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 […]

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 […]

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 […]

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 […]

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 […]

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

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 […]

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 […]

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 $R(x)$ ?

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 […]

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