Page 437 - Schaum's Outline of Theory and Problems of Advanced Calculus

428 INDEX

Fourier transforms, 364–368 (See Functions of a complex variable Half range Fourier sine or cosine
also Fourier integrals) (Cont.): series, 238, 239, 347–351
convolution theorem for, 365 analytic, 393 Hamilton, William Rowen, 150, 158
inverse, 364 Cauchy-Riemann equations, 393, Harmonic functions, 393
Parselval’s identities for, 366, 368, 400 (see Cauchy-Riemann series, 266
373 equations) Heat conduction equation, 357
symmetric form for, 364 continuity of, 393, 399, 400 solution of, by Fourier integrals,
Fractions, 1 definition of, 392 369
Frenet-Serret formulas, 159 derivatives of, 393, 400–403 solution of, by Fourier series, 354,
Fresnel integrals, 387 elementary, 393 355
Frullani’s integral, 333 imaginary part of, 392, 400 Homogeneous functions, Euler’s
Functional determinant, 123, 136 integrals of, 394, 403–406 theorem on, 122
(See also Jacobians) Jacobians and, 422 Hyperbolic coordinates, 218
Functional notation, 39, 116 Laplace transforms and, 423 Hyperbolic functions, 44, 45
Functions, 39–64, 116, 132, 392 limits of, 393, 399, 400 inverse, 41
algebraic, 43 line integrals and, 394 Hyperboloid of one sheet, 127
beta (see Beta functions) multiple-valued, 392 Hypergeometric function or series,
bounded, 40, 41 poles of, 395 276
branches of, 41 real part of, 392, 400 Hypersphere, 116
composite (see Composite residue theorem for (see Residue Hypersurface, 116
functions) theorem)
continuity of (see Continuity) series of, 286, 395, 406–409
decreasing, 41, 42 single-valued, 392 Identity, with respect to addition
definition of, 39, 116 singular points of, 395 and multiplication, 2
derivatives of (see Derivatives) Fundamental theorem: Image or mapping, 124, 416
differential of (see Differentials) of algebra, 43 Imaginary part, of a complex
domain of, 39, 116 of calculus, 94, 104 number, 6
elementary transcendental, 43, 44 of functions of a complex
even, 338, 347–351 variable, 392, 399, 400
explicit and implicit, 123 Gamma functions, 375–391 Imaginary unit, 6
gamma (see Gamma functions) analytic continuation of, 376 Implicit functions, 69, 123
harmonic, 393 asymptotic formulas for, 376, 378 and Jacobians, 135–139
hyperbolic, 44, 45 duplication formula for, 378, 386 Improper integrals, 97, 110, 114,
hypergeometric, 276, 303 infinite product for, 377 306–335
increasing, 41, 42 recurrence formula for, 375, 376 absolute and conditional
inverse (see Inverse functions) Stirling’s formulas and convergence of, 309, 312,
limits of (see Limits of functions) asymptotic series for, 378, 384 319–321
maxima and minima of (see table and graph of, 375 comparison test for, 308, 311
Maxima and minima) Gauss’: containing a parameter, 313
monotonic, 41 differential equation, 276 definition of, 306
multipled-valued (see Multiple- function, 377 of the first kind, 306–308, 317–321
valued function) test, 268, 283 of the second kind, 306, 310–312,
normalized, 342 Geometric integral, 308 321, 322
odd, 238, 347–351 Gibbs, Williard, 150 of the third kind, 306, 313, 322
of a complex variable (see G.l.b (see Greatest lower bound) quotient test for, 304, 311, 315
Functions of a complex Gradient, 158, 161, 162, 172 uniform convergence of, 313, 314,
variable) in curvilinear coordinates, 161 323, 324
of a function (see Composite in cylindrical coordinates, 162 Weierstrass M test for, 313,
function) Graph, of a function of one 324–329
of several variables, 116, 123, 126 variable, 41 Increasing functions, 41
orthogonal, 342, 357, 358 of a function of two variables, monotonic, 41
orthonormal, 342 144, 145 strictly, 41, 47
periodic, 336 Grassman, Herman, 150 Increasing sequences, monotonic
polynomial, 43 Greater than, 3 and strictly, 24
sequences and series of, 269, 270, Greatest limit (see Limit superior) Indefinite integrals, 94 (See also
272, 286, 289 Greatest lower bound, 6 Integrals)
single-valued, 39, 392 of a function, 41 Independence of the path, 212, 213,
staircase of step, 51 of a sequence, 24, 32, 36 243–245, 260
transcendental, 43, 44 Green’s theorem in the plane, 232, Independent variable, 39, 116
types of, 43, 44 240–243, 260 Indeterminate forms, 56, 82–84
vector (see Vector fuctions) in space, (see Divergence theorem) L’Hospital’s rules for (see
Functions of a complex variable, Grouping method, for exact L’Hospital’s rules)
392–423 differentials, 132 Induction, mathematical, 8