page_title: Spherical harmonics
section_title: Rotations
hierarchical_section_title: Spherical harmonics / Symmetry properties / Rotations
context_page_description: 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 complete set of orthogonal functions and thus an orthonormal basis, each function 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 via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by 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, the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO.
  Spherical harmonics originates from solving Laplace's equation in the spherical domains. Functions that solve Laplace's equation are called harmonics.
context_section_description: Consider a rotation  about the origin that sends the unit vector  to . Under this operation, a spherical harmonic of degree  and order  transforms into a linear combination of spherical harmonics of the same degree. That is,
  where  is a matrix of order  that depends on the
  rotation . However, this is not the standard way of expressing this property. In the standard way one writes,
  where  is the complex conjugate of an element of the Wigner D-matrix. In particular when  is a  rotation of the azimuth we get the identity,
  The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The 's of degree  provide a basis set of functions for the irreducible representation of the group SO(3) of dimension . Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.
caption: The rotation of a real spherical function with m = 0 and l = 3. The coefficients are not equal to the Wigner D-matrices, since real functions are shown, but can be obtained by re-decomposing the complex functions