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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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SPHEROIDAL WAVE FUNCTIONS 753
Wave Equation in Oblate Spheroidal Coordinates
21.5.2 may be obtained from 21.5.1 by the
[+&:it, C + F i C .
21.6. Differential Equations for Radial and
Angular Prolate Spheroidal Wave Functions
If in 21.5.1 we put
@=Rrnn(c, t ) S r n n ( C , 9) sin m$
then the “radial solution” R,,(c, f ) and the
“angular solution’’ Smn(c, 7) satisfy the differential
where the separation constants (or eigenvalues)
Xmn are to be determined so that R,,(c, t ) and
Smn(C, r ] ) are finite at t=&1 and r]=&l
angular prolate spheroidal functions satisfy the
same differential equation over different ranges of
(21.6.1 and 21.6.2 are identical.
Differential Equations for Radial and Angular Oblate
(21.6.3 may be obtained from 21.6.1 by the
transformations t+&it, c+?ic; 21.6.4 may be
obtainedfrom21.6.2by the transformation c+Tic.)
21.7. Prolate Angular Functions
S2i(~, 7) =ef d Y (c>f2+r (7)
=Prolate angular function of the first kind
szi(c, 7) = I=-- 2’ ~ Y ( c ) G+ r (71
=Prolate angular function of the second kind
(E(r]) and @ ( r ] ) are associated Legendre
functions of the first and second kinds respectively.
However, for - 15 z 5 1, e ( z ) = ( 1 -z2)m/2drnPn(z)/
dzm (see 8.6.6). The summation is extended over
even values or odd values of r.)
Recurrence Relations Between the Coefficients
( 2 m+k+2) (2 m +k + 1)c2
(2 m+2k+3) (2 m+2k+5)
”“ (2m +2k-3) (2m +2k- 1 )
Transcendental Equation for X,.
U (Xrnn) =U1 (Xrnn) + Uz (Am,) = 0
(The choice of r in 21.7.4 is arbitrary.)
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