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20 Theta FunctionsComputation

§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 \mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right). 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 \mathop{\theta_{{1}}\/}\nolimits\!\left(2-i\middle|i\right) (=\mathop{\theta_{{1}}\/}\nolimits\!\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 \mathop{\theta_{{3}}\/}\nolimits\!\left(z,0.9\right) we use (20.7.32) with q=0.9=e^{{i\pi\tau}}, \tau=-i\mathop{\ln\/}\nolimits\!\left(0.9\right)/\pi. Then \tau^{{\prime}}=-1/\tau=-i\pi/\mathop{\ln\/}\nolimits\!\left(0.9\right) and q^{{\prime}}=e^{{i\pi\tau^{{\prime}}}}=\mathop{\exp\/}\nolimits\!\left(\pi^{2}% /\mathop{\ln\/}\nolimits\!\left(0.9\right)\right)=(2.07\dots)\times 10^{{-41}}. Hence the first term of the series (20.2.3) for \mathop{\theta_{{3}}\/}\nolimits\!\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 \imagpart{\tau}\geq\sqrt{3}/2; see §23.18. In practice a value with, say, \imagpart{\tau}\geq 1/2, \left|q\right|\leq 0.2, is found quickly and is satisfactory for numerical evaluation.

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