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For some time I have been using EllipticTheta[3,-,-] functions, and to me it seems that Mathematica is really not handling them in the best way. Below are two of the instances I have faced so far, and I would appreciate any input to resolve these issues.

1. Plots

Here is a plot I have tried and it works well:

Plot3D[EllipticTheta[3, Pi/2  x, t], {x, 0, 2}, {t, 0, 1}]

enter image description here

I understand the peaks as the Theta function approaches a Dirac comb when t=1.

Then I try

Plot3D[EllipticTheta[3, Pi/2 x, E^(- Pi^2 t)],{x,0,2},{t,0,tmax}]

this is how it looks for tmax=1

enter image description here

by increasing tmax things start looking strange, and eventually the kernel seems to crash (from about tmax=4.5). This is how it looks for tmax=5

enter image description here

It becomes even worse for larger values of tmax. I don't seem to understand what is causing the problem, and how choosing larger intervals for the plot messes up the part of the plot for smaller values of t which were looking fine with tmax=1.

2. Derivatives

I have read that derivatives of the Theta function with respect to its third argument are problematic as discussed here, but I do not understand the reason. The solution suggested in that post is to use the corresponding PDE, i.e.

Derivative[0,0,1][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)] = 
Derivative[0,2,0][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)]

So this means instead of Derivative[0,1,1][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)] it's better to use Derivative[0,3,0][EllipticTheta][3,Pi/2 x,E^(-Pi^2 t)]. To check, I plot it

Plot3D[ Evaluate[
  Derivative[0, 3, 0][EllipticTheta][3, Pi/2 x, E^(-Pi^2 t)]], {x, 0, 
  2}, {t, 0, 1}]

enter image description here

Note that this is quite slow to generate. Q: How to improve the speed?

In addition, I do not know if the values of these higher-order derivatives are something that I could trust.

Also, when I try increasing t to, say, {t,0,2} Mathematica v12.1 jumps out of the notebook. Note that I am actually trying to use these derivatives as part of an integrand (like here) and I'd want to avoid singularities/numerical instabilities even in the cost of (a bit of) accuracy. So maybe I should use a finite sum, instead of the infinite sum in the definition of EllipticTheta?

Here is another example, which (I think) should be identical to zero (adapted from this post)

Plot[ Evaluate[ 
  D[EllipticTheta[3, x, E^(-Pi^2 t)], {x, 3}] /. x -> 0], {t, 0, 1}]

enter image description here

(edit/update: flinty's solution to define a WorkingPrecision resolves this latter one. I had assumed Mathematica does not need that for built-in functions)

Question

Is there a better way to deal with EllipticTheta in Mathematica? I tried implementing it as the direct summation it is defined through, and ran exactly to the same problems. So I guess Mathematica might be actually evaluating the sum when EllipticTheta is called, and maybe that is causing the problems (since it'll run into exponentials with very large negative powers).

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    $\begingroup$ The problem where it starts looking weird for tmax=5 is because the automatic plot range for the z-axis is a very narrow slice around 1.0 , too narrow for rendering precision. A similar issue occurs here and it's resolved by choosing a wider PlotRange . $\endgroup$
    – flinty
    Jul 27, 2020 at 23:18
  • $\begingroup$ @flinty Thank you, I didn't know that. It does seem to partially resolve the issue when I choose PlotRange->{-2,2} for tmax=5, but close to t=0 there is a strange behaviour which seems to be different than the same region for tmax=1. $\endgroup$
    – SaMaSo
    Jul 27, 2020 at 23:24
  • $\begingroup$ Try PerformanceGoal -> "Quality", MaxRecursion -> 6 - it might smooth it out. $\endgroup$
    – flinty
    Jul 27, 2020 at 23:27
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    $\begingroup$ For the part on the derivative, this is just an artifact of $MachinePrecision calculations. Try using WorkingPrecision -> 10 or 20 in the plot options and it's no longer a problem. $\endgroup$
    – flinty
    Jul 27, 2020 at 23:31
  • $\begingroup$ @flinty thank you for the comments. They did resolve the plotting issues. However, I have had a lot of difficulties with the derivatives of EllipticTheta (especially taking derivatives wrt the third argument, or taking derivatives higher than 2 from the second argument) - so I was hoping there is a better way of handling them. $\endgroup$
    – SaMaSo
    Jul 28, 2020 at 23:57

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