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

From specialfunctionswiki
Jump to: navigation, search
 
(One intermediate revision by the same user not shown)
Line 10: Line 10:
 
==Proof==
 
==Proof==
  
==References=
+
==References==
 +
* {{BookReference|An Introduction to Orthogonal Polynomials|1978|T.S. Chihara|prev=Orthogonality relation for cosine on (0,pi)|next=Chebyshev T}} $(1.3)$ (<i>note: only mentions the $m \neq n$ case</i>)
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Latest revision as of 22:36, 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