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Lattice transformations for Jacobi theta functions

Table of contents: Reflection symmetry - Basic modular transformations - General shifts - Helper functions for general modular transformations - General modular transformations - Half parameter - Double parameter - Quadruple parameter

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

Reflection symmetry

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

Basic modular transformations

Single shift

6b2078
θ1 ⁣(z,τ+1)=eπi/4θ1 ⁣(z,τ)\theta_{1}\!\left(z , \tau + 1\right) = {e}^{\pi i / 4} \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z,τ+1)=eπi/4θ2 ⁣(z,τ)\theta_{2}\!\left(z , \tau + 1\right) = {e}^{\pi i / 4} \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z,τ+1)=θ4 ⁣(z,τ)\theta_{3}\!\left(z , \tau + 1\right) = \theta_{4}\!\left(z , \tau\right)
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θ4 ⁣(z,τ+1)=θ3 ⁣(z,τ)\theta_{4}\!\left(z , \tau + 1\right) = \theta_{3}\!\left(z , \tau\right)

Single inversion

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θ1 ⁣(z,1τ)=iτieπiτz2θ1 ⁣(τz,τ)\theta_{1}\!\left(z , \frac{-1}{\tau}\right) = -i \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{1}\!\left(\tau z , \tau\right)
06319a
θ2 ⁣(z,1τ)=τieπiτz2θ4 ⁣(τz,τ)\theta_{2}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{4}\!\left(\tau z , \tau\right)
c4b16c
θ3 ⁣(z,1τ)=τieπiτz2θ3 ⁣(τz,τ)\theta_{3}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{3}\!\left(\tau z , \tau\right)
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θ4 ⁣(z,1τ)=τieπiτz2θ2 ⁣(τz,τ)\theta_{4}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{2}\!\left(\tau z , \tau\right)

General shifts

1fa8e7
θ1 ⁣(z,τ+n)=eπin/4θ1 ⁣(z,τ)\theta_{1}\!\left(z , \tau + n\right) = {e}^{\pi i n / 4} \theta_{1}\!\left(z , \tau\right)
d0dfba
θ2 ⁣(z,τ+n)=eπin/4θ2 ⁣(z,τ)\theta_{2}\!\left(z , \tau + n\right) = {e}^{\pi i n / 4} \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z,τ+n)={θ3 ⁣(z,τ),n evenθ4 ⁣(z,τ),n odd\theta_{3}\!\left(z , \tau + n\right) = \begin{cases} \theta_{3}\!\left(z , \tau\right), & n \text{ even}\\\theta_{4}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}
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θ4 ⁣(z,τ+n)={θ4 ⁣(z,τ),n evenθ3 ⁣(z,τ),n odd\theta_{4}\!\left(z , \tau + n\right) = \begin{cases} \theta_{4}\!\left(z , \tau\right), & n \text{ even}\\\theta_{3}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}
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θ1 ⁣(z,τ+2n)=inθ1 ⁣(z,τ)\theta_{1}\!\left(z , \tau + 2 n\right) = {i}^{n} \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z,τ+2n)=inθ2 ⁣(z,τ)\theta_{2}\!\left(z , \tau + 2 n\right) = {i}^{n} \theta_{2}\!\left(z , \tau\right)
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θ3 ⁣(z,τ+2n)=θ3 ⁣(z,τ)\theta_{3}\!\left(z , \tau + 2 n\right) = \theta_{3}\!\left(z , \tau\right)
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θ4 ⁣(z,τ+2n)=θ4 ⁣(z,τ)\theta_{4}\!\left(z , \tau + 2 n\right) = \theta_{4}\!\left(z , \tau\right)
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θ1 ⁣(z,τ+4n)=(1)nθ1 ⁣(z,τ)\theta_{1}\!\left(z , \tau + 4 n\right) = {\left(-1\right)}^{n} \theta_{1}\!\left(z , \tau\right)
4cf228
θ2 ⁣(z,τ+4n)=(1)nθ2 ⁣(z,τ)\theta_{2}\!\left(z , \tau + 4 n\right) = {\left(-1\right)}^{n} \theta_{2}\!\left(z , \tau\right)
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θ1 ⁣(z,τ+8n)=θ1 ⁣(z,τ)\theta_{1}\!\left(z , \tau + 8 n\right) = \theta_{1}\!\left(z , \tau\right)
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θ2 ⁣(z,τ+8n)=θ2 ⁣(z,τ)\theta_{2}\!\left(z , \tau + 8 n\right) = \theta_{2}\!\left(z , \tau\right)

