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 ζ 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 .

Read more about this topic:  Hurwitz Zeta Function

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