Digital Library of Mathematical Functions
About the Project
NIST
20 Theta FunctionsProperties

§20.4 Values at z = 0

Contents

§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(1-q2n)3,
20.4.3 θ2(0,q) =2q1/4n=1(1-q2n)(1+q2n)2,
20.4.4 θ3(0,q) =n=1(1-q2n)(1+q2n-1)2,
20.4.5 θ4(0,q) =n=1(1-q2n)(1-q2n-1)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(1-q2n)2.
20.4.9 θ2′′(0,q)θ2(0,q) =-1-8n=1q2n(1+q2n)2,
20.4.10 θ3′′(0,q)θ3(0,q) =-8n=1q2n-1(1+q2n-1)2,
20.4.11 θ4′′(0,q)θ4(0,q) =8n=1q2n-1(1-q2n-1)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).