Spheroidal wave equation

In mathematics, the spheroidal wave equation is given by

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

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

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

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)