Formula:KLS:09.08:47

<< Formula:KLS:09.08:46

formula in Jacobi: Special cases

Formula:KLS:09.08:48 >>

${\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x^{2})^{-\frac{1}{2}}T_{m}\left(x\right)T_{n}\left(x\right)\,dx=\left\{\begin{array}[]{ll}\displaystyle\frac{\pi}{2}\,\delta_{m,n},&n\neq 0\\ \pi\,\delta_{m,n},&n=0\end{array}\right.}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle T_{n}}}}$ : Chebyshev polynomial of the first kind : http://dlmf.nist.gov/18.3#T1.t1.r8
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.8 of KLS.

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<< Formula:KLS:09.08:46

formula in Jacobi: Special cases

Formula:KLS:09.08:48 >>