Articles of elementary functions

Expressing the solutions of the equation $ \tan(x) = x $ in closed form.

I know that the equation $ \tan(x) = x $ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t know the problem, and sooner or later, we’ll be able to find a closed-form solution or at least […]

Is each “elementary + finite functions” function “elementary + finite functions”-integrable?

It is known that there exist elementary functions which are not elementary integrable, i.e. there exists no elementary anti derivative. Example: $f(x) = e^{-x^2}$. Let $A$ be the set of elementary functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Then: Add finite many Riemann integrable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ to $A$. Add all compositions of $A$-functions to […]

Prove that an equation has no elementary solution

There are methods proving that a polynomial isn’t solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through elementary functions (Risch Algorithm and differential Galois theory). But I’ve never seen a proof that proves an equation (like $xe^x=1$) not be solvable in terms of […]

Polynomials with degree $5$ solvable in elementary functions?

Quadratic, cubic and quartic polynomials are solvable in radicals, so there is no question here. What about the polynomials of degree $5$ (quintic)? Do we know all such polynomials (classes of polynomials) with all (or some) of the roots expressible as elementary functions of the coefficients? If we do, I would be glad for a […]

Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions

Let $f_n(x)$ be recursively defined as $$f_0(x)=1,\ \ \ f_{n+1}(x)=\sqrt{x+f_n(x)},\tag1$$ i.e. $f_n(x)$ contains $n$ radicals and $n$ occurences of $x$: $$f_1(x)=\sqrt{x+1},\ \ \ f_2(x)=\sqrt{x+\sqrt{x+1}},\ \ \ f_3(x)=\sqrt{x+\sqrt{x+\sqrt{x+1}}},\ \dots\tag2$$ The functions $f_0(x)$, $f_1(x)$ and $f_2(x)$ are integrable in elementary functions, e.g.: $$\int\sqrt{x+\sqrt{x+1}}\,dx=\left(\frac{2\,x}3+\frac{\sqrt{x+1}}6-\frac14\right)\sqrt{x+\sqrt{x+1}}+\frac58\ln\left(2\,\sqrt{x+1}+2\,\sqrt{x+\sqrt{x+1}}+1\right).\tag3$$ Question: Is there an integer $n>2$ such that $f_n(x)$ is integrable in elementary functions? […]

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure of the set of functions whose derivative lies in $E_{i-1}$ with respect to multiplication, inversion, and composition. Does […]

Injective map from real projective plane to $\Bbb{R}^4$

Consider the mapping $\Bbb R^3\rightarrow\Bbb R^4$ given by $$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$ which passes to the quotient and can therefore be viewed as a map from the projective plane $\Bbb P^2=S^3/\sim$. Can we show that then the map is an injection $\Bbb P^2\hookrightarrow\Bbb R^4$ ?

$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$

How do I establish this and similar values of trigonometric functions? $$ 4 \sin 72^\circ \sin 36^\circ = \sqrt 5 $$

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids using complex numbers in the proof. Existence of complex roots will be a corollary of this theorem, if proven successfully. Sorry for the long post. This is my own method and I could not make […]

Proving $x^{\log n} = n^{\log x}$

How could one go about proving that $$x^{\log n} = n^{\log x}\ {?}$$ I’m not really sure how to get started.