Integration with hypergeometric function

In summary, the conversation discusses how to integrate the hypergeometric function ##{}_2F_1## with constants A, B, C, D, and E, using x as the independent variable. The main question is whether it can be written as a power series, and if there is a simpler and more compact method to integrate it. It is noted that the hypergeometric function is defined as a power series, but it is limited to |x|<1. The conversation also mentions a possible solution involving a change of variable from x to X=x².
  • #1
JulieK
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How to integrate:

[itex]_{2}F_{1}(B;C;D;Ex^{2})\,Ax[/itex]

where [itex]_{2}F_{1}(...)[/itex] is the hypergeometric function, x is the independent variable and A, B, C, D, and E are constants.
 
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  • #2
Can you write [tex]{}_2F_1(B,C;D;Ex^2)Ax[/tex] as a (power) series? Do you know how to integrate a power series?
 
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  • #3
I am familiar with the power series method but I am trying first to find a simpler and more compact method.
 
  • #4
JulieK said:
I am familiar with the power series method but I am trying first to find a simpler and more compact method.

Since the hypergeometric function ##{}_2F_1## is defined as a power series, it is hard to imagine a solution that avoids it.
 
  • #5
One problem with the power series is that it is defined only for |x|<1.
 

FAQ: Integration with hypergeometric function

1. What is a hypergeometric function?

A hypergeometric function is a mathematical function that is defined by a series of terms, each of which contains a factorial. It is a special type of confluent hypergeometric function that satisfies a specific differential equation.

2. How is hypergeometric integration different from regular integration?

Hypergeometric integration involves integrating a hypergeometric function, which contains a factorial, while regular integration involves integrating a regular function. In hypergeometric integration, special techniques such as substitution and integration by parts may be necessary to solve the integral.

3. What are the applications of integration with hypergeometric functions?

Integration with hypergeometric functions is used in various fields such as physics, engineering, statistics, and mathematical finance. It is particularly useful in solving problems involving series expansions, probability distributions, and differential equations.

4. How do you solve an integral involving a hypergeometric function?

To solve an integral involving a hypergeometric function, you can use techniques such as substitution, integration by parts, or series expansion. It is also helpful to have a good understanding of the properties of hypergeometric functions, such as their identities and special cases.

5. Can integration with hypergeometric functions be used to solve real-world problems?

Yes, integration with hypergeometric functions can be used to solve real-world problems in various fields. For example, it can be used to calculate probabilities in statistics, model physical systems in physics, and analyze financial data in mathematical finance. It is a powerful tool for solving complex problems that cannot be solved using regular integration methods.

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