Definite integral and Gamma functions

In summary, the definite integral (expressed in terms of gamma functions) is: \int_0^1\frac{dx}{\sqrt{1-x^4}}=\frac{\sqrt{\pi}\Gamma[\frac{5}{4}]}{\Gamma[\frac{3}{4}]}
  • #1
saltydog
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I've been trying to determine how certain definite integrals are expressed in terms of Gamma functions.

Mathematica returns the following:

[tex]\int_0^1 \frac{dx}{\sqrt{1-x^4}}=\frac{\sqrt{\pi}\Gamma[\frac{5}{4}]}{\Gamma[\frac{3}{4}]}[/tex]

(Mapple returns a different but equivalent expression in terms of Gamma)

In general it seems:

[tex]\int_0^1 \frac{dx}{\sqrt{1-x^n}}=\frac{\sqrt{\pi}\Gamma[\frac{n+1}{n}]}{\Gamma[\frac{n+2}{2n}]}[/tex]

Can anyone explain to me how this is determined or provide a hint or a reference?
 
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  • #2
According to Mathematica,the general case is a standard integral for the Gauss hypergeometric function...

Daniel.
 
  • #3
Well, the trick is to convert the given integral into gamma-integrals. If i remember correctly from my calculus course at college, you can do this via PI...though i am not too sure

marlon
 
  • #4
marlon said:
Well, the trick is to convert the given integral into gamma-integrals. If i remember correctly from my calculus course at college, you can do this via PI...though i am not too sure

marlon

Well thanks Marlon and Daniel, but via pi?

You mean, I need to figure out how to convert:

[tex]\int_0^1\frac{dx}{\sqrt{1-x^4}}[/tex]

to some variant of:

[tex]\Gamma[x]=\int_0^{\infty}e^{-t}t^{x-1}dt[/tex]

Well, I'll look in my Calculus books but well, I just don't see it happening. Maybe so though. Think you can give me another hint? It's a very interesting problem and I'd like to know how to figure it out.
 
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  • #5
I don't see how u could possibly put an exponential there.I'm waiting anxiously to be stunned...

Daniel.
 
  • #6
Here we go (on the web under "gamma integral"):

[tex]I=\int_0^1\frac{dx}{(1-x^a)^b}[/tex]

with a>0 and b<1

Letting:

[tex]u=x^a[/tex]

then:

[tex]dx=\frac{1}{a}u^{(1/a)-1}[/tex]

So that we now have:

[tex]\frac{1}{a}\int_0^1 u^{(1/a)-1}(1-u)^{-b}du[/tex]

Now, here's the key: The Beta function is defined:

[tex]\beta(m,n)=\int_0^1 u^{m-1}(1-u)^{n-1}du[/tex]

So that the integral, expressed in the beta function is:

[tex]I=\frac{1}{a}\beta(\frac{1}{a},1-b)[/tex]

Since:

[tex]\beta(m,n)=\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}[/tex]

We finally have:

[tex]I=\frac{\Gamma(\frac{1}{a})\Gamma(1-b)}{a\Gamma(1-b+\frac{1}{a})}[/tex]

Think I need to reiew this with a couple of examples . . .

Edit: Also, made an error with the upper limit in the original post, it should have been from 0 to 1.
 
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  • #7
Your're right about the limits.Your function (with the initial limits) would have had complex values,while the result would have been very real.

Daniel.

BTW,i think it's B (capital beta)-Euler..Else why would it be capital gamma...?(BTW,small gamma is called Euler-Mascheroni's constant).
 

Related to Definite integral and Gamma functions

What is a definite integral?

A definite integral is a mathematical concept used to calculate the area under a curve or the signed area between two points on a graph. It is represented by the symbol ∫ and has a lower and upper limit for the independent variable.

What are gamma functions?

Gamma functions are special functions in mathematics that extend the concept of factorial to non-integer values. They are denoted by the symbol Γ and are widely used in various fields of mathematics, such as statistics and physics.

What is the relationship between definite integrals and gamma functions?

The relationship between definite integrals and gamma functions is that the definite integral of a power function can be expressed in terms of the gamma function. This relationship is known as the fundamental theorem of calculus and is a fundamental tool in the evaluation of integrals.

What are some applications of definite integrals and gamma functions?

Some common applications of definite integrals and gamma functions include calculating areas and volumes in geometry, determining probabilities in statistics, and solving differential equations in physics and engineering.

What are the properties of definite integrals and gamma functions?

Definite integrals and gamma functions have various properties, including linearity, additivity, and the ability to be evaluated using integration techniques such as substitution and integration by parts. They also have special values at specific inputs, such as the gamma function having a value of 1 for an input of 1.

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