I like to know beta function`s property related with my question

Question

Fact 1. ($B$ is beta-function) $$\int_{0}^{1}\frac{z^{-(n+1)}}{(1+\frac{1}{z})^{2n}}=\frac{1}{2}B(n,n)$$

I can find above fact by using MATLAB.

But i like to show above fact using beta function property.

To do this, i try to use "$B(m,n)=2\int_{0}^{1}{x^{2m-1}(1-x^{2})^{n-1}dx}$".

But i fail to show above fact.

Is there any other related beta function property related with my case?

Or how i can show above fact?

Thank you!

Answer 1

Let $I_n$ your integral. First, the change of variable $z=1/t$ give that $\displaystyle I_n=\int_1^{+\infty}\frac{t^{n-1}}{(1+t)^{2n}} dt$. Hence $\displaystyle 2I_n=\int_0^{+\infty}\frac{t^{n-1}}{(1+t)^{2n}} dt$. Now in the last integral, we put $\displaystyle u=\frac{1}{1+t}$. This gives $\displaystyle 2I_n=\int_0^1 u^{n-1}(1-u)^{n-1}du=B(n,n)$.