cox FORM A L TRA NSFORMA TION 

 From (4-2) and (4-5), or from (3-6), 



Tin - ik)T(n + ik) ^ (^2^)^n. (/3) n- 



117 



(6-2) 



= -ik z 



n'.iii - 1)!2 



ro(2«.)"(" - ^0! 



where we have expanded g(v, 6) as given by (6-1) in powers of 

 cos d/ch V and integrated termwise. The notation is (o;)o = 1, («)„ = 

 a{a + 1) • • • (a + « — 1). 



For a right angle corner /3 = 1/2, and a more rapidly convergent series 

 may be obtained by subtracting the sum of the series corresponding to 

 yfe = 0, namely 



log 2 = E 



(l/2)„ 



„=i n\2n 



(6-3) 



Thus for /3 = 1/2 

 R'i' = -ik 



'log.2-Z^"(l-^4„)], 

 „=i n\ln J 



Ai = TT^/sinh -wk, An = Ai IT (1 + ^""^ ^). « > 1 



I 



= 1 



(6-4) 



The rate of convergence of the more general series (6-2) may be increased 

 in a somewhat similar way. It is found that 



R'l' 



J - 2^{\ - AO -^(2 + /3')(1 -Ad 



2^ 

 135 



(23 -f- 20/3' + 2/3') ( 1 - ^3) - 



J = K -\- L 



{^)n 



(6-5) 



K = Z ^^f" = j^ - ^(1 - 0) - .5772 

 „=i w!« 1 — /3 



^ = Z 



(-2/3),„ 



z 



(^)n-. 



^1 (2m)! „=m (» — M)\n 



y (1/2 - /3)^ 



i'i(l/2)„m(w-/8) 



where .5772 • • • is Euler's constant, '^(.v) is the logarithmic derivative of 

 II(.t) = r(.v + 1), and An is given by (6-4). 



