[math, computation] Euler-reflection formula-version 2 : complex integration

This is a English version of my former post in korean .

In the last posting [math, computation] Euler-reflection formula-version 1 : Basel Problem we talk about one simple ways of showing Euler's reflection formula.

And this time, i want to post a second method which using the techniques of complex integration.
To do that first define gamma function and beta function as follows

One important relation between them is

Now let's prove this relation

First by substitution of t=s/s+1 in the Beta function , we have

Then

Actually in the process we used

Anyway from above computation we obtain

Now the problem is computing the integration on right hand side. Here we will use complex integration which you might hear from your complex variable course... this is kind of examples or exercise problems in usual course material. You can obtain this integration without complex integration and actually that is version 3 : Differential equation i am planing to write.

I want to draw some contours .. [i am kinds of novice for drawing figures in latex..] i just attach my notes .

Now taking R infinity and epsilon 0, since the restriction of x is between 0 and 1, we have

Thus upon this condition

Re-organizing

Finally

Now we are done!.

Of course there is a third version by Dedkind using differential equation. I will post it soon~