PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 649 



Pairs Based on the Normal Error Law 



The identical pair (703), exp(— tP), exp(— irg^), is one of the 

 simplest pairs and may well serve as the starting point in the considera- 

 tion of specific coefficient pairs. Each coefficient is the broad impulse 

 of the normal error law. It is remarkable that identical coefficients 

 of this simple form should produce the same identical function when 

 associated with either the cisoidal oscillation or the very different unit 

 impulse. 



If the differential transformation (222), taking the upper signs, is 

 applied to the normal error law pair (703), the infinite series of ^n 

 pairs (702) is obtained. Of these derived pairs, the first eight are 

 written out as pairs (704)-(711). The cisoidal coefficients are alter- 

 nately even and odd functions which oscillate in the neighborhood of 

 the origin, each successive coefficient having an added half oscillation. 

 The 0n pair has (w + 1) half oscillations. Beyond these oscillations, 

 every coefficient in the infinite sequence decreases rapidly and asymp- 

 totically to zero in both directions. The mates of these cisoidal 

 coefficients are identically the same except for a constant coefficient 

 which is i"' and thus goes cyclically through the four values, 1, i, 

 -I, -i. 



The <^n(x) functions are shown by Fig. 2. They are essentially the 

 parabolic cylinder functions of order n. These coefficients may be 

 used for the expansion of every function which, with its first two deriva- 

 tives, is continuous for all positive and negative values of the variable 

 and for which a certain integral exists. This expansion is known as 

 the Gram-Charlier series, which appears in pair (112). 



Starting again with the normal law of error pair (703) in the form 

 (701) and setting p = I/S/tt, and applying the differential transforma- 

 tion (208) repeatedly, we obtain the infinite sequence of pairs (713) 

 of which the first five are listed as pairs (714)-(718). The cisoidal 

 coefficients are the successive integral powers of p multiplied by the 

 normal error exponential. The impulse coefficients are essentially the 

 0„ functions multiplied by the normal error exponentials. These pair, 

 are plotted in Fig. 3 for the special case /3 = a^ = 1. 



Both of the infinite series of pairs derived from the error function 

 and shown in Figs. 2 and 3 are regular throughout, are nowhere infinites 

 and vanish at infinity. 



