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Half range Fourier sine or cosine series, 238, 239, 347 351 Hamilton, William Rowen, 150, 158 Harmonic functions, 393 series, 266 Heat conduction equation, 357 solution of, by Fourier integrals, 369 solution of, by Fourier series, 354, 355 Homogeneous functions, Euler s theorem on, 122 Hyperbolic coordinates, 218 Hyperbolic functions, 44, 45 inverse, 41 Hyperboloid of one sheet, 127 Hypergeometric function or series, 276 Hypersphere, 116 Hypersurface, 116 Identity, with respect to addition and multiplication, 2 Image or mapping, 124, 416 Imaginary part, of a complex number, 6 of functions of a complex variable, 392, 399, 400 Imaginary unit, 6 Implicit functions, 69, 123 and Jacobians, 135 139 Improper integrals, 97, 110, 114, 306 335 absolute and conditional convergence of, 309, 312, 319 321 comparison test for, 308, 311 containing a parameter, 313 de nition of, 306 of the rst kind, 306 308, 317 321 of the second kind, 306, 310 312, 321, 322 of the third kind, 306, 313, 322 quotient test for, 304, 311, 315 uniform convergence of, 313, 314, 323, 324 Weierstrass M test for, 313, 324 329 Increasing functions, 41 monotonic, 41 strictly, 41, 47 Increasing sequences, monotonic and strictly, 24 Inde nite integrals, 94 (See also Integrals) Independence of the path, 212, 213, 243 245, 260 Independent variable, 39, 116 Indeterminate forms, 56, 82 84 L Hospital s rules for (see L Hospital s rules) Induction, mathematical, 8
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INDEX
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Integrals (Cont.): of in nite series of functions, 272, 275 of elementary functions, 96 Schwarz s inequality for, 110 table of, 96 transformations of, 95, 103 108, 299 uniform convergence of, 313, 314, 323, 324 Integral test for in nite series, 267 Integrand, 91 Integrating factor, 223 Integration, applications of, 98, 109, 110, 114 (See also Integrals) by parts, 97 102 contour, 398 interchange of order of, 209 limits, of, 91 of Fourier series, 339, 353 of elementary functions, 96, 97, 107 range of, 91 special methods of, 97, 105 108 under integral sign, 186, 195 Intercepts, 126 Interior point, 117 Intermediate value theorem, 48 Intersection of sets, 12 Intervals: closed, 5 continuity in, 47 in nite, 5 nested, 25, 32 of convergence, 25 open, 5 unbounded, 5 Invariance relations, 181, 182 Invariant, scalar, 182 Fourier transforms, 369 (See also Fourier transforms) Laplace transforms, 315, 423, (See also Laplace transforms) Inverse functions, 41 continuity of, 47 hyperbolic, 45 trigonometric, 44 Inverse, of addition and multiplication, 2, 3 Irrantional algebraic functions, 43 p Irrationality of 2, proof of, 9 Irrational numbers, 2, 9, 10 approximations to, 9 de nition of, 2 (See also Dedekind cut) Isolated singularity, 395 Iterated integrals, 208 210 limits, 119 Jacobian determinant (see Jacobians) Jacobians, 123, 135 139, 161, 162, 174, 175 Jacobians (Cont.): chain rules for, 124 curvilinear coordinates and, 161, 162 functions of a complex variable and, 422 implicit functions and, 135 139 multiple integrals and, 211 of transformations, 124 partial derivatives using, 123 theorems on, 124, 162 vector interpretation of, 160 Kronecker s symbol, 342 Lagrange multipliers, 188, 198, 199 Lagrange s form of the remainder, in Taylor series, 274, 297 Laplace s equation, 129 (See also Laplacian operator) Laplace transforms, 314, 315, 333 convolution theorem for, 334 inverse, 330, 423 relation of functions of a complex variable to, 423 table of, 315 use of, in solving di erential equations, 315, 330 Laplacian operator, 161, 162 (See also Laplace s equation) in curvilinear coordinates, 161 in cylindrical coordinates, 161, 173 in spherical coordinates, 161 Laurent s series, 395, 407, 408 theorem, 408, 409 Least limit (see Limit inferior) Least square approximations, 201 Least upper bound, 6, 32 of functions, 41 of sequences, 24, 36 Left-hand continuity, 47 derivatives, 67, 77, 78 limits, 45 Leibnitz s formula for nth derivative of a product, 89 rule for di erentiating under the integral sign, 186, 194 Leibniz, Gottfried Wilhelm, 65, 90, 265 Lemniscate, 114 Length, of a vector, 150 Less than, 2 Level curves and surfaces, 144, 186 L Hospital s rules, 72, 82 84, 88 proofs of, 82, 83 Limit inferior, 32, 36 Limit points, 5, 12, 117 Bolzano-Weirstrass theorem on (see Bolzano-Weirstrass) Limits of functions, 39 64, 117, 118, 393, 399, 400 de nition of, 43, 118, 119
