Questions tagged [theta-functions]

For questions about $\theta$ functions (special functions of several complex variables).

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13
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1answer
197 views

Summation of $\sum_{n=0}^{\infty}a^nq^{n^2}$

I am trying to find the result for the sum of the form $\sum_{n=0}^{\infty}a^nq^{n^2}$. The special case for $a=1$ is easily given by $\vartheta(0,q)$, where $\vartheta(z,q)$ is the third Jacobi Theta ...
2
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1answer
48 views

What are specific proofs of Jacobi Triple Product Identity?

I am looking for the Special Proofs. Here is a reference from MSE. Motivation for/history of Jacobi's triple product identity I also know that a simple proof via Functional Equation from the book ...
0
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0answers
26 views

Combined Integral transformations (Mellin + Laplace)

I'm looking for the solution of the following integral $$\int\limits_0^\infty \mathrm{d}t\,\mathrm{e}^{-\sigma^2 t}\,t^{s-2} \,\theta_4\left(\frac{1}{2}\mathrm{i}\beta\mu, \mathrm{e}^{-\frac{\beta^2}{...
1
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1answer
27 views

Determine values of $\theta$ for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions

So there is this question that's asking for a "range of values for theta from $-\pi$ to $\pi$, for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions." I'm not really sure ...
20
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1answer
385 views

A problem posed by Ramanujan involving $\sum e^{-5\pi n^2}$

While going through the list of problems posed by Ramanujan in Journal of Indian Mathematical Society I came across this problem involving theta functions: Prove that $$\frac{1}{2}+\sum_{n=1}^{\...
1
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1answer
71 views

How is the integral form of Ramanujan theta function derived?

Ramanujan theta function defined as-$$f(a,b)=\sum_{n=0}^\infty a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}$$ And it's integral representation:$$f(a,b)=1+\int_0^\infty \frac{2ae^{-t^{2}/2}}{\sqrt{2\pi}}\...
1
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0answers
61 views

The estimate for $\sum_{0}^{\infty} e^{-kn^2}-e^{-k(n+1/2)^2}$

I'm trying to give estimate on the following infinite summation: $\sum_{n=-\infty}^{\infty} (e^{-kn^2}-e^{-k(n+\frac{1}{2})^2})$, and k is some fixed positive number. The first term in the summation ...
2
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0answers
45 views

A closed form for these non-elementary, generalized relatives of the geometric series

I usually provide more details to the questions I ask on this site, but for this specific question I can't even wrap my head around how to even attempt solving it. By the way, this question has no ...
0
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0answers
29 views

relation between Riemann theta function and Jacobi theta function

So we know Jacobi 3rd theta function can be defined using different summations such as: \begin{equation} \theta_{3}(a,b)=1+2\sum_{m=1}^{\infty}b^{m^2}\cos(2ma) \end{equation} and I also know that ...
0
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0answers
39 views

$\sum_i^{\infty}e^{(-(2i+1)^2a)}(2i+1)^2$ where a is is a positive real number

I have asked this question two days ago. One helpful person in the comments directed me to the Jacobi Theta functions and I was able to solve the rest myself. Starting from that point, I have: $\sum_{...
1
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0answers
27 views

Resources for learning about theta functions

I would like to learn about the Jacobi theta functions,however, I'm struggling to find a free ebook /website that would introduce the functions in a detailed comprehensible manner for beginners. So is ...
0
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1answer
67 views

evaluation of $\sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}}$ for large z

I want to evaluate the following \begin{equation} \sum_{n=-\infty}^{\infty}e^{-2\pi^{2}z^{2}n^{2}} \end{equation} I know for $z\ll 1$ we can use Euler-Maclaurin formula but in my case z is quite ...
-1
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1answer
78 views

Evaluation of $\sum_{n=1}^{\infty}q^{n^{2}}$?

I would like to evaluate the following summation \begin{equation} \sum_{n=1}^{\infty}q^{n^{2}} \end{equation} assuming $ 0<q<1$ obviously the series converge but can anyone help me how to ...
2
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1answer
76 views

how to prove these formulas about infinite product?

