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Argument transformations for Jacobi theta functions

Table of contents: Reflection symmetry - Periodicity - Quasi-periodicity - Half-period or quarter-period shifts - Theta functions represented in terms of each other - Double argument - Relations involving sums and differences of arguments

This topic lists identities for how Jacobi theta functions θj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right) transform when the argument zz is transformed. This topic mainly covers identities where the lattice parameter τ\tau is fixed. See Lattice transformations for Jacobi theta functions for identities involving transformations of τ\tau. See Jacobi theta functions for other properties of these functions.

Reflection symmetry

Even-odd symmetry

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θ1 ⁣(z,τ)=θ1 ⁣(z,τ)\theta_{1}\!\left(-z , \tau\right) = -\theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z,τ)=θ2 ⁣(z,τ)\theta_{2}\!\left(-z , \tau\right) = \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z,τ)=θ3 ⁣(z,τ)\theta_{3}\!\left(-z , \tau\right) = \theta_{3}\!\left(z , \tau\right)
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θ4 ⁣(z,τ)=θ4 ⁣(z,τ)\theta_{4}\!\left(-z , \tau\right) = \theta_{4}\!\left(z , \tau\right)

Conjugate symmetry

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θj ⁣(z,τ)=θj ⁣(z,τ)\theta_{j}\!\left(\overline{z} , \tau\right) = \overline{\theta_{j}\!\left(z , -\overline{\tau}\right)}

Periodicity

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θ1 ⁣(z+2n,τ)=θ1 ⁣(z,τ)\theta_{1}\!\left(z + 2 n , \tau\right) = \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z+2n,τ)=θ2 ⁣(z,τ)\theta_{2}\!\left(z + 2 n , \tau\right) = \theta_{2}\!\left(z , \tau\right)
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θ1 ⁣(z+n,τ)=(1)nθ1 ⁣(z,τ)\theta_{1}\!\left(z + n , \tau\right) = {\left(-1\right)}^{n} \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z+n,τ)=(1)nθ2 ⁣(z,τ)\theta_{2}\!\left(z + n , \tau\right) = {\left(-1\right)}^{n} \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z+n,τ)=θ3 ⁣(z,τ)\theta_{3}\!\left(z + n , \tau\right) = \theta_{3}\!\left(z , \tau\right)
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θ4 ⁣(z+n,τ)=θ4 ⁣(z,τ)\theta_{4}\!\left(z + n , \tau\right) = \theta_{4}\!\left(z , \tau\right)

Quasi-periodicity

Single shifts

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θ1 ⁣(z+τ,τ)=eπi(2z+τ)θ1 ⁣(z,τ)\theta_{1}\!\left(z + \tau , \tau\right) = -{e}^{-\pi i \left(2 z + \tau\right)} \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z+τ,τ)=eπi(2z+τ)θ2 ⁣(z,τ)\theta_{2}\!\left(z + \tau , \tau\right) = {e}^{-\pi i \left(2 z + \tau\right)} \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z+τ,τ)=eπi(2z+τ)θ3 ⁣(z,τ)\theta_{3}\!\left(z + \tau , \tau\right) = {e}^{-\pi i \left(2 z + \tau\right)} \theta_{3}\!\left(z , \tau\right)
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θ4 ⁣(z+τ,τ)=eπi(2z+τ)θ4 ⁣(z,τ)\theta_{4}\!\left(z + \tau , \tau\right) = -{e}^{-\pi i \left(2 z + \tau\right)} \theta_{4}\!\left(z , \tau\right)

General shifts

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θ1 ⁣(z+m+nτ,τ)=(1)m+neπi(τn2+2nz)θ1 ⁣(z,τ)\theta_{1}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z+m+nτ,τ)=(1)meπi(τn2+2nz)θ2 ⁣(z,τ)\theta_{2}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{m} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z+m+nτ,τ)=eπi(τn2+2nz)θ3 ⁣(z,τ)\theta_{3}\!\left(z + m + n \tau , \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{3}\!\left(z , \tau\right)
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θ4 ⁣(z+m+nτ,τ)=(1)neπi(τn2+2nz)θ4 ⁣(z,τ)\theta_{4}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{4}\!\left(z , \tau\right)

