Spheroidal wave functions are solutions of the

Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...

that are found by writing the equation in spheroidal coordinates and applying the technique of

separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...

, just like the use of

spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

lead to

spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a ...

. They are called ''oblate spheroidal wave functions'' if

oblate spheroidal coordinates Oblate spheroidal coordinates are a three-dimensional orthogonal coordinates, orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinates, elliptic coordinate system about the non-focal axis of the ellipse, i. ...

are used and ''

prolate spheroidal wave functions 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 ...

'' if

prolate spheroidal coordinates Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are loc ...

are used. If instead of the Helmholtz equation, the

Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...

is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions are called ''oblate harmonics'' and with prolate spheroidal coordinates, ''prolate harmonics''. Both type of spheroidal harmonics are expressible in terms of

Legendre functions In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated ...

References

;Notes ;Bibliography * C. Niven ''On the Conduction of Heat in Ellipsoids of Revolution.'' Philosophical transactions of the Royal Society of London, v. 171 p. 117 (1880) * M. Abramowitz and I. Stegun, ''Handbook of Mathematical function'' (US Gov. Printing Office, Washington DC, 1964) * {{mathapplied-stub Partial differential equations Special functions

See also

References