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Specific values of Jacobi theta functions

Table of contents: Values at infinity - Trivial values - Square lattice - Theta constants for non-square lattices

See Jacobi theta functions for other properties of these functions.

Values at infinity

e6b579
lim⁡τ→i∞θ1 ⁣(z,τ)=0\lim_{\tau \to i \infty} \theta_{1}\!\left(z , \tau\right) = 0
1dcf7e
lim⁡τ→i∞θ2 ⁣(z,τ)=0\lim_{\tau \to i \infty} \theta_{2}\!\left(z , \tau\right) = 0
a8ea67
lim⁡τ→i∞θ3 ⁣(z,τ)=1\lim_{\tau \to i \infty} \theta_{3}\!\left(z , \tau\right) = 1
bf747b
lim⁡τ→i∞θ4 ⁣(z,τ)=1\lim_{\tau \to i \infty} \theta_{4}\!\left(z , \tau\right) = 1

Trivial values

8f43ab
θ1 ⁣(0,τ)=0\theta_{1}\!\left(0 , \tau\right) = 0
b3c440
θ1(2r) ⁣(0,τ)=0\theta^{(2 r)}_{1}\!\left(0 , \tau\right) = 0
474c51
θ2(2r+1) ⁣(0,τ)=0\theta^{(2 r + 1)}_{2}\!\left(0 , \tau\right) = 0
d4e418
θ3(2r+1) ⁣(0,τ)=0\theta^{(2 r + 1)}_{3}\!\left(0 , \tau\right) = 0
055b0a
θ4(2r+1) ⁣(0,τ)=0\theta^{(2 r + 1)}_{4}\!\left(0 , \tau\right) = 0

Square lattice

Theta constants for the square lattice

d15f11
θ3 ⁣(0,i)=π1/4Γ ⁣(34)\theta_{3}\!\left(0 , i\right) = \frac{{\pi}^{1 / 4}}{\Gamma\!\left(\frac{3}{4}\right)}
1403b5
θ3 ⁣(0,i)=Γ ⁣(14)2π3/4\theta_{3}\!\left(0 , i\right) = \frac{\Gamma\!\left(\frac{1}{4}\right)}{\sqrt{2} {\pi}^{3 / 4}}
0d4608
θ3 ⁣(0,i)∉Q‾\theta_{3}\!\left(0 , i\right) \notin \overline{\mathbb{Q}}
7d7c65
θ2 ⁣(0,i)=θ4 ⁣(0,i)=[2−1/4]θ3 ⁣(0,i)\theta_{2}\!\left(0 , i\right) = \theta_{4}\!\left(0 , i\right) = \left[{2}^{-1 / 4}\right] \theta_{3}\!\left(0 , i\right)
8697b8
θ3 ⁣(0,i)∈[1.0864348112133080145753161215102234570702057072452±1.89⋅10−50]\theta_{3}\!\left(0 , i\right) \in \left[1.0864348112133080145753161215102234570702057072452 \pm 1.89 \cdot 10^{-50}\right]
66df95
θ4 ⁣(0,i)∈[0.91357913815611682140724259340122208970196391639347±9.67⋅10−52]\theta_{4}\!\left(0 , i\right) \in \left[0.91357913815611682140724259340122208970196391639347 \pm 9.67 \cdot 10^{-52}\right]

Values for simple rational arguments

2f3ed3
θ1 ⁣(n4,i)={0,n≡0(mod4)(−1)⌊n/4⌋θ4 ⁣(0,i),n≡2(mod4)(−1)⌊n/4⌋[2−7/162−1(2+1)1/4]θ3 ⁣(0,i),otherwise\theta_{1}\!\left(\frac{n}{4} , i\right) = \begin{cases} 0, & n \equiv 0 \pmod {4}\\{\left(-1\right)}^{\left\lfloor n / 4 \right\rfloor} \theta_{4}\!\left(0 , i\right), & n \equiv 2 \pmod {4}\\{\left(-1\right)}^{\left\lfloor n / 4 \right\rfloor} \left[{2}^{-7 / 16} \sqrt{\sqrt{2} - 1} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}
dd5f43
θ2 ⁣(n4,i)={(−1)⌊(n+1)/4⌋θ4 ⁣(0,i),n≡0(mod4)0,n≡2(mod4)(−1)⌊(n+1)/4⌋[2−7/162−1(2+1)1/4]θ3 ⁣(0,i),otherwise\theta_{2}\!\left(\frac{n}{4} , i\right) = \begin{cases} {\left(-1\right)}^{\left\lfloor \left( n + 1 \right) / 4 \right\rfloor} \theta_{4}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\0, & n \equiv 2 \pmod {4}\\{\left(-1\right)}^{\left\lfloor \left( n + 1 \right) / 4 \right\rfloor} \left[{2}^{-7 / 16} \sqrt{\sqrt{2} - 1} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}
3fb309
θ3 ⁣(n4,i)={θ3 ⁣(0,i),n≡0(mod4)θ4 ⁣(0,i),n≡2(mod4)[2−7/16(2+1)1/4]θ3 ⁣(0,i),otherwise\theta_{3}\!\left(\frac{n}{4} , i\right) = \begin{cases} \theta_{3}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\\theta_{4}\!\left(0 , i\right), & n \equiv 2 \pmod {4}\\\left[{2}^{-7 / 16} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}
8c4ab4
θ4 ⁣(n4,i)={θ4 ⁣(0,i),n≡0(mod4)θ3 ⁣(0,i),n≡2(mod4)[2−7/16(2+1)1/4]θ3 ⁣(0,i),otherwise\theta_{4}\!\left(\frac{n}{4} , i\right) = \begin{cases} \theta_{4}\!\left(0 , i\right), & n \equiv 0 \pmod {4}\\\theta_{3}\!\left(0 , i\right), & n \equiv 2 \pmod {4}\\\left[{2}^{-7 / 16} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right), & \text{otherwise}\\ \end{cases}

