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20 Theta FunctionsProperties

§20.2 Definitions and Periodic Properties

  1. §20.2(i) Fourier Series
  2. §20.2(ii) Periodicity and Quasi-Periodicity
  3. §20.2(iii) Translation of the Argument by Half-Periods
  4. §20.2(iv) z-Zeros

§20.2(i) Fourier Series

20.2.1 θ1(z|τ) =θ1(z,q)=2n=0(1)nq(n+12)2sin((2n+1)z),
20.2.2 θ2(z|τ) =θ2(z,q)=2n=0q(n+12)2cos((2n+1)z),
20.2.3 θ3(z|τ) =θ3(z,q)=1+2n=1qn2cos(2nz),
20.2.4 θ4(z|τ) =θ4(z,q)=1+2n=1(1)nqn2cos(2nz).

Corresponding expansions for θj(z|τ), j=1,2,3,4, can be found by differentiating (20.2.1)–(20.2.4) with respect to z.

§20.2(ii) Periodicity and Quasi-Periodicity

For fixed τ, each θj(z|τ) is an entire function of z with period 2π; θ1(z|τ) is odd in z and the others are even. For fixed z, each of θ1(z|τ)/sinz, θ2(z|τ)/cosz, θ3(z|τ), and θ4(z|τ) is an analytic function of τ for τ>0, with a natural boundary τ=0, and correspondingly, an analytic function of q for |q|<1 with a natural boundary |q|=1.

The four points (0,π,π+τπ,τπ) are the vertices of the fundamental parallelogram in the z-plane; see Figure 20.2.1. The points

20.2.5 zm,n=(m+nτ)π,

are the lattice points. The theta functions are quasi-periodic on the lattice:

20.2.6 θ1(z+(m+nτ)π|τ) =(1)m+nqn2e2inzθ1(z|τ),
20.2.7 θ2(z+(m+nτ)π|τ) =(1)mqn2e2inzθ2(z|τ),
20.2.8 θ3(z+(m+nτ)π|τ) =qn2e2inzθ3(z|τ),
20.2.9 θ4(z+(m+nτ)π|τ) =(1)nqn2e2inzθ4(z|τ).
See accompanying text See accompanying text
Figure 20.2.1: z-plane. Fundamental parallelogram. Left-hand diagram is the rectangular case (τ purely imaginary); right-hand diagram is the general case. zeros of θ1(z|τ), zeros of θ2(z|τ), zeros of θ3(z|τ), zeros of θ4(z|τ). Magnify

§20.2(iii) Translation of the Argument by Half-Periods


20.2.10 MM(z|τ)=eiz+(iπτ/4),
20.2.11 θ1(z|τ) =θ2(z+12π|τ)=iMθ4(z+12πτ|τ)=iMθ3(z+12π+12πτ|τ),
20.2.12 θ2(z|τ) =θ1(z+12π|τ)=Mθ3(z+12πτ|τ)=Mθ4(z+12π+12πτ|τ),
20.2.13 θ3(z|τ) =θ4(z+12π|τ)=Mθ2(z+12πτ|τ)=Mθ1(z+12π+12πτ|τ),
20.2.14 θ4(z|τ) =θ3(z+12π|τ)=iMθ1(z+12πτ|τ)=iMθ2(z+12π+12πτ|τ).

§20.2(iv) z-Zeros

For m,n, the z-zeros of θj(z|τ), j=1,2,3,4, are (m+nτ)π, (m+12+nτ)π, (m+12+(n+12)τ)π, (m+(n+12)τ)π respectively.