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Jacobi theta functions, $\theta_a(u,q)$ for $a=1,2,3,4$ are defined in the unit disk $|q|<1$.

For some reason that I would like to understand, Mathematica does not evaluate numerically the $q$ derivatives of these functions, regardless the value of $u$ (at least for the dozen of values that I tried).

Plotting their imaginary and real parts shows that all of them undergo the same problem.

Plot[{Re[(EllipticTheta^(0,0,1))[1,-.5,q]],Im[(EllipticTheta^(0,0,1))[1,-.5,q]]},{q,0,1}]
Plot[{Re[(EllipticTheta^(0,0,1))[2,-.5,q]],Im[(EllipticTheta^(0,0,1))[2,-.5,q]]},{q,0,1}]
Plot[{Re[(EllipticTheta^(0,0,1))[3,-.5,q]],Im[(EllipticTheta^(0,0,1))[3-.5,q]]},{q,0,1}]
Plot[{Re[(EllipticTheta^(0,0,1))[4,-.5,q]],Im[(EllipticTheta^(0,0,1))[4,-.5,q]]},{q,0,1}]

Heuristically, the limit point is 0.52830188679244577999999999999999...999...

D[EllipticTheta[1,-0.28I,q],q]/.q->0.52830188679244577999999999

0. +0.909738 I

Evaluation above it returns no result;

D[EllipticTheta[1,-0.28I,q],q]/.q->0.52830188679244578 

(EllipticTheta^(0,0,1))[1,0. -0.28 I,0.528302]
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  • $\begingroup$ Does Plot[{Re[D[EllipticTheta[1, -.5, x], x] /. x -> q], Im[D[EllipticTheta[1, -.5, x], x] /. x -> q]}, {q, 0, 1}] give what you expect? $\endgroup$
    – kglr
    Nov 8, 2014 at 12:17
  • $\begingroup$ Thank you for your proposal; unfortunately it does not work and the plot stops above the 0.52 value as well... $\endgroup$
    – picop
    Nov 8, 2014 at 17:33
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    $\begingroup$ I don't believe Mathematica is able to symbolically deal with derivatives with respect to the nome; a numerical approach can be pursued here, however. Care will be needed when the nome is near the unit circle. $\endgroup$
    – J. M.'s missing motivation ♦
    Aug 3, 2015 at 12:32
  • $\begingroup$ @J.M., So it is structural... I'll get around differently. Thanks for the hint. $\endgroup$
    – picop
    Aug 3, 2015 at 12:48

1 Answer 1

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As it turns out, one can use the PDE satisfied by the Jacobi theta functions to compute the required values:

With[{z = -7 I/25, q = 0.52830188679244577999999999},
     {Derivative[0, 0, 1][EllipticTheta][1, z, q],
      -Derivative[0, 2, 0][EllipticTheta][1, z, q]/(4 q)}]
   {0.9097375919067572916312 I, 0.*10^-23 + 0.90973758336986133322032 I}

(Note that there is a discrepancy between the two evaluations, but I am more inclined to trust the second result instead of the first, since computing the $z$-derivatives for theta functions is much more numerically well-behaved.)

Now, for the case in the OP that could not be numerically evaluated:

With[{z = -7 I/25, q = 0.52830188679244578},
     -Derivative[0, 2, 0][EllipticTheta][1, z, q]/(4 q)]
   9.99066*10^-16 + 0.909738 I
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  • $\begingroup$ Agreed, I had eventually resorted to this as well. $\endgroup$
    – picop
    Mar 3, 2020 at 8:37

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