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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the spheroidal wave equation is given by :(1-t^2)\frac -2(b+1) t\, \frac + (c - 4qt^2) \, y=0 It is a generalization of the
Mathieu differential equation In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu ...
. If y(t) is a solution to this equation and we define S(t):=(1-t^2)^y(t), then S(t) is a
prolate spheroidal wave function The prolate spheroidal wave functions are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated around its long axis, “cigar shape“). Related are ...
in the sense that it satisfies the equationsee Bateman
page 442
/ref> :(1-t^2)\frac -2 t\, \frac + (c - 4q + b + b^2 + 4q(1-t^2) - \frac ) \, S=0


See also

*
Wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...


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) Ordinary differential equations {{mathanalysis-stub