Legendre polynomials
The Legendre polynomials (after Adrien -Marie Legendre ), also called zonal spherical functions, are special polynomials, which form on the interval, an orthogonal function system. They are the particular solutions of the set thresh differential equation. Play an important role in the Legendre polynomials in theoretical physics, particularly in electrodynamics and in quantum mechanics and in the field of filtration technology in the Legendre filters.
- 2.1 Monomdarstellung
- 2.2 Rodrigues formula
- 2.3 Integral representation
- 2.4 recursion Formula
- 3.1 Full orthogonal
- 3.2 Zeroing
- 3.3 Main Features
- 3.4 Generating function
Origin
Construction of orthogonal polynomials
A sequence of real polynomials orthogonal for a given interval and a subsequent weight function, when the orthogonality
Satisfied for all with.
For the interval together with the simplest of all weight functions can be generated iteratively by the monomials such orthogonal polynomials using the Gram-Schmidt orthogonalization 's starting. The Legendre polynomials arise when is additionally required.
Legendre differential equation
The Legendre polynomials are solutions of the set thresh differential equation
Which in the form of
Can be written. The general solution of this differential equation is
With the two linearly independent functions and. It denotes the Legendre polynomials, therefore, as a first kind Legendre functions and Legendre functions as a second nature, because they are no longer polynomials.
In addition, there is still a generalized Legendre differential equation whose solutions are called associated Legendre polynomials.
Legendre polynomials
The -th Legendre polynomial has degree and is formed, that is, it is rational coefficients. For the Legendre polynomials, there are several forms of representation.
Monomdarstellung
The first Legendre polynomials are:
The Legendre polynomial is th
With the Gaussian curve
Rodrigues formula
The Rodrigues formula can be evaluated using the formula of Faa di Bruno and obtains the explicit form of the -th Legendre polynomial again.
Integral representation
For all
Recursion formulas
For the Legendre polynomials the following recursion formulas apply:
The first recursive formula can be represented by means of the substitution in the following, often -to-find manner:
By applying the derivation rule for expressions of the type with, or results in the following recursive representation of the Legendre polynomials, which also takes into account the derivatives of these polynomials:
Are the initial conditions and.
In turn, the above -mentioned formula gives with their initial conditions.
Properties
Complete orthogonal
Consider the Hilbert space of square integrable equipped with the scalar product defined on real-valued functions
The Legendre polynomials form a complete orthogonal family, so they are a special case of orthogonal polynomials. Normalizing this, they form a complete orthonormal system.
It is
Wherein the Kronecker delta designated. This means the completeness that each function (of course in terms of the generated topology) " develop " according to Legendre polynomials can:
With the expansion coefficients
In the physical or technical literature, the completeness is often written as follows as a distribution equation:
Where the Dirac delta distribution is. Such distribution equation should always be read so that both sides of this equation are applied to test functions. Turning to the right side to such a test function on, you get. To use the left hand side you have to multiply by definition, and then integrate over. But then you get exactly above expansion formula ( with instead of ). Orthogonality and completeness can therefore write short and concise as follows:
- Orthogonality: for.
- Completeness: for all (in the sense of convergence ).
Zeros
Has on the interval exactly simple zeros. They are symmetrical to the zero point of the abscissa, as Legendre polynomials are either even or odd. Between two adjacent zeros of exactly one zero of. What is the relationship a zero shares of the interval between two zeros of, or vice versa, except for the outer of, it is quite variable.
The determination of the zeros of the Legendre polynomials is a common task in numerical mathematics, as they play a central role in the Gauss - Legendre quadrature, or referred to in "Complete orthogonal " development " arbitrary " functions by polynomials. While there are numerous table works for this, but often their use is associated with discomfort, because you would have to hold a large number of tables in suitable accuracies for a flexible response. At the root-finding is the knowledge of the interval only of limited value when choosing a Iterationsanfangs, especially as is required or the knowledge of the zeros of another polynomial. A more closely with increasing approximation of the nascent zero of is given by:
For example, as all zeros to at least two decimal places be estimated accurately, with errors between and, while the smallest interval of only zeros is. Three decimal places are already safe, with errors between and, while the best nesting is only by. The maximum estimation error is only for the two fifth zeros from the outside, their exact amount starts.
With such a start value and the first two " recursion " can be personalized with a calculation procedure, both the function value and its derivative determine. Using the Newton method can be all zeros except for the two outer with more than find square convergence, since the zeros are close to the turning points. The two outer converge to zero " only " square, ie an initial distance to zero of decreases after an iteration initially on about, then and.
The specified assessment is part of a very short procedure that calculates both all zeros of a Legendre polynomial and the appropriate weights for the "Gauss - Legendre quadrature ."
Main Features
For each and every apply:
Generating function
For everyone, applies
The power series has 1 on the right side for the radius of convergence
The function is therefore called the generating function of the Legendre polynomials.
The commonly encountered in physics term ( eg in the potentials of the Newtonian gravitation or electrostatics ) can thus be expanded in a power series development for:
Legendre functions second kind
The recursion formulas of Legendre polynomials also apply to the second kind Legendre functions, so this can be determined by specifying the first iteration:
This is to be used which results in singularities in the complex plane in and along branch cuts and the logarithm of the main branch.