Recently , I read one paper titled 'Modular equations and approximations to π' by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ : $$\...
0
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1answer
39 views

Theta functions identites

I have been reading Rick Miranda's Book on Riemann surfaces and to indroduce meromorphic functions on the complex torues $\mathbb{C}$\ $L$ he talks about theta functions. I was able to see that $\...
2
votes
1answer
73 views

Zeros of the Jacobi Theta function

How do you obtain all the zeros in $z$ of the Jacobi Theta function $$\vartheta(z) = \sum_{n} e^{\pi i n^2 \tau + 2\pi i n z} \, ?$$ Probably the easiest way is to just read them of the Jacobi-Triple ...
4
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0answers
41 views

The Four Square Theorem and Integral Apollonian Circle Packings, is there any connection?

I have been studying theta-functions and made an interesting observation which I have a question about QUESTION: Is there a more intuitive, in particular a mostly geometric way, to prove the four ...
0
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0answers
14 views

Proving the immersion part of an Embedding

Trying to see the proof of embedding the Jacobian of a Compact Riemann Surface $X$ using Theta functions. So, using the Theta divisor we have the corresponding line bundle say $L$, we want to prove ...
0
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1answer
30 views

Transformation formula for Theta-series

I am currently reading Weil's book : "Elliptic Functions According to Eisenstein and Kronecker" and in page 56 he uses the well-known transformation formula for theta series $$\sum\limits_{\mu} ...
3
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1answer
303 views

Finding roots of $\sum\limits_{n = - \infty }^ \infty n z^n q^{n^2} =0 $ , $z_k=u_k(q)$

The Jacobi triple product identity is: $$F(z,q)=\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ where $|q|<1$ All roots ...
2
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1answer
91 views

Is $\sum_{n \in \mathbb{Z}} e^{-(n-\mu)^2/2\sigma^2} \le \sum_{n \in \mathbb{Z}} e^{-n^2/2\sigma^2}$ for all $\mu$ and all $\sigma$?

I have been looking at discrete Gaussian distributions and arrived at the following conjecture. I would greatly appreciate a proof (or disproof). Conjecture. Let $\mu \in [0,1]$ and $\sigma^2 > ...
0
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0answers
28 views

Weierstrass elliptic function identity

For a lattice $\Lambda = [\lambda_1, \lambda_2] \subset \mathbb C$, the Weierstrass $\wp$-function defined as \begin{equation} \wp(z) = \frac{1}{z^2} + \sum_{\lambda \in \Lambda \setminus \{0\}} \...
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0answers
37 views

Can this number be expressed in terms of Theta functions?

In this video, a mathematics guy explains how a two-dimensional grid can represent a list of every algebraic number exactly once. Then, he explains how Cantor diagonalisation to construct a real ...
2
votes
2answers
135 views

Estimate the series $\sum_{k=0}^\infty(-1)^k(2k+1)^2 z^{(2k+1)^2}$

I have encounter a series of the form $$\sum_{k=0}^\infty(-1)^k(2k+1)^2 z^{(2k+1)^2}, \ z\in [0,1),$$ which basically comes from the derivative of $$\sum_{k=0}^\infty (-1)^k z^{(2k+1)^2}.$$ The ...
4
votes
1answer
75 views

Prove $\sum_{n\ge1}\frac{1}{q^n+q^{-n}}=\tfrac14(\vartheta_3^2(q)-1)$

Prove that $$\sum_{n\ge1}\frac{1}{q^n+q^{-n}}=\tfrac14(\vartheta_3^2(q)-1),$$ provided by Wolfam. Note that here, we use the notational conventions $$\vartheta_3(z,q)=\sum_{n\in\Bbb Z}q^{n^2}e^{...
0
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0answers
61 views

Umemura's Theorem on Solving any Algebraic Polynomial Equation

I have recently read Bruce King's Beyond the Quartic Equation. In the last pages, he shows a theorem, Umemura's Theorem, which shows how to solve any polynomial equation. I have tried to understand ...
2
votes
0answers
218 views

