Express the following integral in terms of the gamma function

In summary, the conversation discusses finding the nth moment of a random variable using the gamma function. The integral and its equivalent forms are also mentioned, and a solution is attempted using integration by parts but is not successful. The suggestion is to use substitutions to simplify the integral into the form of the gamma function.
  • #1
Ben1220
25
0

Homework Statement


This is actually part of a probability problem I'm thinking about. I'm trying to find the nth moment of a certain random variable in terms of the gamma function, which is basically equivalent to solving the following integral or expressing it in terms of the gamma function. Here is the integral:

Homework Equations



Mathematica:
Integrate[(a/b^a) x^(n + a - 1) Exp[-(x/b)^a], {x, 0, Infinity}]

Plain text:
integral_0^infinity(a x^(n+a-1) e^(-(x/b)^a))/b^a dx

a, b, and n are constants.

wolfram alpha:

http://www.wolframalpha.com/input/?i=integrate+%28%28a%2Fb^a%29*x^%28n%2Ba-1%29*e^%28-1%28x%2Fb%29^a%29%2Cx%2C0%2Cinf%29

The Attempt at a Solution



I tried integrating by parts, taking u = -x^n and dv/dx = ... everything else in the expression (since that can be integrated nicely using derivative is present substitution), I was left with -1 + an even nastier integral, so I'm not convinced this is the right method...
 
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  • #2
Don't actually try to integrate it. Use substitutions that turn the integral into the form of the gamma function.
 

1. What is the gamma function?

The gamma function, denoted by Γ(x), is a mathematical function that generalizes the factorial function to real and complex numbers. It is defined as Γ(x) = ∫0 tx-1e-tdt.

2. How is the gamma function related to integrals?

The gamma function is closely related to integrals through its definition, which involves an integral over the positive real numbers. It is also commonly used in the evaluation of various integrals involving exponential and trigonometric functions.

3. What is the significance of expressing an integral in terms of the gamma function?

Expressing an integral in terms of the gamma function can often simplify its evaluation, especially when dealing with complex or improper integrals. It also allows for the use of various properties and identities of the gamma function to further manipulate the integral.

4. Can the gamma function be evaluated numerically?

Yes, the gamma function can be evaluated numerically using various mathematical software programs or functions. It is also possible to approximate its value using various methods such as the Stirling's approximation.

5. Are there any important applications of the gamma function in science?

The gamma function has various applications in science, particularly in the fields of physics and statistics. It is used in the calculation of probabilities in statistical distributions, as well as in the solution of differential equations in physics and engineering.

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