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:

    “Whoever has a keen eye for profits, is blind in relation to his craft.”
    —Sophocles (497–406/5 B.C.)

    “In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
    —Simone Weil (1909–1943)

    “... 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)

    “My function in life is not to be a politician in Parliament: it is to get something done.”
    —Bernadette Devlin (b. 1947)