PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 651 



Essentially Singular Pairs for Integral Powers of the 



Parameter 



If in Fig. 3, with the value of n held fixed, we allow a to approach the 

 limit 0, the cisoidal coefficient becomes ^" and the impulse coefficient, 

 which is compressed horizontally towards the origin and expanded 

 vertically, with corresponding areas increasing as a~", ultimately 

 vanishes everywhere except at the origin where it acquires an essential 

 oscillating singular point. At the limit, then, a singular pair is ob- 

 tained; it will be designated as ^", ^,i(g). ^n(g) is characterized by 

 having all of its moments about the origin vanish except the wth 

 moment, which is equal to (— !)"«! The dotted graphs on the left of 

 Fig. 3 show p" to the scales indicated. The curves on the right show 

 §>n{g) provided we assume that the horizontal scale is increased with a 

 and the vertical scale increased inversely with a"+^ as a approaches the 

 limit 0. Fig. 3 thus serves to picture the essentially singular function 

 ^„(g). That is, it is sufficient if the coefficient maintains this form 

 while proceeding to the limit. This form is, however, not essential. 

 It is necessary only that the method of approach to the limit give the 

 same set of moments. 



An alternative way of deriving the mate for the positive integral 

 powers p"^ is by means of a linear combination of (w + 1) pairs of the 

 form of (603) with parameters equal to a, 2a, 3a, • • • , {n + l)a, 

 respectively, so that the first term in the power series expansion of the 

 cisoidal coefficient is ^". The corresponding impulse coefficient is a 

 succession of {n + 1) bands, each of width a, the first band beginning 

 at epoch zero, the heights of the successive bands being equal to the 

 binomial coefficients for power n divided by a"+^ but alternately posi- 

 tive and negative. The wth moment of this impulse coefficient is 

 for m < 11, equal to (— l)"w! for m = n, and proportional to a"*"" 

 for m > n. Upon allowing a to approach zero, the cisoidal coefficient 

 approaches ^", and the impulse coefficient approaches ^n(g), since in 

 the limit the same set of moments is obtained as was found above to 

 characterize the wth singularity function. This is pair (402*). 



The special cases for w = 0, 1 are of most frequent occurrence. 

 They are pairs (403*), (404*). ^o is the unit impulse since its 0th 

 moment equals unity; ^i is the doublet with the moment — 1 since 

 its first moment is — 1. ^i and all higher order singular functions 

 are included in the series coefficients of (104*), (106*). 



Fig. 3 may be extended upward step by step from the normal error 

 law pair by dividing by p on the left and integrating with respect to g 

 on the right. At each step a constant of integration is introduced. 

 The first two pairs thus obtained are pairs (725*) and (726*). Choos- 



