Formula:KLS:09.08:40

<< Formula:KLS:09.08:39

formula in Jacobi: Special cases

Formula:KLS:09.08:41 >>

${\displaystyle{\displaystyle{\displaystyle\frac{\Gamma\left(\lambda+1\right)}{\Gamma\left(\lambda+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}\int_{-1}^{1}\frac{C^{\lambda}_{n}\left(y\right)}{C^{\lambda}_{n}\left(1\right)}(1-y^{2})^{\lambda-\frac{1}{2}}{\mathrm{e}^{\mathrm{i}xy}}dy={\mathrm{i}^{n}}2^{\lambda}\Gamma\left(\lambda+1\right)x^{-\lambda}J_{\lambda+n}\left(x\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}$ : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle J_{\nu}}}}$ : Bessel function of the first kind : http://dlmf.nist.gov/10.2#E2

Bibliography

Equation in Section 9.8 of KLS.

URL links

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

<< Formula:KLS:09.08:39

formula in Jacobi: Special cases

Formula:KLS:09.08:41 >>