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20 Theta FunctionsProperties

§20.6 Power Series


20.6.1 |πz|<min|zm,n|,

where zm,n is given by (20.2.5) and the minimum is for m,n, except m=n=0. Then

20.6.2 θ1(πz|τ) =πzθ1(0|τ)exp(j=112jδ2j(τ)z2j),
20.6.3 θ2(πz|τ) =θ2(0|τ)exp(j=112jα2j(τ)z2j),
20.6.4 θ3(πz|τ) =θ3(0|τ)exp(j=112jβ2j(τ)z2j),
20.6.5 θ4(πz|τ) =θ4(0|τ)exp(j=112jγ2j(τ)z2j).

Here the coefficients are given by

20.6.6 δ2j(τ) =n=m=|m|+|n|0(m+nτ)2j,
20.6.7 α2j(τ) =n=m=(m12+nτ)2j,
20.6.8 β2j(τ) =n=m=(m12+(n12)τ)2j,
20.6.9 γ2j(τ) =n=m=(m+(n12)τ)2j,

and satisfy

20.6.10 α2j(τ) =22jδ2j(2τ)δ2j(τ),
β2j(τ) =22jγ2j(2τ)γ2j(τ).

In the double series the order of summation is important only when j=1. For further information on δ2j see §23.9: since the double sums in (20.6.6) and (23.9.1) are the same, we have δ2n=cn/(2n1) when n2.