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 .

Read more about this topic:  Hurwitz Zeta Function

Famous quotes containing the words relation to, relation, jacobi and/or function:

    “To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
    —Bruno Bettelheim (20th century)

    “There is a constant in the average American imagination and taste, for which the past must be preserved and celebrated in full-scale authentic copy; a philosophy of immortality as duplication. It dominates the relation with the self, with the past, not infrequently with the present, always with History and, even, with the European tradition.”
    —Umberto Eco (b. 1932)

    “... spinsterhood [is considered to be] an abnormality of small proportions and small consequence, something like an extra finger or two on the body, presumably of temporary duration, and never of any social significance.”
    —Mary Putnam Jacobi (1842–1906)

    “Science has fulfilled her function when she has ascertained and enunciated truth.”
    —Thomas Henry Huxley (1825–95)