INDEX
Abel’s identity, 131 Bolzano-Weierstrass theorem, 66–68 Abel’s test, 131–132 for sequences, 99–100 Abel’s theorem, 463–464 boundary points, 50–52 absolute convergence, 113–114 bounded functions, 142 implication of conditional bounded sets, 42–43 convergence, 114 absolute value, 7 Cantor function, 229–230 reverse triangle inequality, 9 Cantor set, 80–81, 133–134, 383 triangle inequality, 9 Casorati-Weierstrass theorem, 498–499 algebraic properties of Rk, 11–13 Cauchy completeness, 106–107 algebraic properties of continuity, 184 Cauchy condensation test, 117–119 algebraic properties of limits, 91–92 Cauchy integral theorem algebraic properties of series, 110–111 triangle lemma, see triangle lemma algebraic properties of the derivative, Cauchy principal value, 365–366 244 Cauchy sequences, 104–105 alternating series, 115 convergence of, 105–106 alternating series test, 115–116 Cauchy’s inequalities, 437 analytic functions, 481–482 Cauchy’s integral formula, 428–433 complex analytic functions, 483 converse of, 436–437 counterexamples, 482–483 extension to higher derivatives, 437 identity principle, 486 for simple closed contours, 432–433 zeros of, see zeros of complex analytic on circles, 429–430 functions on open connected sets, 430–432 antiderivatives, 361 on star-shaped sets, 429 for f :[a, b] Rp, 370 → Cauchy’s integral theorem, 420–422 of complex functions, 408 consequence, see deformation of on star-shaped sets, 411–413 contours path-independence, 408–409, 415 Cauchy-Riemann equations Archimedean property, 5–6 motivation, 297–300 argument, 29–30 necessity, 300–302 continuity of, 182–183 sufficiency, 302–304 continuous arguments, 445–446 Cauchy-Schwarz inequality, 15–16 average values, 362–363 for integrals, 384 chain rule, 245–246 balls, 48–49 for f : D C, 296 → closed balls, 48–49 for f : Dk Rp, 283–284 → open balls, 48–49 change of variables Bernoulli numbers, 513 for complex integrals, 389 bijective, see one-to-one correspondence for real integrals, 367
539 540 INDEX circles, 30–31, 391 connected sets, 72–75 closed sets, 58–59 contour-connected sets, 392–393, 450 relatively closed, 61 path-connected sets, 392 closure, 59–60 simply connected sets, see simply codomain, 136 connected sets compact sets, 70–71 continuity Heine-Borel theorem, 71 algebraic properties, 184 comparison test, 117 classification of discontinuities, see limit comparisons, 119–120 discontinuity completeness property of Rk, 13–14 composite functions, 185–186 complex analytic functions, see analytic connection with closure, 189–190 functions definition of, 178–179 complex conjugate, 24–25 examples of, 179–184 complex integrals extending, see continuous extensions change of variables, 389 implication by limit preservation, 190 fundamental theorem of calculus, 390 implication of boundedness, 187–188 inequalities, 388–389 inverse images of open sets, 188–189 line integrals, see complex line left and right continuity, 194–195 integrals Lipschitz continuity, 226 linearity property, 388 of inverses, 195–196 of uniformly convergent sequences, of several variables, 186–187 472 order properties, 191–192 over an interval, 387–388 preservation of compactness, 190–191 complex line integrals, 394–395 preservation of connectedness, 191 algebraic properties of, 395–397 uniform continuity, see uniform independence of parametrization, continuity 398–399 continuous extensions, 203–204 inequalities, 398 of uniformly continuous functions, complex numbers, 19–24 206–207 algebraic properties of, 21–22 pasting theorem, 204–206 field properties of, 22–23 Tietze extension theorem, 220–222 imaginary part, 21 contours, 375–376 integer powers of, 23 closed contours, 375 matrix representation, 34–35 in the complex plane, 390 multiplication, 19 lengths of, 397–398 order properties of, 23 parametrizations of, see powers of, 31 parametrizations real part, 21 reverse of, 377, 399–400 roots of, 31–33 simple contours, 375 conditional convergence, 113–114 smooth contours, 375 arbitrary rearrangements, 124–125 subcontours, 390 relation to absolute convergence, 114 winding numbers of, see winding conformal mapping, 524–530 numbers disks to disks, 529–530, 532 convergence, 85–89 half-planes to disks, 524 absolute convergence, see absolute rectangular regions to hyperbolas, 529 convergence Riemann mapping theorem, 532–533 componentwise, 112 sectors to half-planes, 527–528 conditional convergence, see absolute strips to sectors, 528–529 convergence INDEX 541 convex functions, 385 definitions, see definition of convex sets, 410 derivatives cosine function, see