spheroidal wave equation

In mathematics, the spheroidal wave equation is given by

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

It is a generalization of the Mathieu differential equation.see Abramowitz and Stegun, [http://people.math.sfu.ca/~cbm/aands/page_722.htm page 722]

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 equationsee Bateman, [https://archive.org/details/partialdifferent033549mbp/page/n469, page 442]

:(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

See also

References

{{Reflist}}

;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)

Category:Ordinary differential equations

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