Chamilka's Question from Math Help Forum.

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Sudharaka

Well-known member
MHB Math Helper
Original Title: Please Need an help on this integral!!

chamilka said:
Hi everyone!!

I have to integrate the following function.

$\int_{0}^{1}x^{a+b-1}(1-x)^{c-a}(1-x^b)^{d-1}\,dx$

Here a,b,c and d are constants. I have to integrate the above function with respect to x in the region from zero to infinity. This may seem like a Beta integral function, but there is a slight change that there are three terms x, (1-x) and (1-x^b).

Also I have found in a journal article that the answer for the integral is given as below

$\sum_{i=0}^{\infty}(-1)^{i}\binom{d-1}{i}B(a+b+bi,\,c-a+1)$

But the evaluation methods are not given. They might used a series expansion, but nothing is given there.

Thank you .
Hi chamilka,

$\int_{0}^{1}x^{a+b-1}(1-x)^{c-a}(1-x^b)^{d-1}\,dx$

Using the Binomial series of $$(1-x^b)^{d-1}$$ we get,

\begin{eqnarray}

\int_{0}^{1}x^{a+b-1}(1-x)^{c-a}(1-x^b)^{d-1}\,dx&=&\int_{0}^{1}x^{a+b-1}(1-x)^{c-a}\sum_{i=0}^{\infty} \; {d-1\choose i}\;(-x^b)^{i}\,dx\\

&=&\int_{0}^{1}\left(\sum_{i=0}^{\infty}(-1)^{i}{d-1\choose i}x^{a+b+bi-1}(1-x)^{c-a}\right)\,dx

\end{eqnarray}

The series, $$\displaystyle\sum_{i=0}^{\infty}(-1)^{i}{d-1\choose i}x^{a+b+bi-1}(1-x)^{c-a}$$ is a power series and hence could be integrated term by term. Therefore,

\begin{eqnarray}

\int_{0}^{1}x^{a+b-1}(1-x)^{c-a}(1-x^b)^{d-1}\,dx&=&\sum_{i=0}^{\infty}\left(\int_{0}^{1}(-1)^{i}{d-1\choose i}x^{a+b+bi-1}(1-x)^{c-a}\,dx\right)\\

&=&\sum_{i=0}^{\infty}(-1)^{i}{d-1\choose i}\left(\int_{0}^{1}x^{(a+b+bi)-1}(1-x)^{(c-a+1)-1}\,dx\right)\\

\end{eqnarray}

By the definition of the Beta function, if $$Re(a+b+bi)>0\mbox{ and }Re(c-a+1)>0$$ we get,

$\int_{0}^{1}x^{a+b-1}(1-x)^{c-a}(1-x^b)^{d-1}\,dx=\sum_{i=0}^{\infty}(-1)^{i}{d-1\choose i}B(a+b+bi,\,c-a+1)$

Kind Regards,
Sudharaka.

chamilka

New member
Thank you Sudharaka. Here I got a more clear proof for my question. This is very great!

Btw I have posted a new post

Also quoted here:

Hi everyone!
I got two versions of one particular function and now I need to show those two versions are equivalent.
For that I need to show the follwing,

View attachment 223

Is it possible to show this by using the properties of Beta functions, Gaussian hypergeometric function etc?