The Natural Charge Distribution And Capacitance of a Finite Conical Shell
The Natural Charge Distribution And Capacitance of a Finite Conical Shell
Samuel Karp
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3). If now, ^ ((j, )^^^"^, q > -1/2, as \^\ — ■>oo in the right half plane, then, since K (u. , 9 )'VM. ~'^ . We can assert that the left hand side of (6. 3) vanishes at infinity. ^Tinally if we ass'-jne G (m, ) 'v- m, ""^ as \^\ — •>oo in the left half plane, with p > 1/2, then since K~(|j, Q )«v, (-m-)" as \^\ — ■>oo, we see that the right hand side vanishes at Infinity. Invoking Liouville's theorem we have that the entire function defined by either side of equation (6. 3) is Identically zero.... Hence {6, h) 5^(^) = 8 ine^z" (-|; 9„ )k'' (n, « ) (m. 4) 2 z -1/2 Since K* (n, 9^ ) ~ ^""^^ ^ in the region Re^t > then, cle&rly, ^ ^iii)>^ ^x"^'^ in this region. Also note that G (\i) is given explicitly by u. K~(m. , 9 ) G i^) K-(-|. - 1 (continued on next page) (6. 5) A(u) - 26 - Note, since (K (m, 6 ))~ is zeroless for Rep> - -r, that ^ (m. ) is analytic for 1 ° T" - - . SulDstituting 2. _^(m. ) into (3. 20) page 1^ (setting p. = w - l/2) and solving for A(u) we get 2TTUO .
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