Fourier Series Intro - Laplace Series

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http://mathworld.wolfram.com/LaplaceSeries.html

Laplace Series

The spherical harmonics form a complete orthogonal system, so an arbitrary real function can be expanded in terms of complex spherical harmonics by

(1)

or in terms of real spherical harmonics by

(2)

The representation of a function as such a double series is ageneralized Fourier series known as a Laplace series.

The process of determining the coefficients in (1) is analogous to that of determining the coefficients in a Fourier series, i.e., multiply both sides of (1) by, integrate, and use the orthogonality relationship (◇) to obtain

int_0^(2pi)int_0^pif(theta,phi)Y^__(l^')^(m^')(theta,phi)sinthetadthetadphi =sum_(l=0)^inftysum_(m=0)^lint_0^(2pi)int_0^piA_l^mY_l^m(theta,phi)Y^__(l^')^(m^')(theta,phi)sinthetadthetadphi =sum_(l=0)^inftysum_(m=0)^lA_l^mdelta_(ll^')delta_(mm^') =A_l^m.

(3)

The following sequence of plots shows successive approximations to the function, which is illustrated in the final plot.

SphericalHarmonicSeries

Laplace series can also be written in terms real spherical harmonic as

(4)

Proceed as before, using the orthogonality relationships

int_0^(2pi)int_0^piP_l^m(costheta)cos(mphi)P_(l^')^(m^')(costheta)cos(m^'phi)sin(theta)dthetadphi =-(2pi(l+m)!)/((2l+1)(l-m)!)delta_(mm^')delta_(ll^') int_0^(2pi)int_0^piP_l^m(costheta)sin(mphi)P_(l^')^(m^')(costheta)sin(m^'phi)sinthetadthetadphi =-(2pi(l+m)!)/((2l+1)(l-m)!)delta_(mm^')delta_(ll^').

(5)

So and are given by

(6)

(7)

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