Legendre Polynomials Calculator

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Example

Created on 2024-06-20Asked by Samuel Perez (Solvelet student)

Find the Legendre polynomial

P_2(x)

Solution

The Legendre polynomials

P_n(x)

are given by Rodrigues' formula:

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

For

n = 2

P_2(x) = \frac{1}{2^2 2!} \frac{d^2}{dx^2} \left( (x^2 - 1)^2 \right)

First, calculate

(x^2 - 1)^2

(x^2 - 1)^2 = x^4 - 2x^2 + 1

Next, take the second derivative:

\frac{d}{dx}(x^4 - 2x^2 + 1) = 4x^3 - 4x

\frac{d^2}{dx^2}(x^4 - 2x^2 + 1) = 12x^2 - 4

Therefore:

P_2(x) = \frac{1}{4 \cdot 2} (12x^2 - 4) = \frac{1}{8} (12x^2 - 4) = \frac{3}{2} x^2 - \frac{1}{2}

Thus, the Legendre polynomial

P_2(x)

is:

P_2(x) = \frac{3}{2} x^2 - \frac{1}{2}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Penelope Nelson on Solvelet

1. Compute the Legendre polynomial of degree

3

P_3(x)

,2. Calculate the length of the curve

y = \sqrt{1 + x^2}

from

x = 0

x = 2

DefinitionIn mathematics, Legendre polynomials are a sequence of orthogonal polynomials that are useful for expanding functions defined on the interval [-1,1] with respect to the orthogonal basis of Legendre polynomials. Pn(x) where n is degree of the polynomial. For instance, the initial Legendre polynomials are P0(x)=1, P1(x)=x, and P2(x)=21(3x2−1), etc. They obey the orthogonality relation ∫−11Pm(x)Pn(x)dx=2n+12δmn.

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