Extension of Jacobi triple Product Identity for $\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} h^{n^3}$

I asked a question about Extension of Jacobi triple Product Identity for $\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} h^{n^3}$. I have new idea about zeros for $\sum\limits_{n = - \infty }^ \...
1
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0answers
38 views

Waring problem generalizations and theta-function

My question is twofold: Can the Waring problem be expressed with the Jacobi theta function or some analog (as is the case for $k=2$) for general $k$? Say for $k=4$ or $k=6$, are these able to be ...
5
votes
1answer
110 views

Theta Functions and Partitions

I am reading some papers by Ramanujan on congruence properties of the partition function. At one point he says that he will be using "theta functions" and introduces the following: It can be shewn ...
0
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0answers
78 views

Formula for sequence $\left(r_{k}\right)_{k=0}^{\infty}=(1, 1, 3, 5, 9, 15, 25, 39, 61, 93,\cdots)$

The sequence $$ \left(r_{k}\right)_{k=0}^{\infty}=(1, 1, 3, 5, 9, 15, 25, 39, 61, 93,\cdots) $$ can be found at OEIS as sequence A207641 and is related to Ramanujan theta functions. Unfortunately ...
0
votes
1answer
63 views

modular forms and their fourier coefficients question

I was recently listening to Don Zagiers fourth lecture at ICTP (posted Feb 5, 2015) on mock modular forms. At roughly 22:00 in the lecture he makes two statements: 1. the product of the weight and ...
0
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0answers
36 views

Show that $\theta(z)=\sum_{m=-\infty}^{\infty}e^{-\pi m^2z}$ converges and is analytic in $z\in \mathbb C,Re(z)>0$

Show that $\theta(z)=\sum_{m=-\infty}^{\infty}e^{-\pi m^2z}$ converges and is analytic in $z\in \mathbb C,Re(z)>0$ Possible proof: Let $Re(z)=x$ Firstly note that $|e^{-\pi m^2z}|=e^{-\pi m^2x}\...
12
votes
1answer
203 views

Show that $\prod\limits_{n=1}^\infty \frac{(1-q^{6n})(1-q^n)^2}{(1-q^{3n})(1-q^{2n})}=\sum\limits_{n=-\infty}^\infty q^{2n^2+n}-3q^{9(2n^2+n)+1}$.

Show that $\displaystyle \prod_{n=1}^\infty \frac{(1-q^{6n})(1-q^n)^2}{(1-q^{3n})(1-q^{2n})}=\sum_{n=-\infty}^\infty q^{2n^2+n}-3q^{9(2n^2+n)+1}$. I can't seem to be able to proceed with this ...
0
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0answers
12 views

Understanding θ max

Let C represent total expected costs, p be the probability of skilled worker, w the wage of the worker, q unit of labour and π price for the output. I am trying to figure out the meening of the ...
1
vote
1answer
120 views

How does one express Jacobi theta functions in terms of Elliptic Integrals?

It is known that elliptic functions may be expressed in terms of Jacobi theta functions. Moreover, by construction, the elliptic integrals are inverses of elliptic functions. It therefore seems to be ...
1
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0answers
60 views

Jacobi Theta Function on the Unit Circle - Is there a Limit in the Distribution Sense?

The third Jacobi theta function $$\theta_{3}\left(z,q\right)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\cos\left(2\pi n z\right)$$ appears in the study of path integrals in QM. Specifically in the problem of a ...
0
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0answers
21 views

Is the Jacobi theta function in the Fock space?

Let $\tau$ with $\text{Im} \tau > 0$. Consider the Jacobi theta function given by $$ \nu(z) = \nu(z, \tau) = \sum_{n=-\infty}^\infty e^{i\pi(n^2\tau + 2nz)} $$ $\nu$ is a entire function on $\...
1
vote
1answer
55 views

Does anybody know this relation between Jacobi theta functions?

