448 THE UICLL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



exp [ — ix sin 6 + iy cos 9] + Si 



r^i _^,^, (5.6) 



= (z'/tt)^'" exp [ — ix sin 6 + iy cos 0] / c ''" f/^. 



*/— CO 



which follows from (5.3) and 



exp (-ir) (It = U/iY'-. (5.7) 



/ 



The development leading to (5.5) shows that is satisfies the boundary 

 condition E' = at 77 = 770 f or < ii' < 1. That (5.5) also satisfies the 

 condition for the extended range < w < ^ follows immediately from 



(5.8) 



2i Jl. \ 2 / sni irn 



= —{if-n-y'' exp [ — ix sin 6 + iy cos 6] I exp {-it') dt 



J— 00 



= —exp [ — ix sin 9 + iy cos 9] — Si 



when we note that setting 2' = So reduces Siih) to the left hand side of 

 (5.8) (with z' = zo). 



Equation (5.8) is due to T. M. Cherr}^ who ol)tained it by expressing 

 the cylinder functions as integrals and interchanging the order of in- 

 gration (he works with the function Dn(z) of our equations (9.2)). Sub- 

 stituting the integrals (9.19) for Un(z) and V„(z') in (5.8) and inter- 

 changing the order of integration leads to a similar derivation. Eciuation 

 (5.8) may also be obtained by deforming L2 into Li when < iv < I 

 and into L3 when 1 < w < go . This leads to the two series 



e'" sec ^ Z (-ur/2) Vt7„u)F„(/), (5.9) 



-c'^secl Z (-tW2)1r(n+ DUMWniz') (5.10) 



which may be summed in much the same way as was (5.2) for Si. 



An expression for E which is useful in the study of the current density 

 on the surface of the cylinder may be obtained from (5.5) by combining 

 expression (5.8) for exp [ — ix sin 6 + iy cos 6] + Si with expression (5.4) 



^^ Expansions in Terms of Paral)olic Cylinder Functions, Proc. Edinburgh 

 Math. Soc, Ser. 2, 8, pp. 50-65, 1948. 



