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20 Theta FunctionsNotation

§20.1 Special Notation

(For other notation see Notation for the Special Functions.)

m, n integers.
z() the argument.
τ() the lattice parameter, τ>0.
q() the nome, q=eiπτ, 0<|q|<1. Since τ is not a single-valued function of q, it is assumed that τ is known, even when q is specified. Most applications concern the rectangular case τ=0, τ>0, so that 0<q<1 and τ and q are uniquely related.
qα eiαπτ for α (resolving issues of choice of branch).
S1/S2 set of all elements of S1, modulo elements of S2. Thus two elements of S1/S2 are equivalent if they are both in S1 and their difference is in S2. (For an example see §20.12(ii).)

The main functions treated in this chapter are the theta functions θj(z|τ)=θj(z,q) where j=1,2,3,4 and q=eiπτ. When τ is fixed the notation is often abbreviated in the literature as θj(z), or even as simply θj, it being then understood that the argument is the primary variable. Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21.

Primes on the θ symbols indicate derivatives with respect to the argument of the θ function.

Other Notations

Jacobi’s original notation: Θ(z|τ), Θ1(z|τ), H(z|τ), H1(z|τ), respectively, for θ4(u|τ), θ3(u|τ), θ1(u|τ), θ2(u|τ), where u=z/θ32(0|τ). Here the symbol H denotes capital eta. See, for example, Whittaker and Watson (1927, p. 479) and Copson (1935, pp. 405, 411).

Neville’s notation: θs(z|τ), θc(z|τ), θd(z|τ), θn(z|τ), respectively, for θ32(0|τ)θ1(u|τ)/θ1(0|τ), θ2(u|τ)/θ2(0|τ), θ3(u|τ)/θ3(0|τ), θ4(u|τ)/θ4(0|τ), where again u=z/θ32(0|τ). This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951).

McKean and Moll’s notation: ϑj(z|τ)=θj(πz|τ), j=1,2,3,4. See McKean and Moll (1999, p. 125).

Additional notations that have been used in the literature are summarized in Whittaker and Watson (1927, p. 487).