The practical application of the Fourier integral

The growing practical importance of transients and other non-periodic phenomena makes it desirable to simplify the application of the Fourier integral in particular problems of this kind and to extend the range of problems which can be solved in closed form by this method. Unless the physicist or technician is in a position to evaluate the definite integrals which occur, by mechanical means, he is usually entirely dependent upon the results obtained by the professional mathematician. To facilitate the use of the known closed form evaluations of Fourier integrals many of them have been compiled and tabulated in Table I. They are presented, however, not as definite integrals but as paired functions, one function being the coefficient for the cisoidal oscillation (or complex exponential) and the other function the reciprocally related coefficient for the unit impulse. This arrangement simplifies the table and promises to be most convenient in practical applications, since it is the coefficients of which immediate use is made, just as in the case of the Fourier series. Applications of the tabulated coefficient pairs to 85 transient problems are given, together with all necessary details, in Table II.