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- I need your integrals which are interesting, of the form Int[0,infty]t^(a-1) e^(-t) F(z, e^(-t))dt and that possess known analytical solutions please?

I’m doing some brainstorming for a note I’m writing, I would appreciate it if anybody knows interesting integrals of the form

$$\int_{t=0}^\infty t^{\alpha - 1} e^{-t} F(z, e^{-t})\, dt=G( z, \alpha )$$

where ##z## and ##\alpha## are complex parameters and the solution ##G(z, \alpha )## is known? Even if they seem kinda simple ones, example:

$$\int_{t=0}^\infty t^{\alpha - 1} e^{-t} \cos (t)\, dt= 2^{\tfrac{-\alpha}{2}}\cos \left(\tfrac{\pi}{4}\alpha\right) \Gamma (\alpha )$$

@fresh_42 you always come up with those juicy integrals - if you’re able maybe come post one? Thanks.

If I end up using any of the integrals you guys come up with I will cite this thread and your handle or real name if you prefer?

-Ben

$$\int_{t=0}^\infty t^{\alpha - 1} e^{-t} F(z, e^{-t})\, dt=G( z, \alpha )$$

where ##z## and ##\alpha## are complex parameters and the solution ##G(z, \alpha )## is known? Even if they seem kinda simple ones, example:

$$\int_{t=0}^\infty t^{\alpha - 1} e^{-t} \cos (t)\, dt= 2^{\tfrac{-\alpha}{2}}\cos \left(\tfrac{\pi}{4}\alpha\right) \Gamma (\alpha )$$

@fresh_42 you always come up with those juicy integrals - if you’re able maybe come post one? Thanks.

If I end up using any of the integrals you guys come up with I will cite this thread and your handle or real name if you prefer?

-Ben