Concept

Associated Legendre polynomials

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or equivalently where the indices l and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if l and m are integers with 0 ≤ m ≤ l, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and l integer, these functions are identical to the Legendre polynomials. In general, when l and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of l and m are Legendre functions. In that case the parameters are usually labelled with Greek letters. The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics. These functions are denoted , where the superscript indicates the order and not a power of P. Their most straightforward definition is in terms of derivatives of ordinary Legendre polynomials (m ≥ 0) The (−1)m factor in this formula is known as the Condon–Shortley phase. Some authors omit it. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters l and m follows by differentiating m times the Legendre equation for Pl: Moreover, since by Rodrigues' formula, the P can be expressed in the form This equation allows extension of the range of m to: −l ≤ m ≤ l. The definitions of Pl±m, resulting from this expression by substitution of ±m, are proportional.

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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 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 (sines and cosines) via Fourier series.
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.

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