Page 434 - Schaum's Outline of Theory and Problems of Advanced Calculus

Abel, integral test of, 334 Analytic part, of a Laurent series, Beta functions, 375, 378, 379, 382,
summability, 305 395 384
theorems of, 282, 293 Anti-derivatives, 94 relation, to gamma functions, 379
Absolute convergence: Approximations (see Numerical Bilinear transformation, 422 (See
of integrals, 309, 312, 319–321 methods) also Fractional linear
of series, 268, 283, 300 by partial sums of Fourier series, transformation)
theorems on, 269, 283 352 Binary scale, 16, 21
Absolute maximum or minimum, by use of differentials, 78, 79, 130, system, 2
42 (See also Maxima and 131 Binomial coefficients, 21
minima) least square, 201 series, 275
Absolute value, 3 to irrational numbers, 9 theorem, 21
of complex numbers, 6 using Newton’s method, 74 Bolzano-Weierstrass theorem, 6, 12,
Acceleration, 74, 158 using Taylor’s theorem, 274–275 19, 117
centripetal, 175 Archimedes, 90 Boundary conditions, 339
in cylindrical and spherical Arc length, 99, 109 Boundary point, 117
coordinates, 181 element, 157, 161, 174 Boundary-value problems:
normal and tangential Area, 100, 109 and Fourier integrals, 371
components of, 177 expressed as a line integral, 242 and Fourier series, 339, 356, 357
Accumulation, point of, 5, 117 of an ellipse, 205 in heat conduction, 356, 357
(See also Limit points) of a parallelogram, 155, 168 in vibration of strings, 361
Addition, 1 Argand diagram, 7 separation of variables method
associative law of, 2, 8 Argument, 7 for solving, 356
commutative law of, 2 Arithmetic mean, 10 Bounded functions, 40, 41
of complex numbers, 7, 13 Associative law, 2, 4 sequences, 24, 30–32, 36
of vectors, 151, 152, 163 for vectors, 152, 155 sets, 6
Aerodynamics, 402 Asymptotic series or expansions: Bounds, lower and upper, 6, 12, 13
Aleph-null, 5 for gamma function, 286, 292 Box product, 155
Algebra: Axiomatic foundations: Branches of a function, 41
fundamental theorem of, 43 of complex numbers, 6, 7 Branch line, 417
of complex numbers, 6, 7, 13–15 of real number, 4 Branch point, 396, 417
of vectors, 151, 152, 163–171 of vector analysis, 155
Algebraic functions, 43 Axis, real, 2
Algebraic numbers, 6, 13 x, y and z, 121 Calculus, fundamental theorem of
countability of, 13 integral, 94, 95, 104
Alternating series, 267, 268, 282 Cardinal number of the continuum,
convergence test for, 267, 268 Base, of logarithms, 4 5
error in computations using, 268, Bases, orthonormal, 152 Cardioid, 114
282 Bernoulli, Daniel, 336 Catenary, 113
Amplitude, 7 Bernoulli numbers, 304 Cauchy principal value, 310, 321
Analytic continuation, of gamma Bernoulli’s inequality, 16 Cauchy-Riemann equations, 393,
function, 376 Bessel functions, 276 400–403
Analytic functions, 393 Bessel differential equation, 276 derivation of, 401

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