Legendre Functions and Legendre Polynomials.. Legendre functions

Legendre Functions
and Legendre
Polynomials.

Legendre functions
• Mathematically, Legendre functions are the solutions to the
Legendre’s differential equations.
First introduced in 1785, they were first used as the coefficients while
expanding Newtonian potential (Belinsky, 2013).
These type of equations are commonly found in the boundary value
problems in spheres(Llc, 2010).

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• Consider an equation shown below
+1)y=0………………………………1
The above equation can be rewritten in the form of
Where
And
From p(x) and q(x) it is analytically evident that the two functions of x have a
convergence radius R=1

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Assuming that
For n=0,1, 2,……..
Using the values of n=0, 1, 2 and 3, we obtain

Continuation
From induction formulae, it can be proved that form m=1,2,3,…
From the two equations above, the polynomial y(x) can be rewritten in
the form of

Continuation
Where
In the above equations, when c0=1 and c0=0 , c1=1 and c0=0 and c1=0. In this case y1
and y2 the solution of Legendre equation
• It is noted that whenever alpha in the Legendre equation is non-negative, then either y1
or y2 terminates hence whenever =2m (m=0,1,2,….) is non-negative odd integer then
• Y2(x)=x (=1)
• Y2(x)=x-5/3x^3 (=5)
• Y2(x)=x-14/3x^3+21/5x^5 (=5)
• This forms the basis of equation 1

Legendre Polynomial
• Polynomial solutions represented by of degree n of (4) that satisfies
are referred to Legendre polynomials of degree n
• Assuming that ψ(x)=

Properties of Legendre polynomials
Legendre functions are generating. Given a function F(t,x) defined by
This is an example of a generating function and it can be shown that

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Orthogonality of the Legendre polynomials
For a Legendre polynomial, the following property must hold

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• The Fourier-Legendre series. From the orthogonality property of
Legendre polynomial, any form of pricewise continuous function in
the -1 1 can be expressed by Legendre polynomial terms:
Where
From which

Reference
• Belinsky, R. (2013). Integrals of Legendre Polynomials and Solution of
Some Partial Differential Equations.
Journal of Applied Analysis,
6(2).
• Llc, B. (2010).
Polynomials: Polynomial, Quadratic Equation, Linear
Equation, Legendre Polynomials, Coefficient, Rational Root Theorem,
Discriminant, Hurwitz Polynomial, Abel-Ruffini Theorem, Linear
Function, Root of Unity, Characteristic Polynomial, Sheffer Sequence.
Books LLC, Wiki Series.