Questions tagged [theta-functions]
For questions about $\theta$ functions (special functions of several complex variables).
370 questions
6
votes
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prove $\left(\sum_{n=-\infty}^{+\infty} q^{n^2}\right)^2 = 1 + 4\sum_{n=1}^{\infty} \frac{q^n}{1+q^{2n}}$ [duplicate]
Let
$$
\theta(q) = \sum_{n=-\infty}^{+\infty} q^{n^2}.
$$
So
$$
\theta(q)^2
= \left(\sum_{m\in \mathbb{Z}} q^{m^2}\right)\left(\sum_{n\in \mathbb{Z}} q^{n^2}\right)
= \sum_{N=0}^{\infty} r_2(N)\, q^N,
...
- 61
2
votes
0
answers
177
views
A representation of $S(q)=\sum_{n\ge1} n q^{n^2}$ via Zwegers’ $\mu$-function and its modular completion
I am looking at the Lambert-type series
$$
S(q)=\sum_{n\ge1} n\,q^{n^2}\qquad (q=e^{2\pi i\tau},\ \Im\tau>0),
$$
and I found that it admits a representation connected to Zwegers’ $\mu$-function.
I ...
- 781
0
votes
0
answers
44
views
Argument of the theta function
I am looking at the Jacobi Theta function $\theta_1(z)$ with $\tau\in i\mathbb R$
$$\theta_1(z+1)=-\theta_1(z)\quad \theta_(z+\tau)=-e^{-2\pi iz}e^{-\pi i \tau}\theta_1(z)$$
I take the principal ...
- 41
1
vote
1
answer
57
views
Modular-like behavior of function formally similar to theta function
I've been looking at the function
$$f(z) = \sum_{n=1}^\infty \left(n-\frac{1}{2}\right) e^{i\pi \left(n-\frac{1}{2}\right)^2 z}$$
for $z$ in the upper half-plane, which is formally related to the half-...
- 305
0
votes
0
answers
54
views
Theta function index generalized to complex plane
I am searching for information related to theta function generalization in the sense that index power in exponent is not quadratic but a general complex power of it. I mean:
$$\theta(s,t,x)=\sum_{-\...
- 709
8
votes
0
answers
240
views
Period matrix of hyperelliptic curve
I. Question: Can anyone provide a numerical example, or atleast a brief instruction that is clear and not too confusing for calculating $\Omega$, with the curve $y^2=x(x-1)\left(x^5-10x^3+3x^2-9\right)...
- 9,890
1
vote
0
answers
50
views
Integral involving Jacobi theta functions and vanishing contour contributions
I need help understanding and solving integrals of the form:
\begin{equation}\int_{-\pi/2-i\epsilon}^{\pi/2-i\epsilon} \frac{\vartheta_4(2x,q^2)}{\vartheta_1(x-c,q)\vartheta_1(x+c,q)}dx \end{equation}
...
- 73
1
vote
1
answer
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What's the correct Jacobi triple product?
From the Wikipedia's Jacobi triple product and Wolfram's Jacob triple product.
Consider the following $x=q^{\frac{1}{4}}$ and $z^2=q^{\frac{1}{4}}$, then according to the formulas on the website, the ...
4
votes
0
answers
91
views
Constructing the Hecke Theta Function for a Number Field
I'm working through writing down a proof for the analytic continuation of the Dedekind zeta function of a number field $K$, of degree $d$, signature $(r_{1},r_{2})$ and rank $r_{K} = r_{1}+r_{2}-1$, ...
- 415
1
vote
1
answer
111
views
Linkage problem from complex analysis Saff/Snider book makes me confused.
I am trying to solve (or actually understand) the question from the book Fundamentals of Complex Analysis by Saff and Snider that made me very confused, I tried to find the explanation from the ...
- 368
3
votes
2
answers
460
views
Pythagorean-Identity for Theta function
So I wanted to find a squareroot-less representation for $\sqrt{a \, \vartheta_{10}^2(z,\tau)+b\,\vartheta_{11}^2(z,\tau)}$ and by knowing that $\vartheta_{11}(z+\tau,\tau)=-e^{-i\pi(\tau+2z)}\...
- 360
3
votes
0
answers
135
views
When can we say a theta function is a modular form?
Let $q = e^{2\pi i\tau}$ and $\Theta(x;q) = \sum_{n\in\mathbb{Z}}(-1)^nq^{\binom{n}{2}}x^n$.
I am aware that this is a Jacobi form but I have seen in some papers that people will say that this is a ...
- 65
7
votes
1
answer
297
views
Closed form for $\sum_{n=0}^{\infty} e^{-\pi n^2} \left(1 - 4\pi n^2\right) (-1)^{\frac{1}{2} n(n+1)}$
I was trying to prove if it's true that $\sum_{n=0}^{\infty} e^{-\pi n^2} \left(1 - 4\pi n^2\right) (-1)^{\frac{1}{2} n(n+1)} = \frac{3}{2}$, an "identity" I found online.
The sum of the ...
- 1,518
3
votes
0
answers
69
views
How did Riemann connect plane curves and $\theta$-functions?
Background
While I was learning about bitangents on plane curves, I looked into bitangent numbers on cubic and quartic curves. As I kept searching, the name of a mathematician named Riemann was often ...
- 867
0
votes
0
answers
49
views
The second Kronecker limit formula with a polynomial in the summation
By the second Kronecker limit formula, say, equation (39) in "On advanced analytic number theory",
$$\frac{z-\bar{z}}{-2 \pi i} \sum_{m, n}^{\prime} \frac{e^{2 \pi i(m u+n v)}}{|m+n z|^2}=\...
- 1