Half-period or quarter-period shifts

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θ1 ⁣(z+12,τ)=θ2 ⁣(z,τ)\theta_{1}\!\left(z + \frac{1}{2} , \tau\right) = \theta_{2}\!\left(z , \tau\right)
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θ2 ⁣(z+12,τ)=θ1 ⁣(z,τ)\theta_{2}\!\left(z + \frac{1}{2} , \tau\right) = -\theta_{1}\!\left(z , \tau\right)
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θ3 ⁣(z+12,τ)=θ4 ⁣(z,τ)\theta_{3}\!\left(z + \frac{1}{2} , \tau\right) = \theta_{4}\!\left(z , \tau\right)
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θ4 ⁣(z+12,τ)=θ3 ⁣(z,τ)\theta_{4}\!\left(z + \frac{1}{2} , \tau\right) = \theta_{3}\!\left(z , \tau\right)
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θ1 ⁣(z+12τ,τ)=eπi(z+τ/4)iθ4 ⁣(z,τ)\theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} i \theta_{4}\!\left(z , \tau\right)
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θ2 ⁣(z+12τ,τ)=eπi(z+τ/4)θ3 ⁣(z,τ)\theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z , \tau\right)
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θ3 ⁣(z+12τ,τ)=eπi(z+τ/4)θ2 ⁣(z,τ)\theta_{3}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z , \tau\right)
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θ4 ⁣(z+12τ,τ)=eπi(z+τ/4)iθ1 ⁣(z,τ)\theta_{4}\!\left(z + \frac{1}{2} \tau , \tau\right) = {e}^{-\pi i \left(z + \tau / 4\right)} i \theta_{1}\!\left(z , \tau\right)

Theta functions represented in terms of each other

Theta 1

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θ1 ⁣(z,τ)=θ2 ⁣(z+12,τ)\theta_{1}\!\left(z , \tau\right) = -\theta_{2}\!\left(z + \frac{1}{2} , \tau\right)
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θ1 ⁣(z,τ)=ieπi(z+τ/4)θ4 ⁣(z+12τ,τ)\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{4}\!\left(z + \frac{1}{2} \tau , \tau\right)
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θ1 ⁣(z,τ)=ieπi(z+τ/4)θ3 ⁣(z+12+12τ,τ)\theta_{1}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Theta 2

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θ2 ⁣(z,τ)=θ1 ⁣(z+12,τ)\theta_{2}\!\left(z , \tau\right) = \theta_{1}\!\left(z + \frac{1}{2} , \tau\right)
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θ2 ⁣(z,τ)=eπi(z+τ/4)θ3 ⁣(z+12τ,τ)\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{3}\!\left(z + \frac{1}{2} \tau , \tau\right)
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θ2 ⁣(z,τ)=eπi(z+τ/4)θ4 ⁣(z+12+12τ,τ)\theta_{2}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{4}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Theta 3

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θ3 ⁣(z,τ)=θ4 ⁣(z+12,τ)\theta_{3}\!\left(z , \tau\right) = \theta_{4}\!\left(z + \frac{1}{2} , \tau\right)
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θ3 ⁣(z,τ)=eπi(z+τ/4)θ2 ⁣(z+12τ,τ)\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} \tau , \tau\right)
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θ3 ⁣(z,τ)=eπi(z+τ/4)θ1 ⁣(z+12+12τ,τ)\theta_{3}\!\left(z , \tau\right) = {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Theta 4

