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Di erentiation (Cont.): rules for, 70, 78 80, 87 under the integral sign, 186, 194, 203 Di usivity, 357 Directed line segments, 150 Directional derivatives, 186, 193, 202 Dirichlet conditions, 337, 345 integrals, 379, 385, 389 Dirichlet s test: for integrals, 314 for series, 270, 303 Discontinuities, 47, 119 removable, 56, 119 Distance between points, 165 Distributive law, 2 for cross products, 154 for dot products, 153 Divergence, 158, 159, 172 174 in curvilinear coordinates, 161 in cylindrical coordinates, 161 of improper integrals, 306 309 (See also Improper integrals) of in nite series (see In nite series) Divergence theorem, 236, 249 252, 261 proof of, 249, 250 Divergent integrals, 306 335 sequences, 23 (See also Sequences) series, 25 (See also Series) Division, 1 by zero, 8 of complex numbers, 6, 7 Domain, of a function, 39, 116 of convergence, 272 Dot products, 153, 154, 165, 166 commutative law for, 153 distributive law for, 153 laws for, 153, 154 Double series, 277 Dummy variable, 94 Duplication formula for gamma function, 286, 378, 386 e, 4 Electric eld vector, 181 Electromagnetic theory, 181 Elementary transcendental functions, 43, 71, 95 Elements, of a set, 1 Ellipse, 114 area of, 114 Empty set, 1 Envelopes, 185, 186, 192 Equality, of complex number, 6 of vectors, 158 Equations: di erence, 65 di erential (see Di erential equation) Equations (Cont.): integral, 364, 369, 370 polynomial, 6, 43 Equipotential surfaces, 186 Errors, applications to, 189, 200, 204 in computing sums of alternating series, 266, 282 mean square, 353 Essential singularity, 395 Eudoxus, 90 Euler, Leonhart, 336 Euler s, constant, 296, 378, 388 formulas or indentities, 8, 295 theorem on homogeneous functions, 122 Even function, 338, 347 351 Everywhere dense set, 2 Evolute, 185 Exact di erentials, 122, 131, 132, 231 (See also Di erentials) Expansion of functions: in Fourier series (see Fourier series) in power series, 272 Expansions (see Series) Explicit functions, 123 Exponential function, 42 order, 334 Exponents, 3, 11 Factorial function (see Gamma functions) Fibonacci sequence, 35, 37 Field, 2 conservative, 233 scalar, 153 vector, 153 Fluid mechanics, 402 Fourier coe cients, 337, 345 expansion (see Fourier series) Fourier integrals, 363 374 (See also Fourier transforms) convergence of (see Fourier s integral theorem) solution of boundary-value problems by, 371 Fourier, Joseph, 336 Fourier series, 336 362 complex notation for, 339 convergence of, 338, 354 356 di erentiation and integration of, 339 Dirichlet conditions for convergence of, 337 half range, 338, 347 351 Parseval s identity for, 310, 338, 351, 352 solution of boundary-value problems by, 339, 356 358 Fourier s integral theorem, 363, 364 heuristic demonstration of, 369 proof of, 369
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Degree, of a polynomial equation, 6 of homogeneous fuctions, 122 Del (r), 159 formulas involving, 159 in curl, gradient, and divergence, 159 Deleted neighborhood, 6, 117 De Moivre s theorem, 7, 15 Dense, everywhere, 2 Denumerable set (see Countable set) Dependent variable, 39, 116 Derivatives, 65 89, 75 79 (See also Di erentiation) chain rules for, 70, 124 continuity and, 66, 72, 121, 130 de nition of, 65, 66 directional, 186, 193, 202 evaluation of, 71, 75 89 graphical interpretation of, 66 higher order, 71, 120 of functions of a complex variable, 393, 400 403 of in nite series of functions, 269, 400 403 of elementary functions, 71, 78 80 of vector functions, 157, 171, 172 partial (see Partial derivatives) right- and left-hand, 67, 77, 78, 86 rules for nding, 70 table of, 71 Determinant: for cross