PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 643 



Fourier's fundamental discovery was that the two functions may be 

 transposed in the Fourier integral if the sign of one of the parameters 

 is reversed. Thus, either one of the two functions constituting any 

 coefficient pair may be taken as the coefficient of the cisoidal oscilla- 

 tion, provided only that the proper sign is given the epoch parameter 

 occurring in the other function. For this reason also both functions 

 are thus quite properly regarded as coefficients. 



It is found convenient to call each coefficient of a coefficient pair 

 the mate of the other coefficient, pair and mate being employed just 

 as in the case of gloves. To find the mate of a glove, it is necessary to 

 know all about the given glove including the fact as to whether it is 

 the right or the left one of the pair. In the same way, to find the mate 

 of a coefficient function, it is necessary to know not only the form of the 

 function, but, in addition, whether its variable is the frequency or the 

 epoch. The notation dUG(g), dJlF(f) will be employed to indicate 

 the mate of the particular coefficient G{g), F{f). 



We have now defined and explained the proposed terminology for 

 use in the practical application of the Fourier integral theorem. 

 Before proceeding to practical applications, it is desirable to become 

 familiar with these coefficient pairs considered in their own right. 

 We may well begin by reminding the reader of the dissimilarity be- 

 tween the elementary oscillations. 



The Two Elementary Functions Contrasted 



The dissimilarity between the two elementary functions of the 

 time, the cisoidal oscillation cis(27r//) and the unit impulse ^o(^ — g) 

 is most striking. This is clearly shown by the wire models of Fig. 1 

 where each function is depicted for five values of its parameter. For 

 the value zero the cisoidal oscillation degenerates into an infinite 

 straight line parallel to the time axis and cutting the real axis at 

 X = I. For the same value zero of its parameter g, the unit impulse is 

 zero everywhere except at the origin where it has a vanishingly narrow 

 loop extending to x = -f- oo . 



For other values of the parameter, the cisoidal oscillation is always 

 an infinite cylindrical helix, centered on the time axis, and passing 

 through the point x = 1, while the infinite loop of the impulse 

 function is displaced unchanged along the time axis to t = g. For 

 positive values of the parameter /, the cisoidal oscillation is a right- 

 handed helix, with pitch equal to/~\ and thus decreasing as/ increases. 

 For negative values of /, the pitch is the same but the helix is left- 

 handed. 



Both functions have essential singularities, which are quite dif- 



