Formula:KLS:09.08:47

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∫ - 1 1 ( 1 - x 2 ) - 1 2 ⁢ T m ⁡ ( x ) ⁢ T n ⁡ ( x ) ⁢ 𝑑 x = { π 2 ⁢ δ m , n , n ≠ 0 π ⁢ δ m , n , n = 0 superscript subscript 1 1 superscript 1 superscript 𝑥 2 1 2 Chebyshev-polynomial-first-kind-T 𝑚 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 differential-d 𝑥 cases 2 Kronecker-delta 𝑚 𝑛 𝑛 0 Kronecker-delta 𝑚 𝑛 𝑛 0 {\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
T n subscript 𝑇 𝑛 {\displaystyle{\displaystyle{\displaystyle T_{n}}}}  : Chebyshev polynomial of the first kind : http://dlmf.nist.gov/18.3#T1.t1.r8
δ m , n subscript 𝛿 𝑚 𝑛 {\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.

URL links

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