Difference between revisions of "Orthogonality of Chebyshev T on (-1,1)"

From specialfunctionswiki
Jump to: navigation, search
(Created page with "==Theorem== The following formula holds for $m,n \in \{0,1,2,\ldots\}$: $$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac...")
 
Line 1: Line 1:
 
==Theorem==
 
==Theorem==
 
The following formula holds for $m,n \in \{0,1,2,\ldots\}$:
 
The following formula holds for $m,n \in \{0,1,2,\ldots\}$:
$$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} dx = \left\{ \begin{array}{ll}
+
$$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} \mathrm{d}x = \left\{ \begin{array}{ll}
 
0 &; m \neq n \\
 
0 &; m \neq n \\
 
\dfrac{\pi}{2} &; m=n\neq 0 \\
 
\dfrac{\pi}{2} &; m=n\neq 0 \\

Revision as of 22:30, 19 December 2017

Theorem

The following formula holds for $m,n \in \{0,1,2,\ldots\}$: $$\int_{-1}^1 \dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}} \mathrm{d}x = \left\{ \begin{array}{ll} 0 &; m \neq n \\ \dfrac{\pi}{2} &; m=n\neq 0 \\ \pi &; m=n=0, \end{array} \right.$$ where $T_m$ denotes Chebyshev polynomials of the first kind.

Proof

=References