25 Chebyshev
Chebyshev DE¶
\[ (1-x^2)y'' - xy' + n^2 y =0 \\ n>0 \]
\(x = \pm 1\) are the regular singular points of equation
Chebyshev Polynomials¶
are solutions of Chebyshev DE
Using Frobenius method near \(x=1\), we get the solution
\[ T_n = F \left( n, -n, \frac{1}{2}, \frac{1-x}{2} \right) \]
equation is a finite polynomial of degree \(n\), as \(b=-n\) (-ve integer)
Using transformation \(x=\cos \theta\) in equation, we get another solution
\[ T_n = \cos(n \theta), \quad \theta = \cos^{-1}x \]
Euler’s Theorem¶
\[ \begin{aligned} e^{i\theta} &= \cos \theta + i \sin \theta \\ e^{ni\theta} &= \cos n\theta + i \sin n\theta \\ &= (\cos \theta + i \sin \theta)^n \\ \end{aligned} \]