Theta constants for non-square lattices

Conversion from index 2 and 4 to index 3

47f4ba
θ2 ⁣(0,yi)=1yθ3 ⁣(0,1+iy)\theta_{2}\!\left(0 , y i\right) = \frac{1}{\sqrt{y}} \theta_{3}\!\left(0 , 1 + \frac{i}{y}\right)
e2288d
θ2 ⁣(0,1+yi)=1+i2yθ3 ⁣(0,1+iy)\theta_{2}\!\left(0 , 1 + y i\right) = \frac{1 + i}{\sqrt{2 y}} \theta_{3}\!\left(0 , 1 + \frac{i}{y}\right)
cf7ee3
θ4 ⁣(0,yi)=θ3 ⁣(0,1+yi)\theta_{4}\!\left(0 , y i\right) = \theta_{3}\!\left(0 , 1 + y i\right)
81550a
θ4 ⁣(0,1+yi)=θ3 ⁣(0,yi)\theta_{4}\!\left(0 , 1 + y i\right) = \theta_{3}\!\left(0 , y i\right)

Algebraic ratios for real part 0

4256f0
θ3 ⁣(0,i2)=[2+12⋅21/4]θ3 ⁣(0,i)\theta_{3}\!\left(0 , \frac{i}{2}\right) = \left[\sqrt{\frac{\sqrt{2} + 1}{2}} \cdot {2}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right)
52302f
θ3 ⁣(0,i3)=[(23+3)1/4]θ3 ⁣(0,i)\theta_{3}\!\left(0 , \frac{i}{3}\right) = \left[{\left(2 \sqrt{3} + 3\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right)
7f9273
θ3 ⁣(0,i4)=[1+2−1/41+22+12⋅21/2]θ3 ⁣(0,i)\theta_{3}\!\left(0 , \frac{i}{4}\right) = \left[\frac{1 + {2}^{-1 / 4}}{\sqrt{1 + \sqrt{2}}} \sqrt{\frac{\sqrt{2} + 1}{2}} \cdot {2}^{1 / 2}\right] \theta_{3}\!\left(0 , i\right)
cf3c8e
θ3 ⁣(0,2i)=[2+22]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 2 i\right) = \left[\frac{\sqrt{\sqrt{2} + 2}}{2}\right] \theta_{3}\!\left(0 , i\right)
f12e20
θ3 ⁣(0,3i)=[3+121/4⋅33/8]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 3 i\right) = \left[\frac{\sqrt{\sqrt{3} + 1}}{{2}^{1 / 4} \cdot {3}^{3 / 8}}\right] \theta_{3}\!\left(0 , i\right)
95e9e4
θ3 ⁣(0,4i)=[1+2−1/42]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 4 i\right) = \left[\frac{1 + {2}^{-1 / 4}}{2}\right] \theta_{3}\!\left(0 , i\right)
483e7e
θ3 ⁣(0,5i)=[155−10]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 5 i\right) = \left[\frac{1}{\sqrt{5 \sqrt{5} - 10}}\right] \theta_{3}\!\left(0 , i\right)
cb6c9c
θ3 ⁣(0,5i)=[5+2553/4]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 5 i\right) = \left[\frac{\sqrt{5 + 2 \sqrt{5}}}{{5}^{3 / 4}}\right] \theta_{3}\!\left(0 , i\right)
669765
θ3 ⁣(0,6i)=[(−4+32+35/4+23−33/4+22(33/4))1/32(33/8)((2−1)(3−1))1/6]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 6 i\right) = \left[\frac{{\left(-4 + 3 \sqrt{2} + {3}^{5 / 4} + 2 \sqrt{3} - {3}^{3 / 4} + 2 \sqrt{2} \left({3}^{3 / 4}\right)\right)}^{1 / 3}}{2 \left({3}^{3 / 8}\right) {\left(\left(\sqrt{2} - 1\right) \left(\sqrt{3} - 1\right)\right)}^{1 / 6}}\right] \theta_{3}\!\left(0 , i\right)
72f583
θ3 ⁣(0,7i)=[13+7+7+3714(28)1/8]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 7 i\right) = \left[\sqrt{\frac{\sqrt{13 + \sqrt{7}} + \sqrt{7 + 3 \sqrt{7}}}{14} {\left(28\right)}^{1 / 8}}\right] \theta_{3}\!\left(0 , i\right)
8356db
θ3 ⁣(0,9i)=[1+(2(3+1))1/33]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 9 i\right) = \left[\frac{1 + {\left(2 \left(\sqrt{3} + 1\right)\right)}^{1 / 3}}{3}\right] \theta_{3}\!\left(0 , i\right)
6ade92
θ3 ⁣(0,45i)=[3+5+(3+5+601/4)(2+3)1/3310+105]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 45 i\right) = \left[\frac{3 + \sqrt{5} + \left(\sqrt{3} + \sqrt{5} + {60}^{1 / 4}\right) {\left(2 + \sqrt{3}\right)}^{1 / 3}}{3 \sqrt{10 + 10 \sqrt{5}}}\right] \theta_{3}\!\left(0 , i\right)

