# Questions tagged [elementary-functions]

For questions on elementary functions, functions of one variable built from a finite number of exponentials, logarithms, constants, and $n$th roots through composition and combinations using the four elementary operations $(+, –, ×, ÷)$.

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### primitive roots of unity to solve equations over $\mathbb Z_p$

So, I have been thinking of the equation $x^n-1 \equiv 0 \in \mathbb{Z}_p$, $p$ prime. So, I noticed something weird and I wonder if there is a theory for that. Let $\omega_n$ be the nth primitive ...
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### Are there any elementary functions $\beta(x)$ that follows this integral $\int_{y-1}^{y} \beta(x) dx =\cos(y)$

Are there any simple functions $\beta(x)$ that follows this integral $$\int_{y-1}^{y} \beta(x) dx =\cos(y)$$ I think there is an infinite amount of solutions that are continuous everywhere but how can ...
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### Inequality of modulus of quadratic polymnomials in $\mathbb{C}$

I am trying to work out some stability conditions for ODE methods and during the computations one needs to solve the following inequality: Let $\alpha_1, \alpha_2, \beta_1, \beta_2 \in \mathbb{R}$. ...
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### Does $f(x) = x - \tanh(x)$ have an inverse function that can be expressed in terms of elementary functions?

I find this question relevant in my current study of the tractrix, namely because this expression appears in one parameterization of the curve. I’ve noticed that the plot of the Cartesian equation of ...
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### Reference request: Source for “Cauchy's Theorem” (?) on integration in elementary functions

Buried deep in my notes from a course I took many years ago, I find a reference to the following, which (in my notes) is called "Cauchy's Theorem": Theorem. The integral $\int x^p (1-x)^q dx$ can ...
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### Coprimality of a rational term?

Let $P_0,P_1,P_2,P_3\in\overline{\mathbb{Q}}$, $P_3\neq 0$, and $p(x),q(x)\in\overline{\mathbb{Q}}[x]$ so that $p(x)$ and $q(x)$ are coprime over $\overline{\mathbb{Q}}$. I have the term {\frac{...
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### Find a function with certain property

I am searching for an example of the function with the following property. For given function $f : \mathbb{R}\rightarrow\mathbb{R}$, both $e^{x}f(x)$ and $e^{-f(x)}$ are monotonically decreasing. I ...
### Find number of real roots of the polynomial $x^3+7x^2+6x+5$.
I want to find the number of real roots of the polynomial $x^3+7x^2+6x+5$. Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude ...
### mupliplication of cosine of $\pi/2^k$
Let $x,y\in \{\frac{2\pi i}{2^m}\}_{0\leq i\leq 2^m-1 }$, where $m$ is a positive integer . Q If we have that $\cos A=\cos x\cdot \cos y$, can we say that $A=\pi k$ for some $k\in \mathbb Q$. For \$...