Formula:KLS:09.09:03

From DRMF
Jump to navigation Jump to search


1 2 ⁢ π ⁢ ∫ - ∞ ∞ ( 1 + x 2 ) - N - 1 ⁢ e 2 ⁢ ν ⁢ arctan ⁡ x ⁢ P m ⁡ ( x ; ν , N ) ⁢ P n ⁡ ( x ; ν , N ) ⁢ 𝑑 x = Γ ⁡ ( 2 ⁢ N + 1 - 2 ⁢ n ) ⁢ Γ ⁡ ( 2 ⁢ N + 2 - 2 ⁢ n ) ⁢ 2 2 ⁢ n - 2 ⁢ N - 1 ⁢ n ! Γ ⁡ ( 2 ⁢ N + 2 - n ) ⁢ | Γ ⁡ ( N + 1 - n + i ⁢ ν ) | 2 ⁢ δ m , n 1 2 superscript subscript superscript 1 superscript 𝑥 2 𝑁 1 2 𝜈 𝑥 pseudo-Jacobi-polynomial 𝑚 𝑥 𝜈 𝑁 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 differential-d 𝑥 Euler-Gamma 2 𝑁 1 2 𝑛 Euler-Gamma 2 𝑁 2 2 𝑛 superscript 2 2 𝑛 2 𝑁 1 𝑛 Euler-Gamma 2 𝑁 2 𝑛 superscript Euler-Gamma 𝑁 1 𝑛 imaginary-unit 𝜈 2 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty% }(1+x^{2})^{-N-1}{\mathrm{e}^{2\nu\operatorname{arctan}x}}P_{m}\!\left(x;\nu,N% \right)P_{n}\!\left(x;\nu,N\right)\,dx{}=\frac{\Gamma\left(2N+1-2n\right)% \Gamma\left(2N+2-2n\right)2^{2n-2N-1}n!}{\Gamma\left(2N+2-n\right)\left|\Gamma% \left(N+1-n+\mathrm{i}\nu\right)\right|^{2}}\,\delta_{m,n}}}}

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
P n subscript 𝑃 𝑛 {\displaystyle{\displaystyle{\displaystyle P_{n}}}}  : pseudo Jacobi polynomomal : http://drmf.wmflabs.org/wiki/Definition:pseudoJacobi
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
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 9.9 of KLS.

URL links

We ask users to provide relevant URL links in this space.


Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:09.09:03&oldid=3829"