Fungrim entry: c7f7a5

$\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \cos\!\left(\pi z\right) \ne 0$
TeX:
\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \left|\operatorname{Im}(z)\right| < \left|\operatorname{Im}(\tau)\right| \;\mathbin{\operatorname{and}}\; \cos\!\left(\pi z\right) \ne 0
Definitions:
Fungrim symbol Notation Short description
Pi$\pi$ The constant pi (3.14...)
JacobiTheta$\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Sin$\sin(z)$ Sine
Infinity$\infty$ Positive infinity
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
HH$\mathbb{H}$ Upper complex half-plane
Abs$\left|z\right|$ Absolute value
Im$\operatorname{Im}(z)$ Imaginary part
Cos$\cos(z)$ Cosine
Source code for this entry:
Entry(ID("c7f7a5"),
Formula(Equal(Mul(Div(1, Pi), Div(JacobiTheta(2, z, tau, 1), JacobiTheta(2, z, tau))), Where(Add(Neg(Tan(Mul(Pi, z))), Mul(4, Sum(Mul(Mul(Pow(-1, n), Div(Pow(q, Mul(2, n)), Sub(1, Pow(q, Mul(2, n))))), Sin(Mul(Mul(Mul(2, Pi), n), z))), For(n, 1, Infinity)))), Equal(q, Exp(Mul(Mul(Pi, ConstI), tau)))))),
Variables(z, tau),
Assumptions(And(Element(z, CC), Element(tau, HH), Less(Abs(Im(z)), Abs(Im(tau))), NotEqual(Cos(Mul(Pi, z)), 0))))

Topics using this entry

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