Basic properties of Jacobi's theta function

Question

On the following Wikipedia page about Jacobi's theta function $\vartheta$, it says that the $\vartheta$ satisfies the condition that "at fixed $\tau$, this is a Fourier series for a $1$-periodic function in $z$; I am wondering what the explicit $1$-periodic function is.

Answer 1

Basically, infinite trigonometric sum in $\sin,\cos$ of arguments $2\ n\ x$ and $(2\ n-1)\ x$ with coefficients

$$\{1 , (-1)^n, e^{-\frac{ \pi\ K \ n^2}{K'} } , e^{-\frac{\pi\ K (n-\frac{1}{2})^2}{K'}} \}$$.

$K$ is the complete elliptic integral. $4 K$ the period of a rotating pendulum on the unit circle at a given energy greater 0 in the top position and K' is the second real period in the imaginary direction of the double periodic set of elliptic functions with fixed K.

As Fourier series they are of much better in convergence to periodic functions with nearly pole-like concentrations, eg a pendulum coming to rest at top but after an hour makes a full loop of some seconds or for comets coming around the sun for some months all 4 centuries. Mostly on needs to approximate the Theta series coefficients with small corrections on some coefficients only.

Since the rationals of the four Thetas, via the Jacobi elliptic functions sn, cn, sd, mimic the algebra of sin,cos,tan etc at the unit circle, they are easy to use in cartesian coordinates by table lookup. This was the way they were discovered in the late days of Gauss when Fourier series became the hot topic.

  Plot[Evaluate[EllipticTheta[{1, 2, 3, 4}, x, 0.95] + {0, 1, 2, 3}], 
       {x, 0,2 \[Pi]}, PlotRange -> All, PlotLegends -> "Expressions"]