I just found the following proof of Euler’s reflection formula. The thing I like about it is that allĀ 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

to notice

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.

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