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Spherical Harmonic Differential Equation

In three dimensions, the spherical harmonic differential equation is given by

[1/(sintheta)partial/(partialtheta)(sinthetapartial/(partialtheta))+1/(sin^2theta)(partial^2)/(partialphi^2)+l(l+1)]u=0,

and solutions are called spherical harmonics (Zwillinger 1997, p. 130). In four dimensions, the spherical harmonic differential equation is

u_(xx)+2u_xcotx+csc^2x(u_(yy)+u_ycoty+u_(zz)csc^2y)+(n^2-1)u=0

(Humi 1987; Zwillinger 1997, p. 130).

See also

Spherical Harmonic

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References

Humi, M. "Factorisation of Separable Partial Differential Equations." J. Phys. A: Math. Gen. 20, 4577-4585, 1987.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 130, 1997.

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Spherical Harmonic Differential Equation

Cite this as:

Weisstein, Eric W. "Spherical Harmonic Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalHarmonicDifferentialEquation.html

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