trigonometric differentiability, see differentiability functions directional derivatives, see directional curves, 372 derivatives catenations of, 373 examples in R, 239–242 closed curves, 372, 390 examples of, 243, 259, 277, 281–282 contours, see contours for f : D C, 291–294 → in the complex plane, 390 for f : Dk R, 258 → initial point, 390 for f : Dk Rp, 273–274 → initial points of, 372 gradients, see gradients lengths of, 397–398 higher-order derivatives, 243–244 parametrizations of, see higher-order derivatives in C, 294–295 parametrizations of a power series, 468–469 simple closed curves, 372 of a sequence, 255–256 simple curves, 372, 390 of a series, 256–257 smooth curves, 374, 390 partial derivatives, see partial subdivisions of, 374 derivatives terminal point, 390 uniqueness of, 239, 259, 274–275, 294 terminal points of, 372 derived set, 54 diameter, 68 de Moivre’s formula, 29 differentiability decreasing, 145 analyticity, see analytic functions decreasing sequences, 96 definition of, see definition of Dedekind completeness property, 4–5 derivatives for f : Dk R, 258 definition of derivatives, 237 → , δ version, 235–236, 293 implication of continuity, 242, 276, 294 difference quotient version, 235, in Rk, to Rp, 273 292–293 of implicit functions, see implicit directional derivatives, 264–265 function theorem equivalence of definitions, 238, 293 of inverse functions, see inverse for f : Dk R, 258 function theorem → for f : Dk Rp, 273 relation to the gradient, 263–264 → for complex f : D C, 292 sufficient conditions of, 279–281 → gradients, 263 differentiability classes n linear approximation version, C functions, 289–290 n k 236–237, 292 C functions f : D R, 290 → partial derivatives, 261 differentiation deformation of contours, 423–428 under an integral sign, 435–436 deleted neighborhoods, 49–50 dilation, 520 dense sets, 62–63 directional derivatives, 264–265 derivatives definition of, 264–265 algebraic properties of, 244, 270, 283, examples of, 265 295 Dirichlet’s test, 132 at local extrema, 247–248 disconnected sets, 72–74 at local extrema in Rk, 271 discontinuity Cauchy-Riemann equations, see classification of, 196–197 Cauchy-Riemann equations essential discontinuities, 197 chain rule, 245–246 jump discontinuities, 197 542 INDEX
removable discontinuities, 197 fundamental theorem of algebra, discrete distance, 75–76, 79 442–443 disks, 30–31 fundamental theorem of calculus, distance, 42 360–362 discrete distance, 75–76, 79 for complex integrals, 390 divergence, 85 for complex line integrals, 408 test for, 113 for line integrals, 380 to , 97 for vector-valued functions, 370–371 ±∞ domain, 136 Gauss’s mean value theorem, 430 general LFTs, 521–523 entire functions, 441 geometric series, 115, 458–460 bounded entire functions, 441–442 gradients, 262–264 essential singularities, 494–495 definition of, 263 Casorati-Weierstrass theorem, greatest lower bound, 45–47 498–499 Euler’s formula, 27 ex harmonic functions, 438–439 definition, 128 harmonic conjugates, 454 exponential function Heine-Borel theorem, 71–72 complex, 150–151 Hessian, 329–330 continuity of, 218–219 higher-order derivatives, 243–244 definition, 128 in C, 294–295 derivative of, 241–242 exponents of complex numbers, 175–176 identity principle, 486 extended complex plane, see Riemann image, 139 sphere imaginary part, 21 exterior points, 50–52 implicit function theorem, 318–321 extreme value theorem, see max-min improper integrals, see integrals theorem increasing, 145 increasing sequences, 96 infimum, 45–47 R field properties of , 2–3 infinite series, see series field property of limits, 92–93 injective, see one-to-one fixed point theorem, 312 inner products, 14–15 fixed points, 534 parallelogram equality, 37–39 function polarization identity, 37–38 complex functions, 137 integrable functions, 337–339 functions, 136 absolute values of, 357–358 complex functions, 137 classes of, 349–354 composition of, 143 continuous functions, 351–353 products and quotients of, 143 functions with finitely many real-valued functions of a single real discontinuities, 353–354 variable, 137 integrals of, 337–339 real-valued functions of a vector, 137 monotone functions, 353 real-valued functions of several real on subintervals, 358–359 variables, 137 products of, 356–357 sums, differences, and scalar relation to upper and lower sums, multiples of, 142 347–349 INDEX 543
