Question

Expand the following functions in Legendre series.

$\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{arcsinx}$

Short answer

Step-by-step solution

Concept of Legendre polynomial

Simplify the expression

Substitute for the value of   

$\begin{array}{c}\mathbf{2}{\mathbf{c}}_{\mathbf{0}}\mathbf{=}{\mathbf{\int }}_{\mathbf{-}\mathbf{1}}^{\mathbf{1}}{\mathbf{sin}}^{\mathbf{-}\mathbf{1}}\mathbf{x}{\mathbf{P}}_{\mathbf{0}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{d}\mathbf{x}\\ \mathbf{2}{\mathbf{c}}_{\mathbf{0}}\mathbf{=}{\left\{x{\sin }^{-1}x-\int \frac{x}{\sqrt{1-x^2}}dx\right\}}_{\mathbf{-}\mathbf{1}}^{\mathbf{1}}\\ {\mathbf{c}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\end{array}$

Substitute for the value of m=1

Substitute for the value of m=2

Substitute for the value of m=3