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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: SPHEROIDAL WAVE FUNCTIONS 753 Wave Equation in Oblate Spheroidal Coordinates 21.s.2 (c=a fk) 21.5.2 may be obtained from 21.5.1 by the transformations [+&:it, C + F i C . 21.6. Differential Equations for Radial and Angular Prolate Spheroidal Wave Functions If in 21.5.1 we put cos @=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 equations 21.6.1 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 respectively . Radial and angular prolate spheroidal functions satisfy the same differential equation over different ranges of the variable.) (21.6.1 and 21.6.2 are identical. Differential Equations for Radial and Angular Oblate Spheroidal Functions 21.6.3 (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 21.7.1 S2i(~, 7) =ef d Y (c>f2+r (7) r=O, 1 =Prolate angular function of the first kind 21.7.2 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 21.7.3 (Ykdk+Z+ (&-Xrnn)d+Ykdk-2=o ( 2 m+k+2) (2 m +k + 1)c2 (2 m+2k+3) (2 m+2k+5) Bn=(m+k)(mfk+l) 2(m+k)(~~~+k+1)-2m’--l c2 (2m+2k-l)(2m+2k+3) k(k-l)c2 ”“ (2m +2k-3) (2m +2k- 1 ) Transcendental Equation for X,. 21.7.4 U (Xrnn) =U1 (Xrnn) + Uz (Am,) = 0 (The choice of r in 21.7.4 is arbitrary.) The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.