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Spherical harmonics - Wikipedia

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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. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, certain functions defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role

Spherical harmonics - Wikipedia Jump to content From Wikipedia, the free encyclopedia Special mathematical functions defined on the surface of a sphere "Ylm" redirects here. For other uses, see YLM (disambiguation) . Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of Y ℓ m ( θ , φ ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} in angular direction ( θ , &#x03C

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