114 BELL SYSTEM TECB^tCAL JOURNAL 



since both sides have the value 1/2. In the second, term by term integra- 

 tion is vah'd since the series integrated are uniformly convergent as may be 

 seen from the inequality 



k<"'W I < (^)"XJ^)"''^ + ^^""'^ + 0(yV')], (1.11) 



in which we suppose that y remains finite as w -^ oo . This may be obtained 

 by using the known behavior of Hermite polynomials of large order. ^ 



When Q » rms In so that a is very large the distribution approaches that 

 of a sine wave, namely 



0, I 3' I > ^ 



pi{y) 



\ pi{yi)dyi 



J — 00 



- - + - arc sm ^ , \y\ < a 



2 TT a 



(1.12) 



In order to study the manner in which the Hmiting expressions (1.12) are 

 approached it is convenient to make the change of variable 



x = y - acosd, dd = [a^ - {y - xY]-^'^ dx 



z = X — y -\- a 



in (1.6). We obtain 



Pi(y) = - r\(x) [a' -{y - x)Y"' dx 



T "y-a 



.2a 



(1.13) 





z -\- y — a)[z{2a — z)] ' dz. 



JO 



An asymptotic (as a becomes large) expression for pi{y) suitable for the 

 middle portion of the distribution where a — \y \^ 1 may be obtained from 

 the first integral in (1.13). Since the principal contributions to the value of 

 the integral come from the region around a; = we are led to expand the 

 radical in powers of x and integrate termwise. Legendre polynomials enter 

 naturally since they are sometimes defined as the coefficients in such an 

 expansion. Replacing the limits of integration y -\- a and y — ahy -\-<x) 

 and — 00 , respectively and integrating termwise gives 



Pi(y) 



X L »-i (o^ - f)" J 



{a* - y'r'" [. . 3< + 1 3(35<' + m + 3) , 1 



T I ^ 2(a» - /) "^ 8(a2 - y^y "t" • ' • J 



(1.14) 



•A suitabk asymptotic formula is given in Orthogonal Polynomials, by G. Szego, 

 Am. Math. Soc. CoUoquium, Pub., Vol. 23, (1939), p. 195. 



