148 BELL SYSTEM TECHNICAL JOURNAL 



z = f(w) which carries the guide plus horn in the z plane into the straight 

 guide with walls at 9 = 0, 6 = r in the w plane may be obtained from 



^ = (1 -e'Tb/T (4-1) 



aw 



This gives, upon setting 



= |/'(i + id) p = [1 - le'" cos 26 + e'Tb'/T 



dz 

 dw 



the relation (4-2) 



1 -\- g(v^ ^) = [1 _ 2g2- cos 2d + ey 



from which the a„'s may be obtained in accordance with (1-2). 



5. Expressions for the an s for Horn 



The Fourier coefficients of 1 + g{v, 6) appearing in (1-2) and (2-2) are 

 the same. It may be shown from (4-2) that 



U"''F{-a,-a;\;e''') , v >0 

 1 + flo = jr(l -f 2a)/P(l -fa) , I- = • (5-1) 



[F{-a,-a-\;e') , z;<0 



and 



{^2e'"''-''\-a)rF{-a, r - a; r -^ V, e^''')/r\ , v>0 

 a^r = \ 2(-a).(l + ao),.=o/(l + a). , ^ = (5-2) 



[2e'"{-a)rF{-a, r - a; r + 1; e'")/r[ , v <0 



where the F's denote hypergeometric functions, r = 1, 2, • • • and we have 

 used the notation 



(/3)o = 1, {0)r = i3(/^ + 1) • • • (/3 + r - 1) {S-?>) 



When n is odd, a,i = because of symmetry about d = 7r/2. The expres- 

 sions for i- > in (5-1) and (5-2) may be verilied by expanding the two 

 factors in 



1 + g(iS d) = e {\ - e ) {I - e ) 



by the binominal theorem and picking out the terms containing e""^ When 

 ^ < we use the relation 1 + g{v, B) = e^^^l + g{-v, 6)], and when v = 

 we may sum the hypergeometric series. 

 Differentiation of (5-1) and (5-2) leads to 



r4ae''""F(-«, 1 - a;l;e-'") , i' > 

 ^ (1 + a„) = 2a(l + ao).=o , Z) = (5-4) 



^' \^a-e''F{\ - cc,\ - cc\'^\e\ i'<0 



