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In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form with and . in the vicinity of the regular singular point . One can divide by to obtain a differential equation of the form which will not be solvable ...
The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre [3] as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′.
A Möbius transformation may be applied to move ∞ into the finite part of the complex plane if required, see example on Bessel differential equation below. Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are power series times complex powers (z − a) r near any given a in the complex ...
This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for complicated ordinary differential equations. The solution of the hypergeometric differential equation is very important. For instance, Legendre's differential equation can be ...
In mathematics, the Gaussian or ordinary hypergeometric function 2F1 (a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE).
The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. . The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...
Frobenius formula. In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group Sn. Among the other applications, the formula can be used to derive the hook length formula .
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 [−1, 1] only ...