Ork — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 209 



If all the values of \ for which C{\) - iS{\) is zero have positive 

 imaginary parts, it may be proved similarly that the line integral 



Lt. 



. G{\) cos \x + S(\) sin \x 

 f C{\)-iS(\) 



e-^^"<p(u) du, 



2 



{ip(x-e)+ <j>(.X + e)\, x>0, 



or, 



G(^) 



H^j> 



(7(C0 ) - ^;S'(C0 ) 



In this case we commence, not with (3), but with 

 Lt. 



0. 



'^ {(7(A) + iSjX)} c-'^- + {C{\) - i8{\)]e' 

 -k CX\)-iS(X) 



(10) 



e-'^"<p{io)du, (11) 



where the path of A is a contour lying at infinity on the under side of the 

 axis of real quantities. 



In physical examples, the coefficients in G, S are real, while C contains 

 exclusively terms of odd, and S exclusively terms of even degree, or vice versa ; 

 so that equations (9), (10) are identical, as are also the conditions imposed on 

 the roots of the equations C(X) ± iSiX) ,= 0. In any case, however, one may 

 obtain an equation which appears more symmetrical, by combining (9), (10), 

 in the form 



A- dX ri2f\\ ^rk-\ {^W cos Aw + S[X) sm Xu\ cf)(u)d.ic 



lJ-A ^ ('^) + ^ \^) JO 



or 



■{X) + S^{X) 



^{^l 



X > 0, 



X = 0, (12) 



is neither zero nor infinite ; 



C^(OO) +;S'2(00) 



provided (i) { 6'(oo ) - iS{oo ) } /{ 6'(oo ) + iS{co ) } 

 (ii) all the values of A for which C(X) + i8{X) vanishes have negative 

 imaginary parts ; (iii) all the values of A for which C{X) - iS{X) vanishes 

 have positive imaginary parts. 



As conditions (ii), (iii) appear to hold in all cases of physical interest, it 

 seems unnecessary to write down the additions which must be made to the 

 right-hand members of (9), (10), (12), arising from the residues in case the 

 suppositions made as to the situation of the zeroes of the denominators do not 

 hold. Of course the equations thus modified do not in that case furnish an 

 expansion of the type desired. 



Art. 2. Values of the above Integrals ivhen x is negative. 



In some of the physical problems alluded to, it is necessary to consider 

 the value of the left-hand members of (9), (10), or (12), (mutually equivalent 



