Definition:Jacobi Theta Function/First Type

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Definition

The Jacobi Theta function of the first type is defined as:

$\forall z \in \C: \ds \map {\vartheta_1} {z, q} = 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n q^{\paren {n + \frac 1 2}^2} \sin \paren {2 n + 1} z$

where:

$q = e^{i \pi \tau}$
$\tau \in \C$ such that $\map \Im \tau > 0$


Also see

  • Results about the Jacobi theta functions can be found here.


Source of Name

This entry was named for Carl Gustav Jacob Jacobi.


Sources