I'm proving a theorem that I know is correct, but I would need to proof that $$ \theta{}_{2}\left(z\right)\theta{}_{3}\left(z\right)=\theta'{}_{4}\left(z\right)\theta{}_{1}\left(z\right)-\theta'{}_{1}\...
0
votes
0answers
48 views

What is the derivative of Riemann theta function with characters?

Consider $\theta_{a,b}(z\,|\,\tau)$ defined as: $$ \theta_{a,b}(z\,|\,\tau)=\sum_{n\in\mathcal{Z}} e^{i\pi\tau(n+a)^2}e^{2\pi i(n+a)(z+b)} $$ What is the derivative of $\ln \theta_{a,b}(z\,|\,\tau)$,...
2
votes
1answer
90 views

Why the derivative of the logarithm of a theta function is not an elliptic function?

Let's consider the theta functions periodicity conditions $$\vartheta\left(z+1\right) =\vartheta\left(z\right),$$ $$\vartheta\left(z+\tau\right) =e^{-\pi i\tau-2\pi iz}\vartheta\left(z\right),$$ Those ...
2
votes
2answers
120 views

What is the algorithm complexity in Big-Theta notation?

I need to find the complexity of Code(n) algorithm in terms of Big Theta notation. Thank you in advance. Code(n) ...
2
votes
1answer
64 views

Help showing that a function has order of n^2

Let $f:\mathbb{N}\rightarrow\mathbb{R}^+$ so that $f\in\Theta(n)$. Also, we define the function $g:\mathbb{N}\rightarrow\mathbb{R}^+$ as $$g(n)=\sum_{i=0}^{n}{f(i)}$$ Show that $g\in\Theta(n^2)$ $\...
0
votes
1answer
19 views

For any constant k>1 the function f(n) = 1 + k + k^2 + k^3 + .. + k^n is in Θ(k^n)

I know what theta means we essentially are proving that c1 * g(n) <= f(n) <= c2*g(n) I came across this question while looking for asymptote examples and none of the other examples have this ...
0
votes
0answers
29 views

Integral involving $\theta_3(z,q)$ with respect to $z$

I'm studying Dirichlet kernels and their related identities; I've stumbled upon a "weighted Dirichlet kernel" so to speak. I'd like to find a closed form for the summation. To do so, I'm required to ...
2
votes
1answer
58 views

Factorization of elliptic functions with theta functions

I found in Mumford (tata Lectures on Theta) that every elliptic function can be written as fraction of theta functions $$\prod\frac{\vartheta_{a_{j}}\left(z-z_{j}\right)}{\vartheta_{b_{i}}\left(z-z_{...
0
votes
1answer
75 views

Two Equivalent Equations for the Zeros of the Jacobi Theta Function?

I'm trying to find the 0's for the Jacobi theta function with characteristic: $$ \vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b \in \...
0
votes
1answer
44 views

How to solve a quintic polynomial using elliptic functions with Mathematica?

I followed the exact steps from this forum post to solve quintic polynomials of the form: $x^5 - x + d$ But I got a different answer in number form from Mathematica than the original quintic ...
3
votes
1answer
61 views

Ramanujan theta function on matrices: When does $\sum A^{n(n-1)/2} B^{n(n+1)/2}$ converge?

Assume that $AB=BA$. The infinite sum $$\sum_{n\in\mathbb Z}A^{n(n-1)/2}B^{n(n+1)/2}$$ $$=\cdots+A^3B^1+A^1B^0+A^0B^0+A^0B^1+A^1B^3+A^3B^6+A^6B^{10}+\cdots$$ converges unconditionally if and only ...
5
votes
1answer
100 views

Theta function identity

Let us consider the Theta function $$\theta(\tau) = \sum_{n \in \mathbb{Z}} e^{\pi i n^2 \tau} \text{ for } \mathrm{Im}(\tau)>0.$$ Then it is rather easy to see that $$\theta(\tau+2)=\theta(\tau)$...
0
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0answers
48 views

How many inputs does the following theta function have?

In the book http://renaissance.ucsd.edu/courses/mae207/mech.pdf page 118 The theta function contains only 1 input, isn't it suppose to be 2 inputs? how does one of the theta functions in this page ...

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