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Tag Archives: Integral
Let denote the Riemann zeta function. Evaluate the integral
Based on symmetries,
Let . It follows that
Using the recursion we get that
Let and . Show that
We note that
where denotes the Euler – Mascheroni constant.
Beginning by parts we have,
The result now follows taking and .
Let denote the Bessel function of the first kind. Prove that
We recall that
Using the fact that the looks like an ‘almost periodic’ function with decreasing amplitude. If we denote by the zeros of then as and furthermore
as for each . So the integral converges uniformly in this case justifying the interchange of limit and integral.
The result follows.
Evaluate the integral
We have successively: