Write this integral in terms of gamma function

In summary, the conversation is about solving an integral using the gamma function. The integral in question is \int _{0}^{\infty }e^{-x^4}dx and the hint given is to let x^4 = u. The answer is then expressed as \Gamma \left( \dfrac {5} {4}\right) and the conversation continues with the individual's attempts at solving the integral.
  • #1
eprparadox
138
2
Hey!

So I'm self studying mary boas's mathematical methods book and I've come across this integral:

[tex]
\int _{0}^{\infty }e^{-x^4}dx
[/tex]

and I'm suppose to write this using the gamma function. The hint given states to let x^4 = u. And the answer is:

[tex]
\Gamma \left( \dfrac {5} {4}\right)
[/tex]

I tried substituting u = x^4 and du = 4x^3dx, but that doesn't give the correct answer.

I'm a bit confused as to how the book got that answer. Any ideas would be great.

Thanks!
 
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  • #2
After substitution you should have something you can express as ## t \Gamma(t) ## which you can rewrite using the relation ## t\Gamma(t) = \Gamma(t+1)##.
 

Related to Write this integral in terms of gamma function

1. What is the gamma function and why is it important in integrals?

The gamma function, denoted by Γ(z), is an extension of the factorial function to complex and real numbers. It is defined as Γ(z) = ∫0∞ xz-1e-x dx. The gamma function is important in integrals because it allows us to evaluate integrals involving powers and exponentials, which are commonly encountered in mathematical and scientific applications.

2. How do you write an integral in terms of the gamma function?

To write an integral in terms of the gamma function, you first need to identify the power and exponential terms in the integrand. Then, you can use the property of the gamma function that Γ(z+1) = zΓ(z) to rewrite the integral in terms of Γ(z). Finally, you can use the definition of the gamma function to evaluate the integral.

3. Can any integral be written in terms of the gamma function?

No, not all integrals can be written in terms of the gamma function. The gamma function is only applicable to integrals involving powers and exponentials. Integrals involving other types of functions, such as trigonometric functions or logarithmic functions, cannot be written in terms of the gamma function.

4. Are there any special cases for writing integrals in terms of the gamma function?

Yes, there are some special cases for writing integrals in terms of the gamma function. For example, when the upper limit of the integral is infinity, the integral can be written as a multiple of the gamma function. Additionally, when the power in the integrand is a negative integer, the integral can be written in terms of the gamma function using the property Γ(n) = (n-1)!.

5. How is the gamma function related to other mathematical functions?

The gamma function is closely related to the factorial function, as it is an extension of the factorial function to complex and real numbers. It is also related to the beta function, which is defined as B(x,y) = Γ(x)Γ(y)/Γ(x+y). Additionally, the gamma function has applications in probability and statistics, as it is used to define the gamma distribution.

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