Fungrim home page

Integrals of Jacobi theta functions

Table of contents: Laplace transforms - Mellin transforms - Constant definite integrals - Periodic integrals

This topic lists identities involving integrals of Jacobi theta functions θj ⁣(z,τ)\theta_{j}\!\left(z , \tau\right). See the topic Jacobi theta functions for other properties of these functions.

Laplace transforms

Laplace transforms

8a857c
0eatθ1 ⁣(x,ibt)dt=πabsinh ⁣(2xπab)cosh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{1}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
74be8f
0eatθ2 ⁣(x,ibt)dt=πabsinh ⁣((2x1)πab)cosh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(x , i b t\right) \, dt = -\sqrt{\frac{\pi}{a b}} \frac{\sinh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
026e44
0eatθ3 ⁣(x,ibt)dt=πabcosh ⁣((2x1)πab)sinh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
a46f94
0eatθ4 ⁣(x,ibt)dt=πabcosh ⁣(2xπab)sinh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta_{4}\!\left(x , i b t\right) \, dt = \sqrt{\frac{\pi}{a b}} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}

Laplace transforms of derivatives

321538
0eatθ1 ⁣(x,ibt)dt=2πbcosh ⁣(2xπab)cosh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta'_{1}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\cosh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
f5a15a
0eatθ2 ⁣(x,ibt)dt=2πbcosh ⁣((2x1)πab)cosh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta'_{2}\!\left(x , i b t\right) \, dt = -\frac{2 \pi}{b} \frac{\cosh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\cosh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
64c188
0eatθ3 ⁣(x,ibt)dt=2πbsinh ⁣((2x1)πab)sinh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta'_{3}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\sinh\!\left(\left(2 x - 1\right) \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}
5c054e
0eatθ4 ⁣(x,ibt)dt=2πbsinh ⁣(2xπab)sinh ⁣(πab)\int_{0}^{\infty} {e}^{-a t} \theta'_{4}\!\left(x , i b t\right) \, dt = \frac{2 \pi}{b} \frac{\sinh\!\left(2 x \sqrt{\frac{\pi a}{b}}\right)}{\sinh\!\left(\sqrt{\frac{\pi a}{b}}\right)}

Special cases of Laplace transforms

f42652
0eatθ1 ⁣(0,it)dt=2π1cosh ⁣(πa)\int_{0}^{\infty} {e}^{-a t} \theta'_{1}\!\left(0 , i t\right) \, dt = 2 \pi \frac{1}{\cosh\!\left(\sqrt{\pi a}\right)}
b2f31a
0eatθ2 ⁣(0,it)dt=πatanh ⁣(πa)\int_{0}^{\infty} {e}^{-a t} \theta_{2}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \tanh\!\left(\sqrt{\pi a}\right)
1ee920
0eatθ3 ⁣(0,it)dt=πacoth ⁣(πa)\int_{0}^{\infty} {e}^{-a t} \theta_{3}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \coth\!\left(\sqrt{\pi a}\right)
594cc3
0eatθ4 ⁣(0,it)dt=πa1sinh ⁣(πa)\int_{0}^{\infty} {e}^{-a t} \theta_{4}\!\left(0 , i t\right) \, dt = \sqrt{\frac{\pi}{a}} \frac{1}{\sinh\!\left(\sqrt{\pi a}\right)}

Mellin transforms

9376ec
0ts1θ2 ⁣(0,it2)dt=(2s1)πs/2Γ ⁣(s2)ζ(s)\int_{0}^{\infty} {t}^{s - 1} \theta_{2}\!\left(0 , i {t}^{2}\right) \, dt = \left({2}^{s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta(s)
41631f
0ts1(θ3 ⁣(0,it2)1)dt=πs/2Γ ⁣(s2)ζ(s)\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{3}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta(s)
709905
0ts1(θ4 ⁣(0,it2)1)dt=(21s1)πs/2Γ ⁣(s2)ζ(s)\int_{0}^{\infty} {t}^{s - 1} \left(\theta_{4}\!\left(0 , i {t}^{2}\right) - 1\right) \, dt = \left({2}^{1 - s} - 1\right) {\pi}^{-s / 2} \Gamma\!\left(\frac{s}{2}\right) \zeta(s)

