This is a quality thread! I now know 3 more proofs of this!
To simplependulum: did you think of that yourself?! If so, how long did that take you and what led you to it? Kudos!
At first i was thinking how to solve without using the fact , in other words , I shall give another way to prove this fact starting at this point .
However , i failed , all my attempts eventually led to other troublesome integrals so I gave up , until I was asked to solve another integral which brought some ideas to me in solving the integral I was struggling two weeks before .
Here is my derivation of the first fact, although we need this definition of the zeta function
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So, in the initial integral, let . Then . and . This last bit gives you the new limits of integration.
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Where on the RHS, I've let u = x so that i don't have to use u from now on. Now lets focus on the integrand.
.
The integral becomes
provided both converge. Now on the second piece, let , and to give that the piece is
Combining both pieces now we get