On Integral Relations Involving Products of Spheroidal Functions

Nicholas Chako

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N* s;;('5)(, )sR;(ic)\, ). 3. (k)(^)3j;;(j)\^) (2. 13a) [\i, V|n, v j - ^-^ Vv (^^ . , n s;;^j)(os!;.^^\4)-3^, ^"^?)s!;/^^o (2. 13b) [ti, V}n, v J - -^ ^^ ' . , (J. L^), , These expressions are independent of |j. , vj jj. , v . W ($) is an abbreviation for the Wronskian St. (j)(^)S|X(k)\^^. Sjt(k)(^^g^. (j)\^) .
- 7 - The proof is obvious. Furthermore it follows easily that no other relations of the form (2. 8) and (2. 9) exist.
The operators (2. 7) applied to the solution (2. 3) of the wave eq...ua- tion lead to the following kernels: (2. 1U) (2. 15) (2. 16) We restrict ourselves in the following to spheroidal functions with j = 3, U; this implies no loss of generality because the other cases, (j ■ 1, 2) can be obtained by employing the relations (2. 17) s^J^^'^h^) - s^^^ho * i s^;(2)(u .
V V V The operators (2. 7) may be written in spheroidal coordinates 9S (2. 18) (2. 19) z c 2 2 4 - 1 ^ 3 + 1-^^ _ a \ ■»• c '« as - 1 5^ ' ^ 2 2 ie ±10 7(4^-1) (1-ri 2) a a^ - 8 - (2. 20) M.
^^^ ink-^k''''--?=^ ^ '^ ' ViC'Dii-^ For these three operators respectively we take as functions Up(/| ) \/(?^i)a^ (2.

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