Algebraic ratios for real part 1

4c8873
θ3 ⁣(0,1+i)=[2−1/4]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + i\right) = \left[{2}^{-1 / 4}\right] \theta_{3}\!\left(0 , i\right)
324483
θ3 ⁣(0,1+i2)=[(2−1)2/3(4+32)1/1227/24]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + \frac{i}{2}\right) = \left[\frac{{\left(\sqrt{2} - 1\right)}^{2 / 3} {\left(4 + 3 \sqrt{2}\right)}^{1 / 12}}{{2}^{7 / 24}}\right] \theta_{3}\!\left(0 , i\right)
b58070
θ3 ⁣(0,1+2i)=[2−1/8]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + 2 i\right) = \left[{2}^{-1 / 8}\right] \theta_{3}\!\left(0 , i\right)
6cbce8
θ3 ⁣(0,1+4i)=[2−7/16(2+1)1/4]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + 4 i\right) = \left[{2}^{-7 / 16} {\left(\sqrt{2} + 1\right)}^{1 / 4}\right] \theta_{3}\!\left(0 , i\right)
5384f3
θ3 ⁣(0,1+6i)=[(1+3+2(27)1/4)1/3211/24⋅33/8(3−1)1/6]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + 6 i\right) = \left[\frac{{\left(1 + \sqrt{3} + \sqrt{2} {\left(27\right)}^{1 / 4}\right)}^{1 / 3}}{{2}^{11 / 24} \cdot {3}^{3 / 8} {\left(\sqrt{3} - 1\right)}^{1 / 6}}\right] \theta_{3}\!\left(0 , i\right)
e2bc80
θ3 ⁣(0,1+8i)=[2−7/8(16+15⋅21/4+122+9⋅81/4)1/8]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + 8 i\right) = \left[{2}^{-7 / 8} {\left(16 + 15 \cdot {2}^{1 / 4} + 12 \sqrt{2} + 9 \cdot {8}^{1 / 4}\right)}^{1 / 8}\right] \theta_{3}\!\left(0 , i\right)
390158
θ3 ⁣(0,1+10i)=[27/8(51/4−1)55+5]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + 10 i\right) = \left[\frac{{2}^{7 / 8}}{\left({5}^{1 / 4} - 1\right) \sqrt{5 \sqrt{5} + 5}}\right] \theta_{3}\!\left(0 , i\right)
675f23
θ3 ⁣(0,1+12i)=[2−19/48⋅3−3/8(2−32+35/4+33/4)1/3(2−1)1/12(3+1)1/6(−1−3+2(33/4))1/3]θ3 ⁣(0,i)\theta_{3}\!\left(0 , 1 + 12 i\right) = \left[\frac{{2}^{-19 / 48} \cdot {3}^{-3 / 8} {\left(2 - 3 \sqrt{2} + {3}^{5 / 4} + {3}^{3 / 4}\right)}^{1 / 3}}{{\left(\sqrt{2} - 1\right)}^{1 / 12} {\left(\sqrt{3} + 1\right)}^{1 / 6} {\left(-1 - \sqrt{3} + \sqrt{2} \left({3}^{3 / 4}\right)\right)}^{1 / 3}}\right] \theta_{3}\!\left(0 , i\right)

Other values

c60033
θ3 ⁣(0,6i)=(696π3Γ ⁣(124)Γ ⁣(524)Γ ⁣(724)Γ ⁣(1124)18+122−103−76)1/4\theta_{3}\!\left(0 , \sqrt{6} i\right) = {\left(\frac{\sqrt{6}}{96 {\pi}^{3}} \frac{\Gamma\!\left(\frac{1}{24}\right) \Gamma\!\left(\frac{5}{24}\right) \Gamma\!\left(\frac{7}{24}\right) \Gamma\!\left(\frac{11}{24}\right)}{18 + 12 \sqrt{2} - 10 \sqrt{3} - 7 \sqrt{6}}\right)}^{1 / 4}
799b5e
θ3 ⁣(0,6i)=2πK ⁣((2−3)2(2−3)2)\theta_{3}\!\left(0 , \sqrt{6} i\right) = \sqrt{\frac{2}{\pi} K\!\left({\left(2 - \sqrt{3}\right)}^{2} {\left(\sqrt{2} - \sqrt{3}\right)}^{2}\right)}

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

2021-03-15 19:12:00.328586 UTC