380 BELL SYSTEM TECHNICAL JOURNAL 



No attempt will be made here to discuss the solution of the infinite 

 integral (325), which is usually a problem presenting formidable 

 difficulties, even to the professional mathematician. The general 

 method of solution is by contour integration in the complex plane 

 and the calculus of residues. By this process it has been successfully 

 applied to the solution of special problems, and also to deriving some 

 general forms of solution such as the expansion theorem solution. ^^ 

 Compared, however, with the operational calculus, it has no advantages 

 from the standpoint of rigour, and lacks entirely the remarkable sim- 

 plicity and directness of the Heaviside method. 



In the direct solution of circuit problems, therefore, it is believed 

 that the application of the Fourier integral is attended by few if 

 any advantages, and presents formidable mathematical difficulties. 

 On the other hand, there are certain types of problems encountered 

 in circuit theory, where the Fourier integral is a powerful tool. 

 These will be briefly discussed. 



The Energy Absorbed from Transient Applied Forces 



In many technical problems, the complete solution for the in- 

 stantaneous current due to suddenly applied electromotive forces, 

 although formally straight-forward, involves a prohibitive amount 

 of labor. In yet others, the applied forces may be random and 

 specified only by their mean square values. In such problems a 

 great deal of useful information is furnished by the mean power and 

 mean square current absorbed by the network, and to the calcula- 

 tion of these quantities, the Fourier integral is ideally adapted. 

 Its application depends on the following proposition, due to Ray- 

 leigh (Phil. Mag., Vol. 27, 1889, p. 466), and its corollary. 



Let a function <j)(t), supposed to exist only in the epoch {)<t<T, 

 be formulated as the Fourier integral 



0(/) =-L ( |/(a;) |- cos [cct-d(w)] dw 



TT Jo 



where 



/(a,) =1 r / <t>{t)coswtdt~\+\^ j 4>it) sin cot dt~T 



tan ^(w) = / <f){t)s[n(x)t dt-^ / (f){t) cos wt dt. 

 Jo Jo 



■'Bush, "Suniiiuiry of Wagner's Proof of Hcaviside's Formula." Froc. Inst, 

 of Radio Knginccrs. Oct., 1917. Fry. "The Solution of Circuit Troblcnis." (Phys. 

 Rev. Aug., 1919). 



