# Gamma and Beta Functions

• April 3rd 2013, 01:58 PM
abeyatta
Gamma and Beta Functions
the integral from 0-1 of √(1-x^4) dx = (Γ(1/4)^2)/(6√2π)
Here is how i tried to tackle it...

let t= x^4 , then dt/dx = 4x^3. And also x^3 = t ^(3/4).

Now the integral from 0-1 of (1-t) ^1/2 dt/4x^3

1/4 of the integral from 0-1 of x^(-3) (1-t)^1/2 dt

1/4 of the integral from 0-1 of. t^(-3/4) (1-t) ^1/2 dt

x-1 = -3/4 therefore x= 1/4
y-1 = 1/2 then. y= 3/2

which is 1/4 B(1/4 , 3/2) NOT equal to the LHS of the equation.

i also tried it by substituting x^2 = sin u and end up with a complete different answer of B(1/2, 3/2).

can some one pls point out what I missed.

Any help would be much appreciated
• April 3rd 2013, 03:00 PM
Ruun
Re: Gamma and Beta Functions
$I=\int_{0}^1 \sqrt{1-x^4}dx$

Let $t=x^4$ so $dt=4x^3dx$ this is $x=t^{1/4}$ and $dx=\frac{dt}{4x^3}=\frac{dt}{4t^{3/4}}$

$\int_{0}^1(1-t)^{1/2} t^{-3/4}dt=\int_{0}^1(1-t)^{1/2} t^{-3/4}dt$

The definition of the beta function reads

$B(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt$

so that $y-1=\frac{1}{2}$ and $x-1=\frac{3}{4}$ hence $y=\frac{3}{2}, x=\frac{7}{4}$

so

$\int_{0}^1(1-t)^{1/2} t^{-3/4}dt=B(\frac{7}{4},\frac{3}{2})$

Using that

$B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$

$4I=\frac{\Gamma(\frac{7}{4})\Gamma( \frac{3}{2})}{ \Gamma (\frac{13}{4})}$

which is not yours either
• April 3rd 2013, 03:49 PM
abeyatta
Re: Gamma and Beta Functions
Thanks Ruun

I think we have the same answer if it wasn't for the following

when you work out for x, you did x-1 = 3/4, but it should have been (-3/4) , which gives x = 1/4

But either way the answer should be equal to Γ(1/4)^2/(6√2π).....its 6 root 2pie in the denominator ,
• April 4th 2013, 02:31 AM
BobP
Re: Gamma and Beta Functions
Hi Abeyatta,

I agree with your answer of $\frac{1}{4}B(1/4,3/2).$

It can be converted to the given answer by the use of a few identities.

Start with $B(m,n) = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}.$

That gets you a $\Gamma(7/4)$ in the denominator and you deal with that by using the so called duplication formula

$2^{2x-1}\Gamma(x)\Gamma(x+\frac{1}{2})=\sqrt{\pi} \, \Gamma(2x)$

(Substitute x = 5/4).

Remove the $\Gamma(3/2)$ and $\Gamma(5/2)$ by use of the identities $\Gamma(n+1)=n\Gamma(n)$ and finally $\Gamma(1/2)=\sqrt{\pi}.$

(The $\Gamma(5/4)$ also requires the use of $\Gamma(n+1)=n\Gamma(n)).$
• April 6th 2013, 10:27 AM
abeyatta
Re: Gamma and Beta Functions
Thank you very much Bob, and apologies for late reply.