Spheroidal wave equation

In mathematics, the spheroidal wave equation is given by

[math]\displaystyle{ (1-t^2)\frac{d^2y}{dt^2} -2(b+1) t\, \frac{d y}{dt} + (c - 4qt^2) \, y=0 }[/math]

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

[math]\displaystyle{ (1-t^2)\frac{d^2S}{dt^2} -2 t\, \frac{d S}{dt} + (c - 4q + b + b^2 + 4q(1-t^2) - \frac{b^2}{1-t^2} ) \, S=0 }[/math]

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)