Hurwitz Zeta Function - Relation To Jacobi Theta Function

Relation To Jacobi Theta Function

If is the Jacobi theta function, then

\int_0^\infty \left t^{s/2} \frac{dt}{t}= \pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left

holds for and z complex, but not an integer. For z=n an integer, this simplifies to

\int_0^\infty \left t^{s/2} \frac{dt}{t}= 2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s) =2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).

where ΞΆ here is the Riemann zeta function. Note that this latter form is the functional equation for the Riemann zeta function, as originally given by Riemann. The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the Dirac delta function in z as .

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