I 



PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 663 



TABLE I 



Paired Coefficients for the Cisoidal Oscillation and the Unit Impulse ^ 

 Part I. General Processes for Deriving the Mate 



Pair 

 No. 



Coefficient F(J) for the 

 Cisoidal Oscillation 



cis(27r//) = exp(pt) 



Coefficient G(g) for the 

 Unit Impulse 



o-»-0 O- 



101. 

 102. 



103. 

 104.* 



105.* 



106.* 

 107.* 



108.* 



r G(g)cis{- 2nfg)dg 



tJ—aa 



Xi(^ - Po) + MP - PoY + 



limby 401 



r F{f)cis{2Tfg)df 



tJ—Ob 



G 



D, r F(f)cis(2irfg)p-^df 



Cis (2x/o^)[Xi^iCg) + \2^2(g) + '"1 



Xi .. V . + X2 .. V .. + 



(P - po) (P - Po)' 



\ip + X2/'' + \zp^ + 



Xi- + X2— + X3— + 

 p p^ p^ 



''2! 



limby 408* 



lim by 401 * 



lim by 408^ 



cIs {lirf^g) ( Xi + X2 Y", + Xg 



+ X4I-J +..•), 0< 



Xl^l(g) + \2^2{g) + \,^z{g) + ••• 



g g^ g^ 



Xi + X2 T-| + X3 yj + X4 Yi + * • • » <g 



-fxo + Xx-+X2-+ •• 



<a <\, limby 516* 



V{a 



1 r 1 



^.[Xo + X.-, 



+ X2 



1 



a{a + 1) 



g' + 



, 0<g 



1 The pair for the oscillation cis (— 2irft) and the unit impulse is F(f), G(- g) which differs from the 

 tabulated pair only in the sign of the epoch; similarly, for the oscillation cos {2irft) or sin (lirft.) the only 

 change is the substitution for G(g) of the even part of Gig) or of the odd part of — iG(g), the pairs being 

 Pif), KG{g) + G{- g)], or F{f), - ihLGig) - G{- g)], respectively. Every pair in Table I may be 

 thrown into the form of an evaluated Fourier integral by equating the pair after writing either 



j_« ^f ^^^ O-T^fg) before the coefficient F(J) or J_ dg cis (- 2wfg) before the coefficient G(g), Every 



pair in Table I may also be regarded as an operational expression F{p/i2ir) of the operator p = i2ivf = d/dg 

 with G{g) its explicit expression in g. 



* A star marks a pair as being the limit approached by regular pairs. 



