20 Theta Functions Computation 20.13 Physical Applications 20.15 Tables

§20.14 Methods of Computation

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Keywords:: computation, theta functions
Referenced by:: §22.20(i), §22.20(iv), §22.20(vi), §23.22(i)
Permalink:: http://dlmf.nist.gov/20.14
See also:: Annotations for Ch.20

The Fourier series of §20.2(i) usually converge rapidly because of the factors $q^{(n+\frac{1}{2})^{2}}$ or $q^{n^{2}}$ , and provide a convenient way of calculating values of $\theta_{j}\left(z\middle|\tau\right)$ . Similarly, their -differentiated forms provide a convenient way of calculating the corresponding derivatives. For instance, the first three terms of (20.2.1) give the value of $\theta_{1}\left(2-i\middle|i\right)$ ( $=\theta_{1}\left(2-i,e^{-\pi}\right)$ ) to 12 decimal places.

For values of $\left|q\right|$ near 1 the transformations of §20.7(viii) can be used to replace $\tau$ with a value that has a larger imaginary part and hence a smaller value of $\left|q\right|$ . For instance, to find $\theta_{3}\left(z,0.9\right)$ we use (20.7.32) with $q=0.9=e^{i\pi\tau}$ , $\tau=-i\ln\left(0.9\right)/\pi$ . Then $\tau^{\prime}=-1/\tau=-i\pi/\ln\left(0.9\right)$ and $q^{\prime}=e^{i\pi\tau^{\prime}}=\exp\left(\pi^{2}/\ln\left(0.9\right)\right)=% (2.07\dots)\times 10^{-41}$ . Hence the first term of the series (20.2.3) for $\theta_{3}\left(z\tau^{\prime}\middle|\tau^{\prime}\right)$ suffices for most purposes. In theory, starting from any value of $\tau$ , a finite number of applications of the transformations $\tau\to\tau+1$ and $\tau\to-1/\tau$ will result in a value of $\tau$ with $\Im\tau\geq\sqrt{3}/2$ ; see §23.18. In practice a value with, say, $\Im\tau\geq 1/2$ , $\left|q\right|\leq 0.2$ , is found quickly and is satisfactory for numerical evaluation.

20.13 Physical Applications 20.15 Tables

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