Helper functions for general modular transformations

Index permutations

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Symbol: JacobiThetaPermutation Sj ⁣(a,b,c,d)S_{j}\!\left(a, b, c, d\right) Index permutation in modular transformation of Jacobi theta functions
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Sj ⁣(a,b,c,d)={1,j=1T ⁣(c,d),j=2T ⁣(a+c,b+d),j=3T ⁣(a,b),j=4   where T ⁣(m,n)={1,(m,n)(0,0)(mod2)2,(m,n)(0,1)(mod2)4,(m,n)(1,0)(mod2)3,(m,n)(1,1)(mod2)S_{j}\!\left(a, b, c, d\right) = \begin{cases} 1, & j = 1\\T\!\left(c, d\right), & j = 2\\T\!\left(a + c, b + d\right), & j = 3\\T\!\left(a, b\right), & j = 4\\ \end{cases}\; \text{ where } T\!\left(m, n\right) = \begin{cases} 1, & \left(m, n\right) \equiv \left(0, 0\right) \pmod {2}\\2, & \left(m, n\right) \equiv \left(0, 1\right) \pmod {2}\\4, & \left(m, n\right) \equiv \left(1, 0\right) \pmod {2}\\3, & \left(m, n\right) \equiv \left(1, 1\right) \pmod {2}\\ \end{cases}

Roots of unity

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Symbol: JacobiThetaEpsilon εj ⁣(a,b,c,d)\varepsilon_{j}\!\left(a, b, c, d\right) Root of unity in modular transformation of Jacobi theta functions
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ε1 ⁣(a,b,c,d)={(cd)exp ⁣(πi4[d(bc1)+2]),c even(dc)exp ⁣(πi4[c(a+d+1)3]),c odd\varepsilon_{1}\!\left(a, b, c, d\right) = \begin{cases} \left( \frac{c}{d} \right) \exp\!\left(\frac{\pi i}{4} \left[d \left(b - c - 1\right) + 2\right]\right), & c \text{ even}\\\left( \frac{d}{c} \right) \exp\!\left(\frac{\pi i}{4} \left[c \left(a + d + 1\right) - 3\right]\right), & c \text{ odd}\\ \end{cases}
3c56c7
εj ⁣(a,b,c,d)=1ε1 ⁣(d,b,c,a){exp ⁣(πi4[(c2)d2+2(1c)δd+1]),j=2exp ⁣(πi4[(a+c2)(b+d)3+2(1ac)δb+d+1]),j=3exp ⁣(πi4[(a2)b4+2(1a)δb+1]),j=4   where δn=nmod2\varepsilon_{j}\!\left(a, b, c, d\right) = \frac{1}{\varepsilon_{1}\!\left(-d, b, c, -a\right)} \begin{cases} \exp\!\left(\frac{\pi i}{4} \left[\left(c - 2\right) d - 2 + 2 \left(1 - c\right) {\delta}_{d + 1}\right]\right), & j = 2\\\exp\!\left(\frac{\pi i}{4} \left[\left(a + c - 2\right) \left(b + d\right) - 3 + 2 \left(1 - a - c\right) {\delta}_{b + d + 1}\right]\right), & j = 3\\\exp\!\left(\frac{\pi i}{4} \left[\left(a - 2\right) b - 4 + 2 \left(1 - a\right) {\delta}_{b + 1}\right]\right), & j = 4\\ \end{cases}\; \text{ where } {\delta}_{n} = n \bmod 2
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(εj ⁣(a,b,c,d))4=(1)n   where n={a(b+d)+cd,j=1a(b+d),j=2ad,j=3d(a+c),j=4{\left(\varepsilon_{j}\!\left(a, b, c, d\right)\right)}^{4} = {\left(-1\right)}^{n}\; \text{ where } n = \begin{cases} a \left(b + d\right) + c d, & j = 1\\a \left(b + d\right), & j = 2\\a d, & j = 3\\d \left(a + c\right), & j = 4\\ \end{cases}
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(εj ⁣(a,b,c,d))8=1{\left(\varepsilon_{j}\!\left(a, b, c, d\right)\right)}^{8} = 1

