of 64 of a real-valued function 318 power series 514 of a set of real numbers 24 interval of convergence of 517 Minkowski's inequality 88 radius of convergence of 517 monotonic 66 reciprocal of a 564 function 66 sum and product of 558 sequence 96 uniqueness theorem for 552 Index 769 prime 33 sequence of functions 505 primitive derived sequence of a 534 substitution formula for 642 pointwise convergence of a 505 uniform convergence of a 505 sequences Raabe's test 175 convergence of 142 rational number 23 series 151 ratio test 166 Abel's test for convergence of a 591 modified 167 absolutely converging 157 real number alternating 155, 156 b-adic representation 706 analytic functions 579 ternary representation of 706 Cauchy product, of 177 real numbers 4 comparison tests for 162 axioms 4 conditionally convergent 157 extended 93 convergence of 152 greater than, for 4 Dedekind's test for convergence of a less than, for 4 591 negative 7 Dirichlet's test for convergence of a positive 7 592 product 5 DuBois-Reymond's test for conver- sum of 5 gence of a 590 relation 44 exponential 168 transi ti ve 3 Gauss' test for convergence of 498 relatively prime 30 geometric 154 remainder theorem 448 harmonic 154 Riemann sum 630 hypergeometric 500 Riemann zeta function 162 integral tests for convergence of 738
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