[math, computation] Euler-reflection formula-version 3 : Differential equation

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

In the last two posts (version 1 and version 2) we have discuss derivation of Euler-reflection formula.

Today as a last post of this topic, I'd like to talk about the third method : Differential equations.

This method was first introduced by Dedkin 1853. We can start from this integral

In the last post, version 2, we compute the right hand side using complex integration, but instead of computing integration directly, here we are dealing with left hand side. The strategy here is find particular differential equation for left hand side and solving that differential equation.

The non-linear differential equation for phi(x) is

where ' means differentiation with respect to x. Just by plugging phi(x) = Gamma(x) Gamma(1-x), you can check this indeed holds.

Note that above differential equation is second order, so to solve the equation we need at least two initial condition, from the properties of Gamma(x) we can see

Now for solving this we can slightly modify the differential equation

Plugging these results we have

Now we can integration!

applying initial condition for differentiation we have constant C

Re-arranging

Now integrate over x

and using initial condition and identities for tri-gometric function

This is what i want to show!!!

Up till now we have talked about three different ways to showing Euler-reflection formula. Personally i like the first one, version 1 because it is the simplest one.

What's your favorite version?