Spheroidal wave equation

In mathematics, the spheroidal wave equation is given by

${\displaystyle (1-t^{2}){\frac {d^{2}y}{dt^{2}}}-2(b+1)t\,{\frac {dy}{dt}}+(c-4qt^{2})\,y=0}$

It is a generalization of the Mathieu differential equation.^[1] If ${\displaystyle y(t)}$ is a solution to this equation and we define ${\displaystyle S(t):=(1-t^{2})^{b/2}y(t)}$, then ${\displaystyle S(t)}$ is a prolate spheroidal wave function in the sense that it satisfies the equation^[2]

${\displaystyle (1-t^{2}){\frac {d^{2}S}{dt^{2}}}-2t\,{\frac {dS}{dt}}+(c-4q+b+b^{2}+4q(1-t^{2})-{\frac {b^{2}}{1-t^{2}}})\,S=0}$

• Wave equation

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)

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