# Questions tagged [modular-function]

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

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### 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 ...
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### $\lambda (\tau)$ as a rational function of $j(\tau)$

It is known that $$j(\tau)=\frac{256(1-x)^3}{x^2}$$ where $x=\lambda (\tau)(1-\lambda (\tau))$ and $\lambda (\tau)$ is the modular lambda function. But I came across the following statement (https://...
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### Approximations of real numbers on $(0,1)$ with powers of the form $(3/2)^n \pmod 1$

Let $0<r<1$ a real number which is not a fraction of the form $p/2^n$ for any integers $p,n$. Now, for every integer $n\ge 1$ we can find the closest fraction of the form $p/2^n$ to $r$, which ...
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### What are examples of modular form of level $1$ (i.e. modular form on $SL_2(\mathbb{Z})$) with poles?

I am thinking of Eisenstein series, $$G_k(\tau) = \sum_{(c,d)\in {\mathbb{Z}}^2-\{(0,0)\}}\frac{1}{(c\tau+d)^k}, \tau \in \mathbb{H}$$ because we don't sum over $(0,0)$, so I'd like to call $(0,0)$ a ...
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### Pole order of $\frac{4}{27}\frac{\left(\lambda^2-\lambda+1\right)^3}{\lambda^2\left(1-\lambda\right)^2}\left(=j(\tau)\right)$ [closed]

Concerning the relation $$j=\frac{4}{27}\frac{\left(\lambda^2-\lambda+1\right)^3}{\lambda^2\left(1-\lambda\right)^2},$$ I understand, that the RHS is an element of $\mathbb{C}(j)$, and thus the LHS ...
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### 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 ...