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20 Theta FunctionsProperties

§20.4 Values at z = 0

Contents
  1. §20.4(i) Functions and First Derivatives
  2. §20.4(ii) Higher Derivatives

§20.4(i) Functions and First Derivatives

20.4.1 θ1(0,q)=θ2(0,q)=θ3(0,q)=θ4(0,q)=0,
20.4.2 θ1(0,q) =2q1/4n=1(1q2n)3=2q1/4(q2;q2)3,
20.4.3 θ2(0,q) =2q1/4n=1(1q2n)(1+q2n)2,
20.4.4 θ3(0,q) =n=1(1q2n)(1+q2n1)2,
20.4.5 θ4(0,q) =n=1(1q2n)(1q2n1)2.

Jacobi’s Identity

20.4.6 θ1(0,q)=θ2(0,q)θ3(0,q)θ4(0,q).

§20.4(ii) Higher Derivatives

20.4.7 θ1′′(0,q)=θ2′′′(0,q)=θ3′′′(0,q)=θ4′′′(0,q)=0.
20.4.8 θ1′′′(0,q)θ1(0,q) =1+24n=1q2n(1q2n)2.
20.4.9 θ2′′(0,q)θ2(0,q) =18n=1q2n(1+q2n)2,
20.4.10 θ3′′(0,q)θ3(0,q) =8n=1q2n1(1+q2n1)2,
20.4.11 θ4′′(0,q)θ4(0,q) =8n=1q2n1(1q2n1)2.
20.4.12 θ1′′′(0,q)θ1(0,q)=θ2′′(0,q)θ2(0,q)+θ3′′(0,q)θ3(0,q)+θ4′′(0,q)θ4(0,q).