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θ4 ⁣(z,τ)=θ3 ⁣(z+12,τ)\theta_{4}\!\left(z , \tau\right) = \theta_{3}\!\left(z + \frac{1}{2} , \tau\right)
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θ4 ⁣(z,τ)=ieπi(z+τ/4)θ1 ⁣(z+12τ,τ)\theta_{4}\!\left(z , \tau\right) = -i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{1}\!\left(z + \frac{1}{2} \tau , \tau\right)
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θ4 ⁣(z,τ)=ieπi(z+τ/4)θ2 ⁣(z+12+12τ,τ)\theta_{4}\!\left(z , \tau\right) = i {e}^{\pi i \left(z + \tau / 4\right)} \theta_{2}\!\left(z + \frac{1}{2} + \frac{1}{2} \tau , \tau\right)

Double argument

Theta 1

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θ1 ⁣(2z,τ)=2θ1 ⁣(z,τ)θ2 ⁣(z,τ)θ3 ⁣(z,τ)θ4 ⁣(z,τ)θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)\theta_{1}\!\left(2 z , \tau\right) = \frac{2 \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)}

Theta 2

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θ2 ⁣(2z,τ)=θ24 ⁣(z,τ)θ14 ⁣(z,τ)θ23 ⁣(0,τ)\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{4}\!\left(z, \tau\right) - \theta_{1}^{4}\!\left(z, \tau\right)}{\theta_{2}^{3}\!\left(0, \tau\right)}
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θ2 ⁣(2z,τ)=θ34 ⁣(z,τ)θ44 ⁣(z,τ)θ23 ⁣(0,τ)\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{4}\!\left(z, \tau\right) - \theta_{4}^{4}\!\left(z, \tau\right)}{\theta_{2}^{3}\!\left(0, \tau\right)}
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θ2 ⁣(2z,τ)=θ22 ⁣(z,τ)θ32 ⁣(z,τ)θ12 ⁣(z,τ)θ42 ⁣(z,τ)θ2 ⁣(0,τ)θ32 ⁣(0,τ)\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}
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θ2 ⁣(2z,τ)=θ22 ⁣(z,τ)θ42 ⁣(z,τ)θ12 ⁣(z,τ)θ32 ⁣(z,τ)θ2 ⁣(0,τ)θ42 ⁣(0,τ)\theta_{2}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right)}{\theta_{2}\!\left(0 , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right)}

Theta 3

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θ3 ⁣(2z,τ)=θ14 ⁣(z,τ)+θ34 ⁣(z,τ)θ33 ⁣(0,τ)\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{4}\!\left(z, \tau\right) + \theta_{3}^{4}\!\left(z, \tau\right)}{\theta_{3}^{3}\!\left(0, \tau\right)}
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θ3 ⁣(2z,τ)=θ24 ⁣(z,τ)+θ44 ⁣(z,τ)θ33 ⁣(0,τ)\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{4}\!\left(z, \tau\right) + \theta_{4}^{4}\!\left(z, \tau\right)}{\theta_{3}^{3}\!\left(0, \tau\right)}
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θ3 ⁣(2z,τ)=θ22 ⁣(z,τ)θ32 ⁣(z,τ)+θ12 ⁣(z,τ)θ42 ⁣(z,τ)θ3 ⁣(0,τ)θ22 ⁣(0,τ)\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) + \theta_{1}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right)}
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θ3 ⁣(2z,τ)=θ32 ⁣(z,τ)θ42 ⁣(z,τ)θ12 ⁣(z,τ)θ22 ⁣(z,τ)θ3 ⁣(0,τ)θ42 ⁣(0,τ)\theta_{3}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}^{2}\!\left(0, \tau\right)}

