PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 653 



ing the integration constants so as to make the impulse coefficients 

 alternately odd and even, these two pairs are as shown in Fig. 4. If 

 we now allow a to approach the limit zero, a new series of pairs is 

 obtained of which the first two pairs are shown dotted in Fig. 4 for 

 the particular choice of integration constants there made. The general 

 limiting pair is designated as p-", ^-„(g) and it is shown with its n 

 arbitrary parameters Xi, X2, • • •, Xn as pair (410*). In some ways it 

 is simpler to derive the limiting pair for negative integral powers of p 

 from rational functions of p, which may be accomplished as shown by 

 pair (411*). Special cases are shown by pairs (408*), (409*), (415*), 

 (416*). 



M = 2 



n = 1 



Fig. 4 — Graphs for the family of pairs ^~"exp(— wa^P), a'^Dg'^expi— irg^/a^), 

 with the integration constants chosen so as to make the impulse coefficients alter- 

 nately odd and even. The heavy curves show the cases a = 1, n = 1,2; the dotted 

 curves show the limit a—*-0,n = 1,2. 



The first of the series ^-i(g) is a unit step at epoch from a constant 

 value X — I for all negative epochs to the constant value X + | for all 

 positive epochs. The constant X may have any value ; this is a singu- 

 lar case marked by the failure of the general rule that the choice of the 

 cisoidal coefficient uniquely determines the impulse coefficient. This 

 means that in any well set problem some other condition determines the 

 value of the constant X. In some problems, for example, it is necessary 

 that the epoch coefficient be an odd function, and then X vanishes. In 

 other problems where either the epoch function must be zero for all 

 negative epochs or on the other hand the p occurring in the cisoidal 

 coefficient is actually the limit of ^ + a as a approaches zero through 

 positive values, the constant X equals |. This limiting condition may 

 arise if we assume that resistance may be ignored, as a first approxima- 



