# I need integrals Int[0,infty]t^(a-1) e^(-t) F(z, e^(-t))dt

• I
• benorin
This is a good observation, thank you, but I don't understand how the compound function is accounted for in this? I mean the ##e^{-t}## inside ##F( z, e^{-t})## inside the integral transform in the first post? I am uncertain how to work with this feature, I suppose I could simply define another function ##H( z,t) := F( z, e^{-t} )## and use that somehow. How would I use a table of transforms for this kind of function?You might be able to simply define another function, like H(z, -log(e^-t)), and use that somehow. Alternatively, you could use a tablef

#### benorin

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TL;DR Summary
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

• Delta2
It seems Mellin transform of ##e^{-t} F##.
$$F=e^t M^{-1}\{G\}$$
where ##M^{-1}## is inverse Mellin transform.

Last edited:
• benorin and Delta2
• WWGD, dextercioby, benorin and 1 other person
It seems Mellin transform of ##e^{-t} F##.
$$F=e^t M^{-1}\{G\}$$
where ##M^{-1}## is inverse Mellin transform.
This is a good observation, thank you, but I don't understand how the compound function is accounted for in this? I mean the ##e^{-t}## inside ##F( z, e^{-t})## inside the integral transform in the first post? I am uncertain how to work with this feature, I suppose I could simply define another function ##H( z,t) := F( z, e^{-t} )## and use that somehow. How would I use a table of transforms for this kind of function?

I am not sure whether I got you but
$$e^tM^{-1}\{G{(z,\alpha} )\}:= H(z,t)$$
$$H(z,t)=H(z,-\log(e^{-t}))=F(z,e^{-t})$$.
by substituting t with -log(e^-t) in H.

• benorin