Constant definite integrals

f2a0c7
0θ1 ⁣(0,it)dt=2π\int_{0}^{\infty} \theta'_{1}\!\left(0 , i t\right) \, dt = 2 \pi
ecb406
0θ2 ⁣(0,it)dt=π\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \, dt = \pi
799742
0(θ3 ⁣(0,it)1)dt=π3\int_{0}^{\infty} \left(\theta_{3}\!\left(0 , i t\right) - 1\right) \, dt = \frac{\pi}{3}
f89d5a
0(θ4 ⁣(0,it)1)dt=π6\int_{0}^{\infty} \left(\theta_{4}\!\left(0 , i t\right) - 1\right) \, dt = -\frac{\pi}{6}
ae6718
0(θ1 ⁣(0,it))2dt=(Γ ⁣(14))44π\int_{0}^{\infty} {\left(\theta'_{1}\!\left(0 , i t\right)\right)}^{2} \, dt = \frac{{\left(\Gamma\!\left(\frac{1}{4}\right)\right)}^{4}}{4 \pi}
4f3d2b
0(θ42 ⁣(0,it)1)dt=log(2)\int_{0}^{\infty} \left(\theta_{4}^{2}\!\left(0, i t\right) - 1\right) \, dt = -\log(2)
140815
0(θ4 ⁣(0,it)1)2dt=π3log(2)\int_{0}^{\infty} {\left(\theta_{4}\!\left(0 , i t\right) - 1\right)}^{2} \, dt = \frac{\pi}{3} - \log(2)
fe4967
0θ2 ⁣(0,it)θ4 ⁣(0,it)dt=log ⁣(3+22)\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \theta_{4}\!\left(0 , i t\right) \, dt = \log\!\left(3 + 2 \sqrt{2}\right)
727715
0θ2 ⁣(0,it)θ3 ⁣(0,it)θ4 ⁣(0,it)dt=2\int_{0}^{\infty} \theta_{2}\!\left(0 , i t\right) \theta_{3}\!\left(0 , i t\right) \theta_{4}\!\left(0 , i t\right) \, dt = 2
ea304c
0θ22 ⁣(0,it)θ42 ⁣(0,it)dt=1\int_{0}^{\infty} \theta_{2}^{2}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right) \, dt = 1
02d9e4
0θ24 ⁣(0,it)θ42 ⁣(0,it)dt=1\int_{0}^{\infty} \theta_{2}^{4}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right) \, dt = 1
963daf
0θ42 ⁣(0,it)1+t2dt=1\int_{0}^{\infty} \frac{\theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = 1
e4cdf1
0θ44 ⁣(0,it)1+t2dt=4log(2)π\int_{0}^{\infty} \frac{\theta_{4}^{4}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{4 \log(2)}{\pi}
86d68c
0θ46 ⁣(0,it)1+t2dt=16Gπ223\int_{0}^{\infty} \frac{\theta_{4}^{6}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{16 G}{{\pi}^{2}} - \frac{2}{3}
45267a
0θ48 ⁣(0,it)1+t2dt=20ζ(3)π3\int_{0}^{\infty} \frac{\theta_{4}^{8}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{20 \zeta(3)}{{\pi}^{3}}
5b87f3
0θ24 ⁣(0,it)θ44 ⁣(0,it)1+t2dt=8ζ(3)π3\int_{0}^{\infty} \frac{\theta_{2}^{4}\!\left(0, i t\right) \theta_{4}^{4}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{8 \zeta(3)}{{\pi}^{3}}
1a15f9
0θ24 ⁣(0,it)θ42 ⁣(0,it)1+t2dt=23\int_{0}^{\infty} \frac{\theta_{2}^{4}\!\left(0, i t\right) \theta_{4}^{2}\!\left(0, i t\right)}{1 + {t}^{2}} \, dt = \frac{2}{3}

Periodic integrals

7c78ea
M+1/2N+1/2θ1 ⁣(x,τ)dx=0\int_{M + 1 / 2}^{N + 1 / 2} \theta_{1}\!\left(x , \tau\right) \, dx = 0
f71675
2M2Nθ1 ⁣(x,τ)dx=0\int_{2 M}^{2 N} \theta_{1}\!\left(x , \tau\right) \, dx = 0
cc59e4
MNθ2 ⁣(x,τ)dx=0\int_{M}^{N} \theta_{2}\!\left(x , \tau\right) \, dx = 0
2429b2
M/2N/2θ3 ⁣(x,τ)dx=NM2\int_{M / 2}^{N / 2} \theta_{3}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}
a0955b
M/2N/2θ4 ⁣(x,τ)dx=NM2\int_{M / 2}^{N / 2} \theta_{4}\!\left(x , \tau\right) \, dx = \frac{N - M}{2}

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