§20.14 Methods of Computation

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 ${\theta }_j\left(z\right|\tau )$. Similarly, their $z$-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 ${\theta }_1\left(2-\mathrm{i}\right|\mathrm{i})$ ($={\theta }_1(2-\mathrm{i},{\mathrm{e}}^{-\pi })$) 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 ${\theta }_3(z,0.9)$ we use (20.7.32) with $q=0.9={\mathrm{e}}^{\mathrm{i}\pi \tau }$, $\tau =-\mathrm{i}\ln \left(0.9\right)/\pi$. Then ${\tau }^{\prime }=-1/\tau =-\mathrm{i}\pi /\ln \left(0.9\right)$ and $q^{\prime }={\mathrm{e}}^{\mathrm{i}\pi {\tau }^{\prime }}=\exp \left({\pi }^2/\ln \left(0.9\right)\right)=(2.07\mathrm{\dots })\times {10}^{-41}$. Hence the first term of the series (20.2.3) for ${\theta }_3\left(z{\tau }^{\prime }\right|{\tau }^{\prime })$ 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 $\mathrm{\Im }\tau \ge \sqrt{3}/2$; see §23.18. In practice a value with, say, $\mathrm{\Im }\tau \ge 1/2$, $\left|q\right|\le 0.2$, is found quickly and is satisfactory for numerical evaluation.