uniform limits of, 367–368 Jacobian matrices, 278–279 integrals, 337–339 connection with differentiability, algebraic properties of, 354–356 279–281 approximation by upper and lower Jensen’s inequality, 385 sums, 344–345 Jordan curve theorem, 393 average values, 362–363 Jordan’s inequality, 358 Cauchy principal value, 365–366 change of variables, 367 L’Hospital’s Rule, 254–255 complex integrals, see complex Laurent series, 487 integrals analytic parts, 487 depending on a parameter, 384 integration of, 489 existence of, see integrable functions Laurent’s theorem, see Laurent’s p for f :[a, b] R , 370 theorem → fundamental theorem of calculus, see principal parts, 487 fundamental theorem of calculus region of convergence, 487–489 improper integrals, 363–366, 384 Laurent’s theorem, 490–493 inequalities, 357–358, 371–372 least upper bound, 45–47 Leibniz’s rule, 384 left and right continuity, 194–195 line integrals, see line integrals Leibniz’s rule, 384 of power series, 470–473 LFTs, see linear fractional of uniformly convergent functions, transformations 368–369, 472 limit comparison test, 119–120 over subintervals, 359 limit inferior, 101–104 splitting over subintervals, 345–346 limit points, 53–54 upper and lower integrals, 342–344 derived set, see derived set integration, see integrals of sequences, 98–99 integration by parts, 382 limit superior, 101–104 interior points, 50–52 limits, 85–89 intermediate value theorem, 192–193 algebraic properties, 91–92 intervals algebraic properties of, 163–165 at , 170 closed intervals, 10 ∞ componentwise, 89–90, 161 diameter of, 18–19 field property, 92–93 half-closed intervals, 10 implication of boundedness, 160 half-open intervals, 10 of , 170–171 in R, 10 ∞ of a function, 156 Rk in , 18–19 of rational functions, 162–163 length of, 10 one-sided, 168–170 open intervals, 10 order properties, 166 inverse function theorem, 313–316 order property, 93–94 examples, 318 uniqueness of, 89, 160 for complex functions, 317–318 line integrals inverse functions, 147 complex line integrals, see complex continuity of, 195–196 line integrals differentiability of, see inverse definition of, 377–379 function theorem fundamental theorem of calculus, 380 inversion, 520–521 in reverse, 381 isolated points, 54–55 independence of parametrization, isomorphic fields, 36 378–379 544 INDEX line segments, 391 nested closed bounded intervals linear fractional transformations, 519 theorem, 64–66 conformal mapping examples, see nested closed bounded sets theorem, conformal mapping 68–69 dilation, 520 nested intervals, 63 general LFTs, 521–523 nondecreasing sequences, 96 inversion, 520–521 nonincreasing sequences, 96 Liouville’s theorem, 441–442 norms application to the fundamental equivalence of, 39 theorem of algebra, 442–443 induced norms, 16 Lipschitz continuity, 226 of matrices, see matrix norms local extrema results for derivatives, on C, 24–25 k 247–248 on R , 16 in Rk, 271 nowhere dense sets, 78 logarithm continuous logarithms, 444–446 one-sided limits, see limits derivative of, 246–247 one-to-one, 144 differentiable logarithms, 414 one-to-one correspondence, 147 logarithm function, 151–154 onto, 146 lower bounds, 44–45 open coverings, 69–70 greatest lower bound, 45–47 open sets, 55–57 lower sums, see Riemann sums relatively open, 61 order properties of R, 3–4 Rk M-test, 214–215 order properties of , 13 Weierstrass M-test, 217–218 order property of limits, 93–94 matrix norms, 310–311 and invertibility, 312 p-series, 119 definition of, 275 parallelogram equality, 37–39 equivalence of, 311–312 parametrizations, 372 multiplicative property of, 275 equivalent parametrizations, 376–377, max-min theorem, 192 390 mean value theorem, 249–250 independence of, 378–379, 398–399 Cauchy mean value theorem, 250 induced parametrizations, 375 failure for complex functions, 296 reparametrizing, 373 for f : Dk R, 285 partial derivatives, 260–261 → for f : Dk Rp, 288 definition of, 261 → from Dk Rp, 277–278 mixed derivative theorem, 267–268 → monotone, 145 geometric interpretation, 268–269 monotone sequence theorem, 96–97 higher-order partial derivatives, monotone sequences, 96 266–267 Morera’s theorem, 440–441 mixed derivative theorem, 267–268 multiplication of series, 125–128 relation to ordinary derivatives, 260 partitions, 336 norm of, 336 neighborhood of convergence, see power refinements of, 336 series subintervals determined by, 336 neighborhoods, 49–50 unions of, 336 deleted neighborhoods, 49–50 pasting theorem, 204–206 INDEX 545 path-connected sets, 392 computing at poles, 503 contour-connected sets, 392–393, 450 residue theorem, see residue theorem path-independence, 408–409, 415 reverse triangle inequality, 9 point at infinity, 517–518 Riemann integration, see integration