product, 154 for curl, 159 for scalar triple product, 155 Jacobian (see Jacobian) Dextral system, 152 Di erence equations, 65 Di erentiability, 66, 67, 121 and continuity, 66, 72, 73 continuous, 66 piecewise, 66 Di erential: as a linear function, 68, 121 elements of area, of volume, 160, 163, 212, 213, 233 Di erential equation: Gauss , 276 solution of, by Laplace transforms, 314, 330 Di erential geometry, 158, 181 Di erentials, 67, 68, 69, 78, 120 122 approximations by use of, 78, 79, 120 exact, 122, 131, 132 geometric interpretation of, 68, 69, 121 of functions of several variables, 120, 130 of vector functions, 156 total, 120 Di erentiation (See also Derivatives) of Fourier series, 339, 353
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Fourier transforms, 364 368 (See also Fourier integrals) convolution theorem for, 365 inverse, 364 Parselval s identities for, 366, 368, 373 symmetric form for, 364 Fractions, 1 Frenet-Serret formulas, 159 Fresnel integrals, 387 Frullani s integral, 333 Functional determinant, 123, 136 (See also Jacobians) Functional notation, 39, 116 Functions, 39 64, 116, 132, 392 algebraic, 43 beta (see Beta functions) bounded, 40, 41 branches of, 41 composite (see Composite functions) continuity of (see Continuity) decreasing, 41, 42 de nition of, 39, 116 derivatives of (see Derivatives) di erential of (see Di erentials) domain of, 39, 116 elementary transcendental, 43, 44 even, 338, 347 351 explicit and implicit, 123 gamma (see Gamma functions) harmonic, 393 hyperbolic, 44, 45 hypergeometric, 276, 303 increasing, 41, 42 inverse (see Inverse functions) limits of (see Limits of functions) maxima and minima of (see Maxima and minima) monotonic, 41 multipled-valued (see Multiplevalued function) normalized, 342 odd, 238, 347 351 of a complex variable (see Functions of a complex variable) of a function (see Composite function) of several variables, 116, 123, 126 orthogonal, 342, 357, 358 orthonormal, 342 periodic, 336 polynomial, 43 sequences and series of, 269, 270, 272, 286, 289 single-valued, 39, 392 staircase of step, 51 transcendental, 43, 44 types of, 43, 44 vector (see Vector fuctions) Functions of a complex variable, 392 423 Functions of a complex variable (Cont.): analytic, 393 Cauchy-Riemann equations, 393, 400 (see Cauchy-Riemann equations) continuity of, 393, 399, 400 de nition of, 392 derivatives of, 393, 400 403 elementary, 393 imaginary part of, 392, 400 integrals of, 394, 403 406 Jacobians and, 422 Laplace transforms and, 423 limits of, 393, 399, 400 line integrals and, 394 multiple-valued, 392 poles of, 395 real part of, 392, 400 residue theorem for (see Residue theorem) series of, 286, 395, 406 409 single-valued, 392 singular points of, 395 Fundamental theorem: of algebra, 43 of calculus, 94, 104 Gamma functions, 375 391 analytic continuation of, 376 asymptotic formulas for, 376, 378 duplication formula for, 378, 386 in nite product for, 377 recurrence formula for, 375, 376 Stirling s formulas and asymptotic series for, 378, 384 table and graph of, 375 Gauss : di erential equation, 276  function, 377 test, 268, 283 Geometric integral, 308 Gibbs, Williard, 150 G.l.b (see Greatest lower bound) Gradient, 158, 161, 162, 172 in curvilinear coordinates, 161 in cylindrical coordinates, 162 Graph, of a function of one variable, 41 of a function of two variables, 144, 145 Grassman, Herman, 150 Greater than, 3 Greatest limit (see Limit superior) Greatest lower bound, 6 of a function, 41 of a sequence, 24, 32, 36 Green s theorem in the plane, 232, 240 243, 260 in space, (see Divergence theorem) Grouping method, for exact di erentials, 132
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