PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 655 



The identical pair (916) divided by its value at the origin is shown 

 in Fig. 5 for different real values of its parameter p. For p = + °o , 

 the curve is of the exp(— tx^) or normal law of error form, and is 

 identical pair (703). For p = h the reciprocal hyperbolic cosine 

 identical pair (625) is shown correctly within the width of the line, 

 this being apparently a mere coincidence since pair (916) does not 

 include it as a special case. Finally, for p = 0, the limiting curve 

 coincides with the horizontal axis taken together with unit length of 

 the positive vertical axis. This represents pair (523) divided by its 

 value at the origin, which is infinite. The point to be especially noted 

 is that the area under every curve of the family illustrated by Fig. 5 

 is the same and equal to unity. This must hold for the limit p = 0, 

 when the curve encloses no area within a finite distance of the origin. 



The identical pair \f~^ \, \g~^\ is of great simplicity and it occupies 

 a central position among algebraic pairs. Starting with the minus 

 one-half power of the parameters in both coefhcients, any increase in 

 the power of one parameter requires an equal decrease in the power of 

 the other parameter as is illustrated, for example, by pairs (502*), 

 (516*), (524). 



It is not permissible to specify any relation whatsoever between 

 the two coefficients of a pair; for example, no pair exists for which one 

 coefficient is twice the other. As stated above, the only multiples 

 permissible are the four units 1, i, — 1, — i. For each of these four 

 cases there are an infinite number of solutions. These solutions 

 satisfy the integral equations given in the foot-note to pair (223). 



Practical Applications of Coefficient Pairs 



Fourier gave the first comprehensive method of finding the solution 

 for transients. His method involves three steps: viz., 



I. Spectrum analysis of the cause among all frequencies. 

 II. Solution for all frequencies. 

 III. Spectrum synthesis of the effects for all frequencies. 



Fourier thus substituted three problems for one. With a table of 

 Fourier coefficient pairs, these three steps may be made as follows : 



I. Find the mate of the cause considered as an impulse coefficient. 

 II. Multiply this mate by the admittance for the system. 

 III. Find the mate of this product considered as a cisoidal coefficient. 



These three steps define a perfectly definite result, since every arbi- 

 trarily chosen coefficient has a mate which is unique and determinate, 

 or may be made so by the specification of some suitable passage to a 

 limit. 



42 



