Cannot do the integral of the Hyper-geometric function?

In summary, the conversation discusses the difference between two types of integrals involving the function ##_2F_1(a,b,c,x)## and the possibility of solving the second type of integral. A reference for further information is also mentioned.
  • #1
Chenkb
41
1
Dear friends:
It's strange that Mathematica can do the integral of ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x^2)##, however, fails when it's changed to ##\int_0^\infty dx~x~_2F_1(a,b,c,1-x-x^2)##.
Are there any major differences between this two types? Is it possible to do the second kind of integral?
Best regards!

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  • #2
I don't believe that that particular integral can be solved analytically. Best place to look is in Rydzik and Gradstien (sp)...
 
  • #3
Dr Transport said:
I don't believe that that particular integral can be solved analytically. Best place to look is in Rydzik and Gradstien (sp)...
Many thanks! That reference is excellent!
 

Related to Cannot do the integral of the Hyper-geometric function?

What is the Hyper-geometric function?

The Hyper-geometric function is a special mathematical function that is used to describe complex relationships between variables in a variety of fields such as physics, engineering, and statistics. It is defined as a series of terms that involves the gamma function and the binomial coefficient.

Why is it difficult to integrate the Hyper-geometric function?

The Hyper-geometric function is difficult to integrate because it does not have a closed-form solution. This means that it cannot be expressed as a simple formula using common mathematical operations. Instead, it requires advanced techniques such as contour integration and series expansion to solve.

Can the Hyper-geometric function be approximated?

Yes, the Hyper-geometric function can be approximated using numerical methods such as the Euler-Maclaurin formula or the Gauss-Laguerre quadrature. These methods use a series of numerical calculations to estimate the value of the function, which can be useful in practical applications.

Are there any real-life applications of the Hyper-geometric function?

Yes, the Hyper-geometric function has many real-life applications. For example, it is used in statistics to model the probability distribution of a sample from a population, in physics to describe the behavior of particles in a magnetic field, and in engineering to analyze the stability of a system.

Is there a way to simplify the Hyper-geometric function for easier integration?

Yes, in some cases, the Hyper-geometric function can be simplified using special cases and identities. For example, the confluent Hyper-geometric function can be simplified to a simpler form called the Kummer function. Additionally, some specific values of the parameters in the Hyper-geometric function can lead to simpler solutions.

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