104 BELL SYSTEM TECHNICAL JOURNAL 



The integral ni(r, 0) is then composed of two components consisting 

 of the integrals along AC and DB, respectively, and we may write 



ni(r, 0) = - (Y^^^ [^(^0 - ^(-^2)], (4) 



where 



•00 is* 



lik) = r '* e'^^-'diw'' - \)-^i\ (5) 



Jkis 



We integrate (5) by parts N times and find 

 I{k) 



N ( \n rln loois* 



fxis* pisrw flN+l 



The derivatives may be expressed in terms of Legendre polynomials 

 by means of the relation 



(_)n^(^2 _ l)-l/2 = „!(^2 _ l)-»/2-l/2p / "^ \ . 



When the limits in the integrated portion are inserted and the definition 

 of 5 used we see that 



m)^''-^g[^)'n^.P.(k.ls)+Rjf, (6) 



where 



HN+ 1)! 



Utsr) .„.^^^ 



+ 1)! C^''* \ w '\ e'"'^ , ,., 



An inequality for i^iAr may be obtained by using ^ 



|P,v+i(/)| ^ |/ + V^^^^K+^ 



which holds for all values of / in the / plane cut from — 1 to +1, 

 if arg ^t^ — 1 = when t is real and greater than + 1. For then 

 the absolute value of the Legendre polynomial in the integrand is 

 seen to be less than | (w + l)/(w — 1) | (^+i>/2 when R{w) > 0, and Rm 

 may be compared with an integral having [e"''"'| and powers of the 

 factors I w + 1 1 and \w — 1 j in the integrand. On the path yl C we 

 ^ E. W. Hobson, loc. cit., p. 60. 



