Questions tagged [modular-function]

This tag is for questions relating to Modular Function or, Elliptic Modular Function.

A function is said to be Modular or, elliptic modular if it satisfies the following conditions:

$1.~~ f$ is meromorphic in the upper half-plane $H$,

$2.~~ f(\bf A\tau)=f(\tau)$ for every matrix $\bf A$ in the modular group Gamma,

$3.~~$ The Laurent series of $f$ has the form

$$ f(\tau)=\sum_{n=-m}^{\infty}a(n)e^{2\pi i n\tau} $$

A modular function is a function that, like a modular form, is invariant with respect to the modular group, but without the condition that $f (z)$ be holomorphic in the upper half-plane. Instead, modular functions are meromorphic.

References:

https://en.wikipedia.org/wiki/Modular_form#Modular_functions

http://mathworld.wolfram.com/ModularFunction.html

83 questions
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Closed form of $\frac{e^{-\frac{\pi}{5}}}{1+\frac{e^{-\pi}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-3\pi}}{1+\ddots}}}}$

It is well known that $$\operatorname{R}(-e^{-\pi})=-\cfrac{e^{-\frac{\pi}{5}}}{1-\cfrac{e^{-\pi}}{1+\cfrac{e^{-2\pi}}{1-\cfrac{e^{-3\pi}}{1+\ddots}}}}=\frac{\sqrt{5}-1}{2}-\sqrt{\frac{5-\sqrt{5}}{2}}$$ where $\operatorname{R}$ is the…
6
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Is $\prod_{n=0}^\infty \left(1-\frac{1}{\cosh ^2((n+1/2)\pi)}\right)=\frac{1}{\sqrt[4]{2}}$ true?

The infinite product $$\prod_{n=0}^\infty \left(1-\frac{1}{\cosh  ^2((n+1/2)\pi)}\right)$$ agrees with $\frac{1}{\sqrt[4]{2}}$ to at least 100 decimal places. The "identity" is reminiscent of $$\sqrt[4]{1-\lambda (i)}=\frac{1}{\sqrt[4]{2}}$$ where…
6
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1 answer

Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind for which the image of $\Delta$ is anything else?
5
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1 answer

Ramanujan Identity related to JacobiFunction

The following identity is allegedly due to Ramanujan $$\int_0^\infty \frac{{\rm d}x}{(1+x^2)(1+r^2x^2)(1+r^4x^2)\cdots} = \frac{\pi/2}{\sum_{n=0}^\infty r^{\frac{n(n+1)}{2}}} \, $$but how do you prove this? The denominator of the right side is…
Diger
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4
votes
2 answers

Plotting graphs of Modular Forms

After watching all the 8 parts of "“Introduction to Modular Forms,” by Keith Conrad" on YouTube, I got "extremely intrigued" by plotting graphs of Modular Forms ( on SL(2,Z) ). So after watching all those videos, I tried the following approach by…
4
votes
1 answer

Explicit equations for $Y(N)$ for small $N$

Consider the congruence subgroup $$\Gamma(N) = \left\{\left(\begin{array}{cc} a & b \\ c & d\end{array} \right) \in SL_2(\mathbb{Z})\ ;\ \left(\begin{array}{cc} a &…
4
votes
1 answer

What are the simplest known classes of bijections in $\mathbb{Z}/n\mathbb{Z}$, where n is a power of 2?

What are some of the simplest known bijections in $\mathbb{Z}/n\mathbb{Z}$? Offhand, the following classes of primitive bijections come to mind: Addition/subtraction (+/–) of any constant Multiplication ($\times$) by an odd constant (and division…
4
votes
0 answers

Showing $E_4(z)$ and $E_6(z)$ are algebraically independent over $\mathbb{C}$

Let $E_4(z)= - \frac{B_4}{8}+ \sum_{n=1}^\infty \sigma_3(n) q^n$ and $E_6(z)= - \frac{B_6}{12}+ \sum_{n=1}^\infty \sigma_5(n) q^n$ How does one show they are algebraically independant over $\mathbb{C}$ ?
usere5225321
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4
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2 answers

What is the derivative of Rogers-Ramanujan Continued Fraction?

We have many useful formulae about the derivatives of modular functions. For example, \begin{eqnarray} &&j'(\tau)=-\frac{E_6}{E_4} j(\tau), \\ &&\eta'(\tau)=\frac{1}{24}E_2 \eta(\tau),…
3
votes
0 answers

Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)

Denote the complete elliptic integral of the first kind by $$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$ and $$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$ Question: Given a real $x$, $0\lt x\lt 1$, is there an algorithm…
3
votes
2 answers

Derive singular value $\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$

Does anyone know how to prove that the following special value of the Modular Lambda Function is correct? $$\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$$ I have a somewhat promising observation that might help us derive this special value, but it hasn't…
3
votes
0 answers

Non-modular coverings of modular curve

Let $Y$ be a Riemann surface, and $$T=\sum_{k=0}^n a_k y^k \in K(Y)[y]$$ be a polynomial in $y$ of degree $n$ over the field $K(Y)$ of meromorphic functions on $Y$. If we denote by $O$ the discrete subset of $Y$ containing all poles of the…
3
votes
1 answer

$q$-expansion of Klein's absolute invariant using infinite products

Given that $$j=\frac{1}{13824q^2}\left(2^8q^2\prod_{k\gt 0}(1+q^{2k})^{16}+\prod_{k\gt 0}(1+q^{2k-1})^{16}+\prod_{k\gt 0}(1-q^{2k-1})^{16}\right)^3,$$ how can I show that $$j=\frac{1}{1728q^2}(1+c_1 q^2+c_2 q^4+\cdots)$$ where $c_1,\, c_2,\, \ldots$…
3
votes
0 answers

Modular Forms and Hypergeometric Functions

In "The 1-2-3 of Modular Forms" Zagier gives, as an example of proposition 21 (pg 61), the identity $$\vartheta_3(z)^2 = \sum_{n=0}^{\infty}\begin{pmatrix}2n\\n\end{pmatrix}\left(\frac{\lambda(z)}{16}\right)^n =…
3
votes
1 answer

$N$-th root of modular forms

In my research, I have encountered many $q$-series of functions that turn out to be the Fourier expansions of roots of modular forms. Examples are $n$-th roots of Eisenstein series and the $j$-invariant. These seem to be related in certain instances…
El Rafu
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