Articles of rational functions

Inverse Nasty Integral

Intrigued by the original question on a nasty integral, one wonders what functions $f(x)$ exist such that $$\int_0^\infty f(x)\; dx=\frac {\pi}2$$ Something to do with the area of a half-circle with unit radius perhaps? Edited To Add It should be specified that $f(x)$ should not contain $\pi$, and is preferably a rational function. The intention […]

How to integrate $\frac{1}{(1 + x^5)(1 + x^7)}$

My cousin who is in high school asked me if it is possible to integrate $$ \int \frac{1}{(1 + x^5)(1 + x^7)} \, dx $$ I checked the list of integrals of rational functions on Wikipedia link and it doesn’t seem to be here. Is this not possible to do using elementary functions? Wolfram Alpha […]

Using Modularity Theorem and Ribet's Theorem to disprove existence of rational solutions

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet’s theorem, and distill these down to an undergrad math level of a way to check if certain rational polynomials have no non-trivial solutions? For instance, I do not understand the details of either theorem, but if I can […]

Minimal polynomial of $x$ over $ K\left(\frac{p(x)}{q(x)}\right) \subset K(x) $

Let $K$ be a field, and let’s consider the field of rationals functions $K(x)$. Let $t\in K(x)$ be the rational function $\frac{P(x)}{Q(x)}$, where $P,Q$ have no common factors. I have to prove that the extension of fields $K(t) \subset K(x) $ has degree $\max(\deg P,\deg Q)$. I have to prove that the minimal polynomial of […]

When is $\infty$ a critical point of a rational function on the sphere?

In his “Iteration of Rational Functions” Beardon defines a critical point of a rational function (mapping the Riemann sphere to itself) as a point such that the function is not injective in any neighborhood of that point. I’m trying to square Beardon’s definition with the other definition, in which a critical point is a point […]

When a field extension $E\subset F$ has degree $n$, can I find the degree of the extension $E(x)\subset F(x)?$

This question already has an answer here: Why does $K \leadsto K(X)$ preserve the degree of field extensions? 5 answers

Range of a Rational Function

How to find the Range of function $$f(x)= \frac{x^2-3x-4}{x^2 – 3x +4}$$ I tried to equate the expression to $y$, then cross multiplied $$ y= \frac{x^2-3x-4}{x^2 – 3x +4}$$ $$ y(x^2 – 3x +4)= x^2-3x-4 $$ bought the terms to one side so it becomes a quadratic and made Discriminant to zero , but i […]

Is there a rational surjection $\Bbb N\to\Bbb Q$?

The question is in the title. Is there a one-dimensional rational function $f\in\Bbb R(X)$ which restricts to $\Bbb N\to\Bbb Q$, which is a surjection onto $\Bbb Q$? My guess is no. Expanding the scope a little (at the expense of precision), are there any “nice” functions that enumerate $\Bbb Q$? Here “nice” is meant to […]

Integral with logarithm – residue

Let $R(x)$ be rational function. It is any general method to calculate $\int_{0}^{\infty}R(x) \log(x)dx$ ? I can do it in special cases, but I am looking for a general method. What should be a minimal assumptions about $R(x)$ ?

A rational orbit that's provably dense in the reals?

Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is “probably” dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational whose orbit is provably dense in $\mathbb{R}$? NB: “Rational mapping” here means simply a function from rationals to rationals, not the definition in […]