In mathematics, the spheroidal wave equation is given by
( 1 − t 2 ) d 2 y d t 2 − 2 ( b + 1 ) t d y d t + ( c − 4 q t 2 ) y = 0 {\displaystyle (1-t^{2}){\frac {d^{2}y}{dt^{2}}}-2(b+1)t\,{\frac {dy}{dt}}+(c-4qt^{2})\,y=0}
It is a generalization of the Mathieu differential equation. If y ( t ) {\displaystyle y(t)} is a solution to this equation and we define S ( t ) := ( 1 − t 2 ) b / 2 y ( t ) {\displaystyle S(t):=(1-t^{2})^{b/2}y(t)} , then S ( t ) {\displaystyle S(t)} is a prolate spheroidal wave function in the sense that it satisfies the equation
( 1 − t 2 ) d 2 S d t 2 − 2 t d S d t + ( c − 4 q + b + b 2 + 4 q ( 1 − t 2 ) − b 2 1 − t 2 ) S = 0 {\displaystyle (1-t^{2}){\frac {d^{2}S}{dt^{2}}}-2t\,{\frac {dS}{dt}}+(c-4q+b+b^{2}+4q(1-t^{2})-{\frac {b^{2}}{1-t^{2}}})\,S=0}
See also Wave equation
References
Bibliography M. Abramowitz and I. Stegun, Handbook of Mathematical function (US Gov. Printing Office, Washington DC, 1964) H. Bateman, Partial Differential Equations of Mathematical Physics (Dover Publications, New York, 1944)