Theta Function - Jacobi Theta Function

There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them. One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula

\vartheta(z; \tau) = \sum_{n=-\infty}^\infty \exp (\pi i n^2 \tau + 2 \pi i n z) = 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} \cos(2\pi n z) = \sum_{n=-\infty}^\infty q^{n^2}\zeta^n

where q = exp(πiτ) and ζ = exp(2πiz). It is a Jacobi form. If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity

The function also behaves very regularly with respect to its quasi-period τ and satisfies the functional equation

where a and b are integers.

Read more about this topic:  Theta Function

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