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20 Theta FunctionsProperties

§20.5 Infinite Products and Related Results

Contents

§20.5(i) Single Products

20.5.1 θ1(z,q)=2q1/4sinzn=1(1-q2n)(1-2q2ncos(2z)+q4n),
20.5.2 θ2(z,q)=2q1/4coszn=1(1-q2n)(1+2q2ncos(2z)+q4n),
20.5.3 θ3(z,q)=n=1(1-q2n)(1+2q2n-1cos(2z)+q4n-2),
20.5.4 θ4(z,q)=n=1(1-q2n)(1-2q2n-1cos(2z)+q4n-2).
20.5.5 θ1(z|τ)=θ1(0|τ)sinzn=1sin(nπτ+z)sin(nπτ-z)sin2(nπτ),
20.5.6 θ2(z|τ)=θ2(0|τ)coszn=1cos(nπτ+z)cos(nπτ-z)cos2(nπτ),
20.5.7 θ3(z|τ)=θ3(0|τ)n=1cos((n-12)πτ+z)cos((n-12)πτ-z)cos2((n-12)πτ),
20.5.8 θ4(z|τ)=θ4(0|τ)n=1sin((n-12)πτ+z)sin((n-12)πτ-z)sin2((n-12)πτ).

Jacobi’s Triple Product

20.5.9 θ3(πz|τ)=n=-p2nqn2=n=1(1-q2n)(1+q2n-1p2)(1+q2n-1p-2),

where p=eiπz, q=eiπτ.

§20.5(ii) Logarithmic Derivatives

When |z|<πτ,

20.5.10 θ1(z,q)θ1(z,q)-cotz=4sin(2z)n=1q2n1-2q2ncos(2z)+q4n=4n=1q2n1-q2nsin(2nz),
20.5.11 θ2(z,q)θ2(z,q)+tanz=-4sin(2z)n=1q2n1+2q2ncos(2z)+q4n=4n=1(-1)nq2n1-q2nsin(2nz).

The left-hand sides of (20.5.10) and (20.5.11) are replaced by their limiting values when cotz or tanz are undefined.

When |z|<12πτ,

20.5.12 θ3(z,q)θ3(z,q)=-4sin(2z)n=1q2n-11+2q2n-1cos(2z)+q4n-2=4n=1(-1)nqn1-q2nsin(2nz),
20.5.13 θ4(z,q)θ4(z,q)=4sin(2z)n=1q2n-11-2q2n-1cos(2z)+q4n-2=4n=1qn1-q2nsin(2nz).

With the given conditions the infinite series in (20.5.10)–(20.5.13) converge absolutely and uniformly in compact sets in the z-plane.

§20.5(iii) Double Products

20.5.14 θ1(z|τ) =zθ1(0|τ)limNn=-NNlimMm=-M|m|+|n|0M(1+z(m+nτ)π),
20.5.15 θ2(z|τ) =θ2(0|τ)limNn=-NNlimMm=1-MM(1+z(m-12+nτ)π),
20.5.16 θ3(z|τ) =θ3(0|τ)limNn=1-NNlimMm=1-MM(1+z(m-12+(n-12)τ)π),
20.5.17 θ4(z|τ) =θ4(0|τ)limNn=1-NNlimMm=-MM(1+z(m+(n-12)τ)π).

These double products are not absolutely convergent; hence the order of the limits is important. The order shown is in accordance with the Eisenstein convention (Walker (1996, §0.3)).