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20 Theta FunctionsComputation

§20.15 Tables

Theta functions are tabulated in Jahnke and Emde (1945, p. 45). This reference gives θj(x,q), j=1,2,3,4, and their logarithmic x-derivatives to 4D for x/π=0(.1)1, α=0(9)90, where α is the modular angle given by

20.15.1 sinα=θ22(0,q)/θ32(0,q)=k.

Spenceley and Spenceley (1947) tabulates θ1(x,q)/θ2(0,q), θ2(x,q)/θ2(0,q), θ3(x,q)/θ4(0,q), θ4(x,q)/θ4(0,q) to 12D for u=0(1)90, α=0(1)89, where u=2x/(πθ32(0,q)) and α is defined by (20.15.1), together with the corresponding values of θ2(0,q) and θ4(0,q).

Lawden (1989, pp. 270–279) tabulates θj(x,q), j=1,2,3,4, to 5D for x=0(1)90, q=0.1(.1)0.9, and also q to 5D for k2=0(.01)1.

Tables of Neville’s theta functions θs(x,q), θc(x,q), θd(x,q), θn(x,q) (see §20.1) and their logarithmic x-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for ε,α=0(5)90, where (in radian measure) ε=x/θ32(0,q)=πx/(2K(k)), and α is defined by (20.15.1).

For other tables prior to 1961 see Fletcher et al. (1962, pp. 508–514) and Lebedev and Fedorova (1960, pp. 227–230).