Approximation of "the" Jacobi theta function using the first nn
factors in its triple product version
jacobiTheta(z, tau, nn = 30L)
| 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) |
The value of the function for z and tau.
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.
Other maths:
blaschkeProd(),
juliaNormal(),
mandelbrot()
# \donttest{ 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 # } # \donttest{ phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/2-1/2), pType = "p", xlim = c(-2, 2), ylim = c(-2, 2), nCores = 1) # } # \donttest{ phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/3+1/3), pType = "p", xlim = c(-2, 2), ylim = c(-2, 2), nCores = 1) # } # \donttest{ phasePortrait(jacobiTheta, moreArgs = list(tau = 1i/4+1/2), pType = "p", xlim = c(-2, 2), ylim = c(-2, 2), nCores = 1) # }