Can Special Functions Express This Complex Integral?

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SUMMARY

The integral \(\int_0^1 \exp[ax - b/(x)] x^{-3/2} dx\) does not have a closed form and diverges for certain values of \(a\) and \(b\). However, it may be expressible in terms of special functions, such as hypergeometric functions. The discussion highlights the use of the residue theorem as a potential method for evaluation, although the original poster is unfamiliar with this technique. Additionally, Monte Carlo simulations indicate a consistent curve shape that suggests a relationship with special functions.

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\int_0^1\exp[ax-b/(x)]x^{-3/2}dx, where 0<x<1, a>0, b>0.
I know there is no closed form, and it goes to infinity for some value of a and b, but is it able to expresse it by some special functions, like hypergeometric functions?
I checked the math handbooks, could not find any similar forms. Some people suggested using residual theorem which I am not familar with. I tried Monte Carlo simulation and it has nice curve shape that I feel can be expressed by some special functions.

Thanks very much if anyone can help!
 
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Sorry, when I say nice curve shaple, I mean as a function of b given certain a values.
spinblue said:
\int_0^1\exp[ax-b/(x)]x^{-3/2}dx, where 0<x<1, a>0, b>0.
I know there is no closed form, and it goes to infinity for some value of a and b, but is it able to expresse it by some special functions, like hypergeometric functions?
I checked the math handbooks, could not find any similar forms. Some people suggested using residual theorem which I am not familar with. I tried Monte Carlo simulation and it has nice curve shape that I feel can be expressed by some special functions.

Thanks very much if anyone can help!
 

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