I just found the following proof of Euler’s reflection formula. The thing I like about it is that all it requires is dumbly marching forwards, there is no trickery.
When and are both positive, that is, when ,
I’m going to join those integrals, so giving different names to each copy of ,
Now change variables and . The inverse transformation is and and the Jacobian is , then
To evaluate this integral, split into the two parts
Now use the series for the denominators
and now change the order of the sums and integrals
Now define the function
and observe that
and use Euler’s products for the sine and cosine
now take the logarithmic derivative
the right hand side is what we had previously found with the variable instead of , therefore
We have the equality when , and since these functions are holomorphic, and they coincide in an open set they must be equal.