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

In summary: 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 table
  • #1
benorin
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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
 
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  • #2
It seems Mellin transform of ##e^{-t} F##.  
[tex]F=e^t M^{-1}\{G\}[/tex]
where ##M^{-1}## is inverse Mellin transform.
 
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  • #4
anuttarasammyak said:
It seems Mellin transform of ##e^{-t} F##.  
[tex]F=e^t M^{-1}\{G\}[/tex]
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?
 
  • #5
I am not sure whether I got you but
[tex]e^tM^{-1}\{G{(z,\alpha} )\}:= H(z,t) [/tex]
[tex]H(z,t)=H(z,-\log(e^{-t}))=F(z,e^{-t})[/tex].
by substituting t with -log(e^-t) in H.
 
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1. What is the purpose of the integral in the equation?

The integral is used to find the area under the curve of the function t^(a-1) e^(-t) F(z, e^(-t)) from 0 to infinity. This can be useful in solving various mathematical problems and in applications such as physics, engineering, and economics.

2. How do I solve this integral?

This integral can be solved using various methods, such as integration by parts, substitution, or using special functions such as the Gamma function. The specific method used will depend on the values of a and z in the function F(z, e^(-t)).

3. What are the possible values of a and z in the function F(z, e^(-t))?

The values of a and z can vary depending on the specific problem or application. In general, a can be any real number and z can be a complex number. However, the specific values needed for a particular problem will depend on the context in which the integral is being used.

4. Can this integral be solved analytically?

In most cases, this integral cannot be solved analytically, meaning that it cannot be expressed in terms of elementary functions. However, it can be evaluated numerically using various numerical integration techniques.

5. What are some common applications of this integral?

This integral is commonly used in probability and statistics, particularly in the calculation of moments and generating functions. It also has applications in physics, such as in the calculation of partition functions in thermodynamics, and in economics, such as in the calculation of expected utility in decision making.

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