Introduction to the theory of Fourier's series and integrals, by H.S. Carslaw.

GENERAL INDEX

321

Definite integrals, ordinary (Chapter IV.); the sums S and s, 77; Darboux's
theorem, 79; definition of upper and lower integrals, 81; definition of, 81;
necessary and sufficient conditions for existence, 82; some properties of, 87;
first theorem of mean value, 92; considered as functions of the upper limit, 93;
second theorem of mean value, 94. See also Dirichlet's integrals, Fourier's
integrals, Infinite integrals and Poisson'sintegral.
Differentiation, of series, 143; of power series, 148; of ordinary integrals, 170;
of infinite integrals, 182; of Fourier's series, 261.
Dirichlet's conditions, definition of, 206.
Dirichlet's integrals, 200.
Discontinuity, of functions, 64; classification of, 65. See also Infinite discontinuity
and Points of infinite discontinuity.
Divergence, of sequences, 37; of series, 41; of functions, 51; of integrals, 98.
Fejer's theorem, 234.
Fejer's theorem and the convergence of Fourier's series, 240, 258.
Fourier's constants, definition of, 196.
Fourier's integrals (Chapter X.); simple treatment of, 284; more general conditions for, 287; cosine and sine integrals, 292; Sommerfeld's discussion
of, 293.
Fourier's series (Chapters VII. and VIII.); definition of, 196; Lagrange's treatment of, 198; proof of convergence of, under certain conditions, 210; for
even functions (the cosine series), 215; for odd functions (the sine series),
220; for intervals other than (-wr, 7r), 228; Poisson's discussion of, 230;
Fejer's theorem, 234, 240; order of the terms in, 248; uniform convergence
of, 253; differentiation and integration of, 261; more general theory of, 262.
Functions of a single variable, definition of, 49; bounded in an interval, 50; upper
and lower bounds of, 50; oscillation in an interval, 50; limits of, 50; continuous, 59; discontinuous, 64; monotonic, 66; inverse, 68; integrable,
84; of bounded variation, 207.
Functions of several variables, 71.
General principle of convergence, of sequences, 34; of functions, 56.
Gibbs's phenomenon in Fourier's series (Chapter IX.); 264.
Hardy's theorem, 239.
Harmonic analyser (Kelvin's), 295.
Harmonic analysis (Appendix I.), 295.
Improper integrals, definition of, 113.
Infinite aggregate. See Aggregate.
Infinite discontinuity. See Points of infinite discontinuity.
Infinite integrals (integrand function of a single variable), integrand bounded and
interval infinite, 98; necessary and sufficient condition for convergence of,
100; with positive integrand, 101; absolute convergence of, 103; t/-test for
convergence of, 104; other tests for convergence of, 106; mean value theorems
for, 109.
Infinite integrals (integrand function of a single variable), integrand infinite, 111;
/z-test and other tests for convergence of, 114; absolute convergence of, 115.
Infinite integrals (integrand function of two variables), definition of uniform convergence of, 174; tests for uniform convergence of, 174; continuity, integration and differentiation of, 179.
Infinite sequences and series. See Sequences and Series.
Infinity of a function, definition of, 65.
Integrable functions, 84.