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Inequalities, 3, 10 Inequality, 3 Bernoulli s, 16 Cauchy s, 422 Schwarz s, 10, 18, 110 Inferior limit (see Limit inferior) In nite: countably, 5 interval, 5 In nite product, 277 for gamma function, 376 In nite series, 25, 33, 37, 265 305 (See also Series) absolute convergence of, 268, 283, 300 comparison test for, 267, 279, 280 conditional convergence of, 268, 283 convergence tests for, 266 268 functions de ned by, 276 Gauss test for, 268 integral test for, 267, 280 283 nth root test for, 268 of complex terms, 276 of functions, 269, 270, 276, 277 partial sums of, 25, 266 quotient test for, 267, 278 Raabe s test for, 268, 285 ratio test for, 268, 284, 300 rearrangement of terms in, 269 special, 270 uniform convergence of, 269, 270 (See also Uniform convergence) Weierstrass M test for, 270, 289 In nitesimal, 89 In nity, 25, 46 In ection, point of, 74 Initial point, of a vector, 150 Integers, positive and negative, 1 Integrable, 91 Integral equations, 367, 372, 373 Integral formulas of Cauchy, 394, 403 406 Integrals, 90 115, 207 228, 306 335, 363 374, 394, 398, 409 423 (See also Integration) de nite, 90, 91 (See also De nite integrals) Dirichlet, 379, 385, 389 double, 207, 213 219 evaluation of, 314, 325 327, 398, 412 416 Fresnel, 387 Frullani s, 333 improper, 97 (see Improper integrals) inde nite, 94 iterated, 208 210 line (see Line integrals) mean value theorems for, 72, 92 multiple (see Multiple integrals) of functions of a complex variable, 392 423
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Limits of functions (Cont.): iterated, 119, 208 of a complex variable, 393, 399, 400 proofs of theorems on, 54 56 right- and left-hand, 45 special, 46 theorems on, 45 Limits of integration, 91 Limits of sequences, 23, 24, 25, 27 de nition of, 23 of functions, 45, 269 theorems of, 23, 24, 28 30 Limits of vector functions, 156 Limit superior, 32, 36 Linear dependence of vectors, 182 Linear transformations, 148 fractional (see Fractional linear transformation) Line integrals, 229 231, 238 240, 259 evaluation of, 231 independence of path of, 232, 238, 243 245 properties of, 231 relation of, to functions of a complex variable, 394 vector notation for, 230 Line, normal (see Normal line) tangent (see Tangent line) Logarithms, 4, 10, 11, 351 as multiple-valued functions, 392 base of, 4 Lower bound, 6, 12, 13 of functions, 40 of sequences, 24 Lower limit (see Limit inferior) L.u.b. (see Least upper bound) Maclaurin series, 274 Magnetic eld vector, 181 Magnitude, of a vector, 150 Many-valued function (see Multiple-valued function) Mappings, 124 (See also Transformations) conformal, 417 Mathematical induction, 8, 15 Maxima and minima, 42, 73, 174, 185, 187, 196 198 absolute, 42 Lagrange s multiplier method for, 188, 198, 199, 204 of functions of several variables, 187, 188, 196 198 relative, 42 Taylor s theorem and, 276, 277, 297, 298 Maximum (see Maxima and minima) Maxwell s equations, 181 Mean square error, 353 Mean value theorems: for derivatives, 72, 80 82, 87, 125, 141 for integrals, 93, 104, 112 Measure zero, 91, 103 Mechanics, 158 uid, 402 Members, of a set, 1 Minimum (see Maxima and minima) Moebius strip, 248 Moment of inertia, 101 polar, 213, 219 Monotonic functions, 41 Monotonic sequences, 24, 30 32 fundamental theorem on, 24 Multiple integrals, 207 228 improper, 316 in curvilinear coordinates, 211, 212, 221, 222 in cylindrical coordinates, 211 in spherical coordinates, 212 Jacobians and, 211 transformations of, 211 213 Multiple-valued functions, 39, 117, 392 logarithm as a, 392 Multiplication, 2 associative law of, 2 involving vectors, 153 155 of complex numbers, 6, 7 Multiply-connected regions, 117 Natural base of logarithms, 3 Natural numbers, 4 Negative integers, 1 numbers, 1, 2 Neighborhoods, 6, 117 Nested intervals, 25, 32 Newton, Isaac, 65, 90, 265 rst and second laws, 68 Newton s methods, 74 Normal component of acceleration, 177 Normalized vectors and functions, 342 Normal line: parametric equations for, 184, 201 principal, 177, 180 to a surface, 184, 189 191 Normal plane, 184, 185, 191, 192 nth root test, 268 Null set, 1 vector, 151 Number, cardinal, 5 Numbers, 1 22 algebraic (see Algebraic number) Bernoulli, 304 complex (see Complex numbers) history, 2, 5 irrational (see Irrational numbers) natural, 1
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