Formula:5800

Digital Library of Mathematical Functions ID 15.9.E13

${\displaystyle{\displaystyle f(t)=\frac{1}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\widetilde{f}(\mathrm{i}\lambda)\Phi^{(\alpha,\beta)}_{\mathrm{i}\lambda}(t)\frac{\Gamma\left(\tfrac{1}{2}(\alpha+\beta+1+\lambda)\right)\Gamma\left(\tfrac{1}{2}(\alpha-\beta+1+\lambda)\right)}{\Gamma\left(\alpha+1\right)\Gamma\left(\lambda\right)2^{\alpha+\beta+1-\lambda}}\mathrm{d}\lambda,}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}$:
• ${\displaystyle{\displaystyle f(t)}}$: function
• ${\displaystyle{\displaystyle\Phi^{(\alpha,\beta)}_{\lambda}(t)}}$: function (locally)
 

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