Finding a substitution for an exponential integral

In summary, the conversation discusses the process of making a change of variable to express the Gamma function in a specific form. The functions f(s), A(y), and \zeta(s) are identified, but the speaker is still struggling to find an appropriate substitution. Suggestions for using a log substitution have been made, but the speaker is still unable to solve the problem.
  • #1
paco_uk
22
0

Homework Statement



Starting from the Gamma function:

[tex]

\Gamma (s) = \int^{\infty}_{0} dx \, x^{s-1} e^{-x}

[/tex]

Make a change of variable to express it in the form:

[tex]

\Gamma (s) = f(s) \int^{\infty}_{0} dy \, \exp{\frac{-A(y)}{\zeta(s)}}

[/tex]And identify the functions f(s), A(y), [tex]\zeta[/tex](s).

Homework Equations


The Attempt at a Solution



I've tried various solutions along the lines of [tex]x^{s}[/tex] and [tex] e^y [/tex] but I can't find anything that works and I don't know any general methods for finding an appropriate substitution. Can anyone suggest where to start?
 
Last edited:
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  • #2
Have you tried using

[tex] x^{s-1} = e^{(s-1) \ln x} ?[/tex]

I haven't quite solved this but a log substitution seems to almost work.
 
  • #3
Thanks for your suggestion. I think that let's me rewrite the integral as:

[tex]

\Gamma (s)&=&\int^{\infty}_{0} dx \, e^{(s-1) \ln{x}-x}

[/tex]

but I still can't find a way to get it in the form required. For example, the substitution [tex]y= \ln{x}[/tex] gives:

[tex]

\Gamma (s)&=&\int^{\infty}_{-\infty} dy \, e^{sy -e^{y}}

[/tex]

... still no use. Is there some clever trick I'm missing?
 

1. What is an exponential integral?

An exponential integral is a mathematical function that represents the integral of the exponential function. It is commonly denoted as Ei(x) and is defined as the integral of e^t/t from 0 to x.

2. Why is finding a substitution for an exponential integral important?

Finding a substitution for an exponential integral is important because it allows for the simplification of complex integrals involving the exponential function. This can make solving mathematical problems and equations more efficient and accurate.

3. What are some common substitutions for an exponential integral?

Some common substitutions for an exponential integral include u-substitution, integration by parts, and using trigonometric identities. The choice of substitution depends on the specific integral being solved.

4. Is it always possible to find a substitution for an exponential integral?

No, it is not always possible to find a substitution for an exponential integral. In some cases, the integral may be unsolvable or can only be solved using advanced mathematical techniques.

5. Are there any limitations to using a substitution for an exponential integral?

Yes, there can be limitations to using a substitution for an exponential integral. For example, the substitution may only work for a certain range of values or may not be applicable to all types of integrals. It is important to carefully consider the limitations when using a substitution for an exponential integral.

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