Page 438 - Schaum's Outline of Theory and Problems of Advanced Calculus

INDEX 429

Inequalities, 3, 10 Integrals (Cont.): Jacobians (Cont.):
Inequality, 3 of infinite series of functions, 272, chain rules for, 124
Bernoulli’s, 16 275 curvilinear coordinates and, 161,
Cauchy’s, 422 of elementary functions, 96 162
Schwarz’s, 10, 18, 110 Schwarz’s inequality for, 110 functions of a complex variable
Inferior limit (see Limit inferior) table of, 96 and, 422
Infinite: transformations of, 95, 103–108, implicit functions and, 135–139
countably, 5 299 multiple integrals and, 211
interval, 5 uniform convergence of, 313, 314, of transformations, 124
Infinite product, 277 323, 324 partial derivatives using, 123
for gamma function, 376 Integral test for infinite series, 267 theorems on, 124, 162
Infinite series, 25, 33, 37, 265–305 Integrand, 91 vector interpretation of, 160
(See also Series) Integrating factor, 223
absolute convergence of, 268, 283, Integration, applications of, 98, 109,
300 110, 114 (See also Integrals) Kronecker’s symbol, 342
comparison test for, 267, 279, 280 by parts, 97–102
conditional convergence of, 268, contour, 398 Lagrange multipliers, 188, 198, 199
283 interchange of order of, 209 Lagrange’s form of the remainder,
convergence tests for, 266–268 limits, of, 91 in Taylor series, 274, 297
functions defined by, 276 of Fourier series, 339, 353 Laplace’s equation, 129 (See also
Gauss’ test for, 268 of elementary functions, 96, 97, Laplacian operator)
integral test for, 267, 280–283 107 Laplace transforms, 314, 315, 333
nth root test for, 268 range of, 91 convolution theorem for, 334
of complex terms, 276 special methods of, 97, 105–108 inverse, 330, 423
of functions, 269, 270, 276, 277 under integral sign, 186, 195 relation of functions of a complex
partial sums of, 25, 266 Intercepts, 126 variable to, 423
quotient test for, 267, 278 Interior point, 117 table of, 315
Raabe’s test for, 268, 285 Intermediate value theorem, 48 use of, in solving differential
ratio test for, 268, 284, 300 Intersection of sets, 12 equations, 315, 330
rearrangement of terms in, 269 Intervals: Laplacian operator, 161, 162 (See
special, 270 closed, 5 also Laplace’s equation)
uniform convergence of, 269, 270 continuity in, 47 in curvilinear coordinates, 161
(See also Uniform convergence) infinite, 5 in cylindrical coordinates, 161, 173
Weierstrass M test for, 270, 289 nested, 25, 32 in spherical coordinates, 161
Infinitesimal, 89 of convergence, 25 Laurent’s series, 395, 407, 408
Infinity, 25, 46 open, 5 theorem, 408, 409
Inflection, point of, 74 unbounded, 5 Least limit (see Limit inferior)
Initial point, of a vector, 150 Invariance relations, 181, 182 Least square approximations, 201
Integers, positive and negative, 1 Invariant, scalar, 182 Least upper bound, 6, 32
Integrable, 91 Fourier transforms, 369 (See also of functions, 41
Integral equations, 367, 372, 373 Fourier transforms) of sequences, 24, 36
Integral formulas of Cauchy, 394, Laplace transforms, 315, 423, Left-hand continuity, 47
403–406 (See also Laplace transforms) derivatives, 67, 77, 78
Integrals, 90–115, 207–228, 306–335, Inverse functions, 41 limits, 45
363–374, 394, 398, 409–423 continuity of, 47 Leibnitz’s formula for nth derivative
(See also Integration) hyperbolic, 45 of a product, 89
definite, 90, 91 (See also Definite trigonometric, 44 rule for differentiating under the
integrals) Inverse, of addition and integral sign, 186, 194
Dirichlet, 379, 385, 389 multiplication, 2, 3 Leibniz, Gottfried Wilhelm, 65, 90,
double, 207, 213–219 Irrantional algebraic functions, 43 265
p ffiffiffi
evaluation of, 314, 325–327, 398, Irrationality of 2,proof of, 9 Lemniscate, 114
412–416 Irrational numbers, 2, 9, 10 Length, of a vector, 150
Fresnel, 387 approximations to, 9 Less than, 2
Frullani’s, 333 definition of, 2 (See also Dedekind Level curves and surfaces, 144, 186
improper, 97 (see Improper cut) L’Hospital’s rules, 72, 82–84, 88
integrals) Isolated singularity, 395 proofs of, 82, 83
indefinite, 94 Iterated integrals, 208–210 Limit inferior, 32, 36
iterated, 208–210 limits, 119 Limit points, 5, 12, 117
line (see Line integrals) Bolzano-Weirstrass theorem on
mean value theorems for, 72, 92 (see Bolzano-Weirstrass)
multiple (see Multiple integrals) Jacobian determinant (see Jacobians) Limits of functions, 39–64, 117, 118,
of functions of a complex Jacobians, 123, 135–139, 161, 162, 393, 399, 400
variable, 392–423 174, 175 definition of, 43, 118, 119