12.2. Modular transformations of the i( ) functions. Let us denote by i(z,) the theta functions of...

12.2. Modular transformations of the θi(τ ) functions.

Let us denote by θi(z,τ) the theta functions of Jacobi, with θ1(τ ) = 0, while the others have the infinite series and product representations

θ2(τ ) =

n∈Z q(n+1/2)2/2 = 2q1/8

∞



n=1

(1−qn)(1+qn)2

θ3(τ ) =

n∈Z qn2/2 =

∞



n=1

(1−qn)(1+qn+1/2)2

θ4(τ ) =

n∈Z

(−1)n qn2/2 =

∞



n=1

(1−qn)(1−qn−1/2)2.

Use these expressions and the Poisson resummation formula (12.2.55) to prove

θ2(τ +1) = eiπ/4 θ2(τ ), θ2(−1/τ ) =

√

−iτ θ4(τ )

θ3(τ +1) = θ4(τ ), θ3(−1/τ ) =

√

−iτ θ3(τ )

θ4(τ +1) = θ3(τ ), θ4(−1/τ ) =

√

−iτ θ2(τ ).