# Integral involving product of derivatives of Legendre polynomials

• hanson
In summary, the purpose of an integral involving product of derivatives of Legendre polynomials is to evaluate the product of derivatives of these orthogonal polynomials, which have many important properties and are commonly used in mathematical and scientific applications. The integral can be calculated using various integration techniques and may arise in situations such as solving differential equations or studying physical systems. There are also special techniques that can be used to simplify or evaluate these integrals, such as using recurrence relations or exploiting the orthogonality properties of Legendre polynomials. It is important to have a good understanding of the properties and special cases of Legendre polynomials in order to effectively solve these integrals.
hanson
Anyone how to evaluate this integral?

$\int_{-1}^{1} (1-x^2) P_{n}^{'} P_m^{'} dx$, where the primes represent derivative with respect to $x$?

I tried using different recurrence relations for derivatives of the Legendre polynomial, but it didn't get me anywhere...

Use the facts

$$\left((1-x^2)P_n^\prime \right)^\prime=-n(n+1)P_n$$

and

$$\int_{-1}^1 P_m P_n \text{ dx}=\dfrac{2}{2n+1} \delta_{mn}$$

to integrate by parts

or just use

$$P_n=\frac{1}{(2n)!} \dfrac{d^n}{dx^n} (x^2-1)^n$$

Last edited:
Thank you very much!

## 1. What is the purpose of an integral involving product of derivatives of Legendre polynomials?

The purpose of this integral is to evaluate the product of derivatives of Legendre polynomials, which can be used in various mathematical and scientific applications, such as solving differential equations or studying physical systems.

## 2. What are Legendre polynomials and why are they important?

Legendre polynomials are a family of orthogonal polynomials that are commonly used in mathematical physics and engineering. They have many important properties, including being solutions to certain types of differential equations, and are useful in solving problems in many fields of science and engineering.

## 3. How do you calculate the integral involving product of derivatives of Legendre polynomials?

The integral can be calculated using various integration techniques, such as integration by parts or substitution. It is important to have a good understanding of the properties of Legendre polynomials and their derivatives in order to successfully solve the integral.

## 4. In what situations would you encounter an integral involving product of derivatives of Legendre polynomials?

This type of integral can arise in many different scenarios, such as when solving differential equations involving Legendre polynomials or when studying physical systems with spherical or cylindrical symmetry. It may also appear in the context of Fourier analysis or in problems related to mathematical physics or engineering.

## 5. Are there any special techniques for solving integrals involving product of derivatives of Legendre polynomials?

Yes, there are some special techniques that can be used to simplify or evaluate these types of integrals, such as using recurrence relations or exploiting the orthogonality properties of Legendre polynomials. It is also helpful to have a good understanding of the properties and special cases of Legendre polynomials in order to tackle these integrals effectively.

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