# jacobiTheta: Jacobi theta function In viscomplexr: Phase Portraits of Functions in the Complex Number Plane

## Description

Approximation of "the" Jacobi theta function using the first `nn` factors in its triple product version

## Usage

 `1` ```jacobiTheta(z, tau, nn = 30L) ```

## Arguments

 `z` Complex number; the point in the complex plane to which the output of the function is mapped `tau` Complex number; the so-called half-period ratio, must have a positive imaginary part `nn` Integer; number of factors to be used when approximating the triple product (default = 30)

## Details

This function approximates the Jacobi theta function theta(z; tau) which is the sum of exp(pi*i*n^2*tau + 2*pi*i*n*z) for n in -Inf, Inf. It uses, however, the function's triple product representation. See https://en.wikipedia.org/wiki/Theta_function for details. This function has been implemented in C++, but it is only slightly faster than well-crafted R versions, because the calculation can be nicely vectorized in R.

## Value

The value of the function for `z` and `tau`.

Other maths: `blaschkeProd()`, `juliaNormal()`, `mandelbrot()`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25``` ```phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/2-1/4), pType = "p", xlim = c(-2, 2), ylim = c(-2, 2), nCores = 1) # Max. two cores on CRAN, not a limit for your use phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/2-1/2), pType = "p", xlim = c(-2, 2), ylim = c(-2, 2), nCores = 1) phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/3+1/3), pType = "p", xlim = c(-2, 2), ylim = c(-2, 2), nCores = 1) phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/4+1/2), pType = "p", xlim = c(-2, 2), ylim = c(-2, 2), nCores = 1) ```