156 BELL SYSTEM TECHNICAL JOURNAL 



We shall illustrate this procedure by examining the integral 



I = f de \ dxx Qx^\ -x"" + 2a cos 6 + 2hx sin ^ + c sin' 6 (A3-1) 



Expanding that part of the exponential which contains the trigonometrical 

 terms and integrating termwise gives 



T =T TT ^'^^'"^^^r(^ + ^ + ^) 



^n4^^w!^!(^ + w + w)!r(w+ i) 

 where we have used 



2'"r(w+ J)«! = \/i(2n)! 

 We next make the substitution 



{l + m + n)\ 2iri Jc /^+-+"+i ^ ^ 



where the path of integration C is a circle chosen large enough to ensure the 

 convergence of the series obtained when the order of summation and integra- 

 tion is changed. The summations may now be performed: 



2iia 



m=0 



1 c r"'(f rV'^ 

 = lf ^-JL^A^ e'^^'" dt 

 2iJc t - c -h"- 



C encloses the pole at c + 6^ ^nd the branch point at c as well as the origin. 

 When a- is zero the integral may be reduced still further. Let c be com- 

 plex and b such that the point c -\- b does not lie on the line joining to c. 

 Deform C until it consists of an isolated loop about c -\- b^ and a loop about 

 and c, the latter consisting of small circles about and c joined by two 

 straight f)ortions running along the line joining to c. The contributions 

 of the small circles about and c vanish in the limit. Along the portion 

 starting at and running to c, arg (/ — c) = — tt + arg c, and along the por- 

 tion starting at c and running to 0, arg (/ — c) = tt -f arg c. On both 

 portions arg / = arg c. Bearing this in mind and setting / = c sin^ d on the 

 two portions gives 



Jo b^ -\- c cos'^ 6 

 The integral may be expressed in terms of the function 



Ie{k,x) = f e~''Io{ku)du 

 Jo 



