APPENDIX D

TABLES OF FOURIER EXPANSIONS AND LAPLACE-TRANSFORM PAIRS

Table D-l. Fourier Coefficients and Mean-square Values of Periodic Functions

Table D-2. Fourier Transforms

Table D-3. Fourier Cosine Transforms

Table D-4. Fourier Sine Transforms

Table D-5. Hankel Transforms

Table D-6. Laplace-transform Pairs Involving Rational Algebraic Functions

Table D-7. Laplace Transforms

D-l. Tables D-l to D-7 present a number of Fourier expansions (Sec. 4.11-4), Hankel transforms (Sec. 8.6-4), and Laplace transform pairs (Sees. 8.2-1, 8.2-6, and 8.4-1) for reference.

D-2. Fourier-transform Pairs and Laplace-transform Pairs. (a)

For suitable functions f(t) (Sees. 4.11-4 and 8.2-6),

so that tables of Laplace-transform pairs (2) (Tables D-6 and D-7) may be used to obtain many Fourier-transform pairs (1):

    1.Given f(t), obtain, if possible,

F(s) ≡[f(t)] and F₁(s) ≡ [f(-t)]. If F(s) and F₁(s) are analytic for σ > 0, then

    2. Given C(ω) or c(v) ≡ F_F(iω),

provided that this expression exists, so that C(s/i) or F_F(s) is analytic for σ ≥ 0, and f(t) = 0 for t <0.

(b) The following procedures permit one to obtain many Laplace-transform pairs (2) from tables of Fourier-transform pairs (1) (Table D-2 and Ref.8.1):

    1. Given f(t) such that f(t) is real and f(t) = 0 for t < 0, use the table of Fourier-transform pairs (or any other method) to obtain C(ω) or c(ν) = F_F(iω). Then

    2. Given f(t) such that f(t) is real and even, f (—t) = f(t),*

    3. Given F(s) analytic for σ ≥ 0, obtain ^_1[F(s)] for t > 0 as the function f(t) corresponding to

* Note that every function can be rewritten as the sum of even and odd parts (Sec. 4.3-2), and that f(–t) is even whenever f(t) is odd.

Table D-1. Fourier Coefficients and Mean-square Values of Periodic Functions [Sec. 4.11-4a; ]

Table D-2a. Fourier-transform Pairs*

Table D-2b. Fourier Transforms*

Table D-3. Fourier cosine Transforms*†

Table D-4. Fourier Sinc Transforms*†

Table D-5. Hankel Transforms*

* From I. A. Sneddon, Fourier Transforms, McGraw-Hill, New York, 1951.

Table D-6. Laplace-transform Pairs Involving Rational Algebraic Functions F(s) = D₁(s)/D(s)

Each formula holds for complex as well as for real polynomials D₁(s) and D(s) (See. 8.4-4); but the latter case is of greater practical interest. In this case the roots of D(s) = 0 are either real or they occur as pairs of complex conjugates, and the functions f(t) are real.

Table D-7. Table of Laplace Transforms†

FIG D-1

FIG D-2

FIG D-3

BIBLIOGRAPHY

      Campbell, C. A., and Foster: Fourier Integrals for Practical Applications, Van Nos- trand, Princeton, N.J., 1948.

      Erdélyi, A.: Integral Transforms, vols. 1 and 2 (Bateman Project), McGraw-Hill, New York, 1954.

      Oberhettinger, F.: Tabellen zur Fourier Transformation, Springer, Berlin, 1957.

      Smith, J. J.: Tables of Green's Functions, Fourier Series, and Impulse Functions for Rectangular Coordinates, Trans. AIEE, 70, 22, 1951.

      Ditkin, V. A., and A. P. Prudnikov: Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.