Theta 4

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θ4 ⁣(2z,τ)=θ44 ⁣(z,τ)θ14 ⁣(z,τ)θ43 ⁣(0,τ)\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{4}^{4}\!\left(z, \tau\right) - \theta_{1}^{4}\!\left(z, \tau\right)}{\theta_{4}^{3}\!\left(0, \tau\right)}
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θ4 ⁣(2z,τ)=θ34 ⁣(z,τ)θ24 ⁣(z,τ)θ43 ⁣(0,τ)\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{3}^{4}\!\left(z, \tau\right) - \theta_{2}^{4}\!\left(z, \tau\right)}{\theta_{4}^{3}\!\left(0, \tau\right)}
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θ4 ⁣(2z,τ)=θ12 ⁣(z,τ)θ32 ⁣(z,τ)+θ22 ⁣(z,τ)θ42 ⁣(z,τ)θ4 ⁣(0,τ)θ22 ⁣(0,τ)\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right)}
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θ4 ⁣(2z,τ)=θ12 ⁣(z,τ)θ22 ⁣(z,τ)+θ32 ⁣(z,τ)θ42 ⁣(z,τ)θ4 ⁣(0,τ)θ32 ⁣(0,τ)\theta_{4}\!\left(2 z , \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right)}

Relations involving sums and differences of arguments

Cross-products with two factors and double lattice parameter

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θ1 ⁣(z,τ)θ1 ⁣(w,τ)=θ3 ⁣(z+w,2τ)θ2 ⁣(zw,2τ)θ2 ⁣(z+w,2τ)θ3 ⁣(zw,2τ)\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right) - \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right)
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θ1 ⁣(z,τ)θ2 ⁣(w,τ)=θ1 ⁣(z+w,2τ)θ4 ⁣(zw,2τ)+θ4 ⁣(z+w,2τ)θ1 ⁣(zw,2τ)\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) + \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)
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θ2 ⁣(z,τ)θ2 ⁣(w,τ)=θ2 ⁣(z+w,2τ)θ3 ⁣(zw,2τ)+θ3 ⁣(z+w,2τ)θ2 ⁣(zw,2τ)\theta_{2}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) = \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) + \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)
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θ3 ⁣(z,τ)θ3 ⁣(w,τ)=θ3 ⁣(z+w,2τ)θ3 ⁣(zw,2τ)+θ2 ⁣(z+w,2τ)θ2 ⁣(zw,2τ)\theta_{3}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) + \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)
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θ3 ⁣(z,τ)θ4 ⁣(w,τ)=θ4 ⁣(z+w,2τ)θ4 ⁣(zw,2τ)θ1 ⁣(z+w,2τ)θ1 ⁣(zw,2τ)\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(w , \tau\right) = \theta_{4}\!\left(z + w , 2 \tau\right) \theta_{4}\!\left(z - w , 2 \tau\right) - \theta_{1}\!\left(z + w , 2 \tau\right) \theta_{1}\!\left(z - w , 2 \tau\right)
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θ4 ⁣(z,τ)θ4 ⁣(w,τ)=θ3 ⁣(z+w,2τ)θ3 ⁣(zw,2τ)θ2 ⁣(z+w,2τ)θ2 ⁣(zw,2τ)\theta_{4}\!\left(z , \tau\right) \theta_{4}\!\left(w , \tau\right) = \theta_{3}\!\left(z + w , 2 \tau\right) \theta_{3}\!\left(z - w , 2 \tau\right) - \theta_{2}\!\left(z + w , 2 \tau\right) \theta_{2}\!\left(z - w , 2 \tau\right)

