Integral representation of incomplete gamma function

Bessel function and has tried using substitutions but was unsuccessful. He is looking for help and resources to solve this problem. In summary, Jason is seeking assistance with verifying an integral representation of incomplete gamma function using Bessel functions and has tried using substitutions but was unsuccessful.
  • #1
patric44
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TL;DR Summary
verifying the integral representation of incomplete gamma function
hi guys
I was trying to verify the integral representation of incomplete gamma function in terms of Bessel function, which is represented by
$$\gamma(a,x) = x^{\frac{a}{2}}\;\int_{0}^{∞}e^{-t}t^{\frac{a}{2}-1}J_{a}(2\sqrt{xt})dt\;\;a>0$$
i was thinking about taking substitutions in order to reach the generating function of Bessel, but it took me nowhere.
i will appreciate any help
 
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  • #3
the formula is correct but I guess it is not that famous
 
  • #4
I would be tempted to start with an integral representation of the Bessel function then switch the order of integration. You might need to try several different ones. It might not lead anywhere, but might be worth a try. My go-to online resource for formulas
https://dlmf.nist.gov

your formula is Equation 8.6.2
https://dlmf.nist.gov/8.6

jason
 

1. What is the integral representation of the incomplete gamma function?

The integral representation of the incomplete gamma function is a special function used in mathematics and physics to represent the incomplete gamma function as an integral. It is defined as Γ(s,x) = ∫x∞ ts-1e-tdt, where s is the shape parameter and x is the lower limit of the integration.

2. How is the integral representation of the incomplete gamma function used in real-world applications?

The integral representation of the incomplete gamma function is used in various fields such as statistics, physics, and engineering to solve problems related to probability, heat transfer, and signal processing. It is also used in the calculation of various mathematical functions, such as the beta function and the hypergeometric function.

3. Can the integral representation of the incomplete gamma function be simplified?

Yes, the integral representation of the incomplete gamma function can be simplified using various techniques such as integration by parts, substitution, and series expansion. However, the simplified form may not always be applicable or convenient for certain calculations.

4. What is the relationship between the integral representation and other representations of the incomplete gamma function?

The integral representation of the incomplete gamma function is equivalent to other representations such as the series representation, continued fraction representation, and recurrence relation representation. This means that any of these representations can be used to calculate the incomplete gamma function and will yield the same result.

5. Are there any special properties of the integral representation of the incomplete gamma function?

Yes, the integral representation of the incomplete gamma function has several special properties, including symmetry, monotonicity, and convexity. These properties make it useful for solving various mathematical problems and for proving theorems in analysis and number theory.

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