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20 Theta FunctionsProperties

§20.5 Infinite Products and Related Results

Contents
  1. §20.5(i) Single Products
  2. §20.5(ii) Logarithmic Derivatives
  3. §20.5(iii) Double Products

§20.5(i) Single Products

20.5.1 θ1(z,q)=2q1/4sinzn=1(1q2n)(12q2ncos(2z)+q4n),
20.5.2 θ2(z,q)=2q1/4coszn=1(1q2n)(1+2q2ncos(2z)+q4n),
20.5.3 θ3(z,q)=n=1(1q2n)(1+2q2n1cos(2z)+q4n2),
20.5.4 θ4(z,q)=n=1(1q2n)(12q2n1cos(2z)+q4n2).
20.5.5 θ1(z|τ)=θ1(0|τ)sinzn=1sin(nπτ+z)sin(nπτz)sin2(nπτ),
20.5.6 θ2(z|τ)=θ2(0|τ)coszn=1cos(nπτ+z)cos(nπτz)cos2(nπτ),
20.5.7 θ3(z|τ)=θ3(0|τ)n=1cos((n12)πτ+z)cos((n12)πτz)cos2((n12)πτ),
20.5.8 θ4(z|τ)=θ4(0|τ)n=1sin((n12)πτ+z)sin((n12)πτz)sin2((n12)πτ).

Jacobi’s Triple Product

20.5.9 θ3(πz|τ)=n=p2nqn2=n=1(1q2n)(1+q2n1p2)(1+q2n1p2),

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=1q2n12q2ncos(2z)+q4n=4n=1q2n1q2nsin(2nz),
20.5.11 θ2(z,q)θ2(z,q)+tanz=4sin(2z)n=1q2n1+2q2ncos(2z)+q4n=4n=1(1)nq2n1q2nsin(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=1q2n11+2q2n1cos(2z)+q4n2=4n=1(1)nqn1q2nsin(2nz),
20.5.13 θ4(z,q)θ4(z,q)=4sin(2z)n=1q2n112q2n1cos(2z)+q4n2=4n=1qn1q2nsin(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=1MM(1+z(m12+nτ)π),
20.5.16 θ3(z|τ) =θ3(0|τ)limNn=1NNlimMm=1MM(1+z(m12+(n12)τ)π),
20.5.17 θ4(z|τ) =θ4(0|τ)limNn=1NNlimMm=MM(1+z(m+(n12)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)).