Can Special Functions Express This Complex Integral?

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In summary, the conversation discusses the integral \int_0^1\exp[ax-b/(x)]x^{-3/2}dx, where 0<x<1, a>0, and b>0. The speaker mentions that there is no closed form for this integral and it goes to infinity for certain values of a and b. They then ask if it can be expressed using special functions, such as hypergeometric functions, and mention that they were unable to find any similar forms in math handbooks. Some people suggested using the residual theorem, which the speaker is not familiar with. They also mention trying a Monte Carlo simulation, which resulted in a nice curve shape that they believe can be expressed using special functions.
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spinblue
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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!
 

FAQ: Can Special Functions Express This Complex Integral?

What is an integral problem?

An integral problem involves finding the area under a curve or the accumulation of a quantity over a given interval. It is a fundamental concept in calculus and is used to solve a variety of real-world problems.

How do I solve an integral problem?

To solve an integral problem, you need to use integration techniques such as substitution, integration by parts, or partial fractions. You also need to have a good understanding of the fundamental properties of integrals, such as the power rule, constant multiple rule, and sum/difference rule.

Can anyone solve an integral problem?

Yes, anyone with a good understanding of calculus and integration techniques can solve an integral problem. However, it may require practice and patience to master the skills needed to solve complex integral problems.

Why is it important to learn how to solve integral problems?

Integrals are used in many fields of science and engineering, including physics, chemistry, and economics. Being able to solve integral problems allows us to analyze and solve real-world problems, make predictions, and understand the behavior of systems in various disciplines.

What are some common mistakes when solving integral problems?

Some common mistakes when solving integral problems include forgetting to add the constant of integration, making algebraic errors, and using incorrect integration techniques. It is important to double-check your work and practice regularly to avoid these mistakes.

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