Cross-products with four factors

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θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ1 ⁣(z+w,τ)θ2 ⁣(zw,τ)=θ1 ⁣(z,τ)θ2 ⁣(z,τ)θ3 ⁣(w,τ)θ4 ⁣(w,τ)+θ3 ⁣(z,τ)θ4 ⁣(z,τ)θ1 ⁣(w,τ)θ2 ⁣(w,τ)\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{2}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) + \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{2}\!\left(w , \tau\right)
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θ2 ⁣(0,τ)θ4 ⁣(0,τ)θ1 ⁣(z+w,τ)θ3 ⁣(zw,τ)=θ1 ⁣(z,τ)θ3 ⁣(z,τ)θ2 ⁣(w,τ)θ4 ⁣(w,τ)+θ2 ⁣(z,τ)θ4 ⁣(z,τ)θ1 ⁣(w,τ)θ3 ⁣(w,τ)\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) + \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)
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θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ1 ⁣(z+w,τ)θ4 ⁣(zw,τ)=θ1 ⁣(z,τ)θ4 ⁣(z,τ)θ2 ⁣(w,τ)θ3 ⁣(w,τ)+θ2 ⁣(z,τ)θ3 ⁣(z,τ)θ1 ⁣(w,τ)θ4 ⁣(w,τ)\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{1}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right) + \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right)
dfea7d
θ2 ⁣(0,τ)θ3 ⁣(0,τ)θ2 ⁣(z+w,τ)θ3 ⁣(zw,τ)=θ2 ⁣(z,τ)θ3 ⁣(z,τ)θ2 ⁣(w,τ)θ3 ⁣(w,τ)θ1 ⁣(z,τ)θ4 ⁣(z,τ)θ1 ⁣(w,τ)θ4 ⁣(w,τ)\theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{3}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right)
9973ef
θ2 ⁣(0,τ)θ4 ⁣(0,τ)θ2 ⁣(z+w,τ)θ4 ⁣(zw,τ)=θ2 ⁣(z,τ)θ4 ⁣(z,τ)θ2 ⁣(w,τ)θ4 ⁣(w,τ)θ1 ⁣(z,τ)θ3 ⁣(z,τ)θ1 ⁣(w,τ)θ3 ⁣(w,τ)\theta_{2}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{2}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{2}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{2}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{3}\!\left(w , \tau\right)
077394
θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ3 ⁣(z+w,τ)θ4 ⁣(zw,τ)=θ3 ⁣(z,τ)θ4 ⁣(z,τ)θ3 ⁣(w,τ)θ4 ⁣(w,τ)θ1 ⁣(z,τ)θ2 ⁣(z,τ)θ1 ⁣(w,τ)θ2 ⁣(w,τ)\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(z + w , \tau\right) \theta_{4}\!\left(z - w , \tau\right) = \theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right) \theta_{3}\!\left(w , \tau\right) \theta_{4}\!\left(w , \tau\right) - \theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right) \theta_{1}\!\left(w , \tau\right) \theta_{2}\!\left(w , \tau\right)

Cross-products of squares

45165c
θ1 ⁣(z+w,τ)θ1 ⁣(zw,τ)θ22 ⁣(0,τ)=θ12 ⁣(z,τ)θ22 ⁣(w,τ)θ22 ⁣(z,τ)θ12 ⁣(w,τ)=θ42 ⁣(z,τ)θ32 ⁣(w,τ)θ32 ⁣(z,τ)θ42 ⁣(w,τ)\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{2}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)
75cb8c
θ1 ⁣(z+w,τ)θ1 ⁣(zw,τ)θ32 ⁣(0,τ)=θ12 ⁣(z,τ)θ32 ⁣(w,τ)θ32 ⁣(z,τ)θ12 ⁣(w,τ)=θ42 ⁣(z,τ)θ22 ⁣(w,τ)θ22 ⁣(z,τ)θ42 ⁣(w,τ)\theta_{1}\!\left(z + w , \tau\right) \theta_{1}\!\left(z - w , \tau\right) \theta_{3}^{2}\!\left(0, \tau\right) = \theta_{1}^{2}\!\left(z, \tau\right) \theta_{3}^{2}\!\left(w, \tau\right) - \theta_{3}^{2}\!\left(z, \tau\right) \theta_{1}^{2}\!\left(w, \tau\right) = \theta_{4}^{2}\!\left(z, \tau\right) \theta_{2}^{2}\!\left(w, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right) \theta_{4}^{2}\!\left(w, \tau\right)