polar notation, 27–29 Riemann mapping theorem, 532–533 polarization identity, 37–38 Riemann sphere, 515–518 poles, 494–495 arithmetic on, 519 behavior near, 499–501 linear fractional transformations, see polygonal contours, 391–392 linear fractional transformations rectangular contours, 392 point at infinity, 517–518 vertices, 391 stereographic projection, see polynomials, 148 stereographic projection power series, 456 Riemann sums, 336–337 Abel’s theorem, 463–464 properties of, 341–342 absolute convergence of, 456–457 recognizing sequences as, 366 algebraic manipulations of, 464–468 upper and lower sums, 339–340 circle of convergence, 456 Riemann-Lebesgue lemma, 385 differentiation of, 468–469 right continuity, see left and right examples of, 458–460 continuity integration of, 470–473 Rolle’s theorem, 249 interval of convergence, 456 failure for complex functions, 296 neighborhood of convergence, in Rk, 272 456–457 root test, 122 radius of convergence, 456–457, 460–463 saddle points, 329 ratio and root tests, 460–463 scalar multiplication, 12 Taylor series, see Taylor series Schwarz lemma, 531–532 uniform convergence of, 456–457 second derivative test, 329–330 preimage, 139 sequences, 84 bounded, 90–91 radius of convergence, see power series Cauchy sequences, 104–105 range, 136 convergence of, 105–106 ratio test, 120–122 limits of, 85–89 real part, 21 monotone sequences, 96 rearrangements of series, 123 of functions, see sequences of of absolutely convergent ones, 125 functions of conditionally convergent ones, supremums of, 94–95 124–125 sequences of functions rectangular contours, 392 convergence, 208–209 approximation to other contours, derivatives of, 255–256 416–417 uniform convergence, see uniform complex line integrals over, 417–420 convergence relatively closed sets, 61 series, 108–110 relatively open sets, 61 algebraic properties, 110–111 removable singularities, 494–498 alternating series, see alternating residue theorem, 504–506 series applications to real integrals, 507–512 components, 112 fractional residue theorem, 506–507 convergence of, 108–110 residues, 502–503 divergence of, 108 546 INDEX
geometric series, 115 Taylor polynomials, 251 Laurent series, see Laurent series Taylor series, 473–474 multiplication of, see multiplication of examples, 477–481 series Taylor’s theorem of functions, see series of functions for complex functions, 475–477 p-series, 119 for real functions, 474–475 power series, see power series Taylor’s theorem with remainder, rearrangements of, see 252–253 for f : Dk Rp, 289 rearrangements of series → for f : Dk R, 285–288 with nonnegative terms, 116–117 → series of functions, 216 test for divergence, 113 convergence, 216 Tietze extension theorem, 220–222 derivatives of, 256–257 topological properties of continuity, uniform convergence, see uniform 188–191 convergence triangle inequality, 9 Weierstrass M-test, 217–218 triangle lemma, 400–402 refinement of, 403–404 simply connected sets, 454, 530–531 trigonometric functions, 154–155 sine function, see trigonometric functions singularities, 493–494 at infinity, 495–496 uniform continuity Casorati-Weierstrass theorem, definition, 198 498–499 examples of, 199–201 essential singularities, 494–495 extensions to the closure, 206–207 isolated singularities, 493–494 implication by continuity on compact nonisolated singularities, 493–494 sets, 201–202 poles, 494–495, 499–501 preservation of boundedness, 203 removable singularities, 494–498 preservation of Cauchy sequences, residues at, see residues 202–203 spheres, 47–48 uniform convergence, 210–211, 216–217 square root function, 149–150 and integration, 367–369 squeeze theorem M-test, 214–215 for functions, 167–168 of complex differentiable functions, 452–453 for sequences, 95 relationship with continuity, 211–212, star-shaped sets, 410 218 antiderivatives, 411–413 uniformly Cauchy, 212–213 star-center of, 410 uniformly Cauchy sequences, 212–213 stereographic projection, 515–518 upper bounds, 44 continuity of, 518 least upper bound, 45–47 strictly decreasing, 145 upper sums, see Riemann sums strictly increasing, 145 strictly monotone, 145 subfields, 35 vertices, 391 subsequences, 97–98 limit points, 98–99 Weierstrass M-test, 217–218 limits of, 98 well-ordered property, 6 supremum, 45–47 winding numbers, 404–406 of a sequence, 94–95 geometric interpretation of, 446–447 surjective, see onto of rectangles, 407 INDEX 547
of simple closed contours, 432, 448–450 on connected subsets, 406–407 zeros of complex analytic functions, 484 isolated zeros, 484–486 multiplicities, 484–486