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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+12)2 or qn2, and provide a convenient way of calculating values of θj(z|τ). 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 θ1(2-i|i) (=θ1(2-i,e-π)) to 12 decimal places.

For values of |q| near 1 the transformations of §20.7(viii) can be used to replace τ with a value that has a larger imaginary part and hence a smaller value of |q|. For instance, to find θ3(z,0.9) we use (20.7.32) with q=0.9=eiπτ, τ=-iln(0.9)/π. Then τ=-1/τ=-iπ/ln(0.9) and q=eiπτ=exp(π2/ln(0.9))=(2.07)×10-41. Hence the first term of the series (20.2.3) for θ3(zτ|τ) suffices for most purposes. In theory, starting from any value of τ, a finite number of applications of the transformations ττ+1 and τ-1/τ will result in a value of τ with τ3/2; see §23.18. In practice a value with, say, τ1/2, |q|0.2, is found quickly and is satisfactory for numerical evaluation.