General modular transformations

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θj ⁣(z,aτ+bcτ+d)=εj ⁣(a,b,c,d)vieπicvz2θSj ⁣(a,b,c,d) ⁣(vz,τ)   where v=cτ+d\theta_{j}\!\left(z , \frac{a \tau + b}{c \tau + d}\right) = \varepsilon_{j}\!\left(a, b, c, d\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(a, b, c, d\right)}\!\left(v z , \tau\right)\; \text{ where } v = c \tau + d
100d3c
θj ⁣(z,τ)=εj ⁣(d,b,c,a)vieπicvz2θSj ⁣(d,b,c,a) ⁣(vz,aτ+bcτ+d)   where v=1cτ+d\theta_{j}\!\left(z , \tau\right) = \varepsilon_{j}\!\left(-d, b, c, -a\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(-d, b, c, -a\right)}\!\left(v z , \frac{a \tau + b}{c \tau + d}\right)\; \text{ where } v = -\frac{1}{c \tau + d}

Half parameter

Theta constants

59fd23
θ22 ⁣(0,τ2)=2θ2 ⁣(0,τ)θ3 ⁣(0,τ)\theta_{2}^{2}\!\left(0, \frac{\tau}{2}\right) = 2 \theta_{2}\!\left(0 , \tau\right) \theta_{3}\!\left(0 , \tau\right)
de7918
θ32 ⁣(0,τ2)=θ22 ⁣(0,τ)+θ32 ⁣(0,τ)\theta_{3}^{2}\!\left(0, \frac{\tau}{2}\right) = \theta_{2}^{2}\!\left(0, \tau\right) + \theta_{3}^{2}\!\left(0, \tau\right)
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θ3 ⁣(0,τ2)θ4 ⁣(0,τ2)=θ42 ⁣(0,τ)\theta_{3}\!\left(0 , \frac{\tau}{2}\right) \theta_{4}\!\left(0 , \frac{\tau}{2}\right) = \theta_{4}^{2}\!\left(0, \tau\right)
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θ42 ⁣(0,τ2)=θ32 ⁣(0,τ)θ22 ⁣(0,τ)\theta_{4}^{2}\!\left(0, \frac{\tau}{2}\right) = \theta_{3}^{2}\!\left(0, \tau\right) - \theta_{2}^{2}\!\left(0, \tau\right)
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θ1 ⁣(0,τ2)θ2 ⁣(0,τ2)=2θ1 ⁣(0,τ)θ4 ⁣(0,τ)\theta'_{1}\!\left(0 , \frac{\tau}{2}\right) \theta_{2}\!\left(0 , \frac{\tau}{2}\right) = 2 \theta'_{1}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)

