144 BELL SYSTEM TECHNICAL JOURNAL 



The integrals in (7.23) may be evaluated in terms of the exponential inte- 

 gral Ei{x) defined by, for x real, 



/x 00 n 



e' dt/t = C + i loge x' + Z -^ 

 00 n=i n\n 



n=0 



where C — .S11 • • • is Euler's constant and Cauchy's principal value of the 

 integral is to be taken when a: > 0. We set / = ^ — w and obtain 



Ud +^)J 



yi = e-"" iEi[p/k\ - 2Ei[p{\ - k)/k] + Ei 



where we have again expressed ^ and X in terms of p and k. 



A power series for yi which converges when —1/3 < ^ < 1 may be ob- 

 tained by expanding the denominators of the integrands in (7.23) in powers 

 of m/^ and integrating termwise: 



yi = r^i - 2^-" + e-'] 



+ 1 !r'[l - 2(1 + p/1 1)^-" + (1 + X/1 !).-'] (7.24) 



+ 2!r'[l - 2(1 + p/1! + p'me-' + (1 + X/1! + XV2!)e-'] 

 + ... 

 The following special values may be obtained from the equation given 

 above. When p = 



yi = -loge (1 - k^) 



(7.25) 

 y2= 



Thisresult may also be obtained by evaluating the integral obtained when 

 we set C = 0, 2i = ri cos di, Z3 = ^i sin ^1, z^ = ^2 cos 62, Zi = ^2 sin di 

 in (7.11) and (7.12). 

 Near^ = 1, 



yi - e-'[Ei{p) -C- loge p(l - ¥)] 



3^2 - pyi - 1 + (1 + p)e"'' 



(7.26) 



Near k = 0, 



yi « ^(1 - e-'f/p, y2 - yi (7.27) 



except when p = in which case yi is approximately k^. 

 When p is large 



^^ n ^ n2 ^ n3 ^ „4 ^ 



P P' P* p' 



^ p p2 



(7.28) 



except near k = 1 where both yi and yz have logarithmic infinities. The 

 asymptotic expansion (7.3) for 12(t), which was obtained by the first method 



