# What is Fourier Legendre series?

## What is Fourier Legendre series?

Download Notebook. Because the Legendre polynomials form a complete orthogonal system over the interval with respect to the weighting function , any function may be expanded in terms of them as. (1) To obtain the coefficients in the expansion, multiply both sides by and integrate.

### How do you find the Fourier Legendre series?

1: Fourier-Legendre series. f(x)={0,−1. We find A0=12∫10P0(x)dx=12,A1=32∫10P1(x)dx=14,A2=52∫10P2(x)dx=0,A3=72∫10P3(x)dx=−716.

#### What is Rodrigues formula for Legendre polynomial?

In mathematics, Rodrigues’ formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827).

**What is the Legendre formula?**

In mathematics, Legendre’s formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac’s formula, after Alphonse de Polignac.

**Where does Rodrigues formula come from?**

## What is the solution of Legendre differential equation?

Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind.

### What is generating function of Legendre polynomial?

The Legendre polynomials can be alternatively given by the generating function ( 1 − 2 x z + z 2 ) − 1 / 2 = ∑ n = 0 ∞ P n ( x ) z n , but there are other generating functions.

#### What is Fourier series formula?

The Fourier series formula gives an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. It is used to decompose any periodic function or periodic signal into the sum of a set of simple oscillating functions, namely sines and cosines.

**What is the difference between Fourier series and Fourier transform?**

The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials.

**What is meant by Legendre polynomial?**

In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications.

## What is the orthogonal property of Legendre’s polynomial?

Abstract We give a remarkable additional othogonality property of the classical Legendre polynomials on the real interval [−1, 1]: polynomials up to degree n from this family are mutually orthogonal under the arcsine measure weighted by the nor- malized degree-n Christoffel function.

### What is Fourier series example?

Note: this example was used on the page introducing the Fourier Series. Note also, that in this case an (except for n=0) is zero for even n, and decreases as 1/n as n increases….Example 1: Special case, Duty Cycle = 50%

n | an |
---|---|

0 | 0.5 |

1 | 0.6366 |

2 | 0 |

3 | -0.2122 |

#### What is Rodrigues formula used for?

In the theory of three-dimensional rotation, Rodrigues’ rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation.

**What is application of Legendre polynomial?**

For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom [3], [4] and in the determination of potential functions in the spherically symmetric geometry [5], etc.

**How to calculate the Fourier–Legendre series with Legendre polynomials?**

So we can form a generalized Fourier series (known as a Fourier–Legendre series) involving the Legendre polynomials, and As an example, let us calculate the Fourier–Legendre series for ƒ ( x ) = cos x over [−1, 1]. Now, which differs from cos x by approximately 0.003, about 0.

## How do Legendre polynomials relate to multipole expansion?

Legendre polynomials in multipole expansions. Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged. This expansion is the basis of interior multipole expansion .

### What is a generalized Fourier series expansion?

A generalized Fourier series is a series expansion of a function based on a system of orthogonal polynomials. By using this orthogonality, a piecewise continuous function can be expressed in the form of generalized Fourier series expansion:

#### What is the difference between a generalized Fourier series and polynomials?

A polynomial sequence where is the degree of is said to be a sequence of orthogonal polynomials if where are given constants and is the Kronecker delta. A generalized Fourier series is a series expansion of a function based on a system of orthogonal polynomials.