General arguments

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θ1 ⁣(z,τ2)=2θ1 ⁣(z,τ)θ4 ⁣(z,τ)θ2 ⁣(0,τ2)\theta_{1}\!\left(z , \frac{\tau}{2}\right) = \frac{2 \theta_{1}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}\!\left(0 , \frac{\tau}{2}\right)}
a9cdda
θ2 ⁣(z,τ2)=2θ2 ⁣(z,τ)θ3 ⁣(z,τ)θ2 ⁣(0,τ2)\theta_{2}\!\left(z , \frac{\tau}{2}\right) = \frac{2 \theta_{2}\!\left(z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{2}\!\left(0 , \frac{\tau}{2}\right)}
e6d333
θ3 ⁣(z,τ2)=θ42 ⁣(z,τ)θ12 ⁣(z,τ)θ4 ⁣(0,τ2)\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{4}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \frac{\tau}{2}\right)}
69b32e
θ3 ⁣(z,τ2)=θ22 ⁣(z,τ)+θ32 ⁣(z,τ)θ3 ⁣(0,τ2)\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \frac{\tau}{2}\right)}
c92a6f
θ4 ⁣(z,τ2)=θ42 ⁣(z,τ)+θ12 ⁣(z,τ)θ3 ⁣(0,τ2)\theta_{4}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{4}^{2}\!\left(z, \tau\right) + \theta_{1}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \frac{\tau}{2}\right)}
95e508
θ4 ⁣(z,τ2)=θ32 ⁣(z,τ)θ22 ⁣(z,τ)θ4 ⁣(0,τ2)\theta_{4}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) - \theta_{2}^{2}\!\left(z, \tau\right)}{\theta_{4}\!\left(0 , \frac{\tau}{2}\right)}

Double parameter

Theta constants

9a2054
2θ2 ⁣(0,2τ)θ3 ⁣(0,2τ)=θ22 ⁣(0,τ)2 \theta_{2}\!\left(0 , 2 \tau\right) \theta_{3}\!\left(0 , 2 \tau\right) = \theta_{2}^{2}\!\left(0, \tau\right)
21c2f7
2θ22 ⁣(0,2τ)=θ32 ⁣(0,τ)θ42 ⁣(0,τ)2 \theta_{2}^{2}\!\left(0, 2 \tau\right) = \theta_{3}^{2}\!\left(0, \tau\right) - \theta_{4}^{2}\!\left(0, \tau\right)
c3d8c2
2θ32 ⁣(0,2τ)=θ32 ⁣(0,τ)+θ42 ⁣(0,τ)2 \theta_{3}^{2}\!\left(0, 2 \tau\right) = \theta_{3}^{2}\!\left(0, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right)
f14471
θ42 ⁣(0,2τ)=θ3 ⁣(0,τ)θ4 ⁣(0,τ)\theta_{4}^{2}\!\left(0, 2 \tau\right) = \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)
46f244
2θ1 ⁣(0,2τ)θ4 ⁣(0,2τ)=θ1 ⁣(0,τ)θ2 ⁣(0,τ)2 \theta'_{1}\!\left(0 , 2 \tau\right) \theta_{4}\!\left(0 , 2 \tau\right) = \theta'_{1}\!\left(0 , \tau\right) \theta_{2}\!\left(0 , \tau\right)

General arguments

e13fe9
θ1 ⁣(2z,2τ)=θ1 ⁣(z,τ)θ2 ⁣(z,τ)θ4 ⁣(0,2τ)\theta_{1}\!\left(2 z , 2 \tau\right) = \frac{\theta_{1}\!\left(z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{4}\!\left(0 , 2 \tau\right)}
7137a2
θ2 ⁣(2z,2τ)=θ22 ⁣(z,τ)θ12 ⁣(z,τ)2θ3 ⁣(0,2τ)\theta_{2}\!\left(2 z , 2 \tau\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) - \theta_{1}^{2}\!\left(z, \tau\right)}{2 \theta_{3}\!\left(0 , 2 \tau\right)}
db4e29
θ2 ⁣(2z,2τ)=θ32 ⁣(z,τ)θ42 ⁣(z,τ)2θ2 ⁣(0,2τ)\theta_{2}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) - \theta_{4}^{2}\!\left(z, \tau\right)}{2 \theta_{2}\!\left(0 , 2 \tau\right)}
f12569
θ2 ⁣(2z,2τ)=θ1 ⁣(14z,τ)θ1 ⁣(14+z,τ)θ4 ⁣(0,2τ)\theta_{2}\!\left(2 z , 2 \tau\right) = \frac{\theta_{1}\!\left(\frac{1}{4} - z , \tau\right) \theta_{1}\!\left(\frac{1}{4} + z , \tau\right)}{\theta_{4}\!\left(0 , 2 \tau\right)}
3479be
θ3 ⁣(2z,2τ)=θ12 ⁣(z,τ)+θ22 ⁣(z,τ)2θ2 ⁣(0,2τ)\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right)}{2 \theta_{2}\!\left(0 , 2 \tau\right)}
7e0002
θ3 ⁣(2z,2τ)=θ32 ⁣(z,τ)+θ42 ⁣(z,τ)2θ3 ⁣(0,2τ)\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}^{2}\!\left(z, \tau\right) + \theta_{4}^{2}\!\left(z, \tau\right)}{2 \theta_{3}\!\left(0 , 2 \tau\right)}
0a9ec2
θ3 ⁣(2z,2τ)=θ3 ⁣(14z,τ)θ3 ⁣(14+z,τ)θ4 ⁣(0,2τ)\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}\!\left(\frac{1}{4} - z , \tau\right) \theta_{3}\!\left(\frac{1}{4} + z , \tau\right)}{\theta_{4}\!\left(0 , 2 \tau\right)}
686ce0
θ4 ⁣(2z,2τ)=θ3 ⁣(z,τ)θ4 ⁣(z,τ)θ4 ⁣(0,2τ)\theta_{4}\!\left(2 z , 2 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{4}\!\left(0 , 2 \tau\right)}

Quadruple parameter

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θ2 ⁣(2z,4τ)=θ3 ⁣(z,τ)θ4 ⁣(z,τ)2\theta_{2}\!\left(2 z , 4 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) - \theta_{4}\!\left(z , \tau\right)}{2}
53fef4
θ3 ⁣(2z,4τ)=θ3 ⁣(z,τ)+θ4 ⁣(z,τ)2\theta_{3}\!\left(2 z , 4 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) + \theta_{4}\!\left(z , \tau\right)}{2}
27b169
θ1 ⁣(4z,4τ)=θ1 ⁣(z,τ)θ1 ⁣(14z,τ)θ1 ⁣(14+z,τ)θ2 ⁣(z,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ3 ⁣(14,τ)\theta_{1}\!\left(4 z , 4 \tau\right) = \frac{\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(\frac{1}{4} - z , \tau\right) \theta_{1}\!\left(\frac{1}{4} + z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}
a255e1
θ2 ⁣(4z,4τ)=θ2 ⁣(18z,τ)θ2 ⁣(18+z,τ)θ2 ⁣(38z,τ)θ2 ⁣(38+z,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ3 ⁣(14,τ)\theta_{2}\!\left(4 z , 4 \tau\right) = \frac{\theta_{2}\!\left(\frac{1}{8} - z , \tau\right) \theta_{2}\!\left(\frac{1}{8} + z , \tau\right) \theta_{2}\!\left(\frac{3}{8} - z , \tau\right) \theta_{2}\!\left(\frac{3}{8} + z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}
0096a8
θ3 ⁣(4z,4τ)=θ3 ⁣(18z,τ)θ3 ⁣(18+z,τ)θ3 ⁣(38z,τ)θ3 ⁣(38+z,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ3 ⁣(14,τ)\theta_{3}\!\left(4 z , 4 \tau\right) = \frac{\theta_{3}\!\left(\frac{1}{8} - z , \tau\right) \theta_{3}\!\left(\frac{1}{8} + z , \tau\right) \theta_{3}\!\left(\frac{3}{8} - z , \tau\right) \theta_{3}\!\left(\frac{3}{8} + z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}
fc3c44
θ4 ⁣(4z,4τ)=θ4 ⁣(z,τ)θ4 ⁣(14z,τ)θ4 ⁣(14+z,τ)θ3 ⁣(z,τ)θ3 ⁣(0,τ)θ4 ⁣(0,τ)θ3 ⁣(14,τ)\theta_{4}\!\left(4 z , 4 \tau\right) = \frac{\theta_{4}\!\left(z , \tau\right) \theta_{4}\!\left(\frac{1}{4} - z , \tau\right) \theta_{4}\!\left(\frac{1}{4} + z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-19 15:10:20.037976 UTC