Questions tagged [elementary-functions]

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

In mathematics, an elementary function is a function of one variable built from a finite number of constants, exponentials and logarithms through composition, combinations using the four field operations $(+, –, ×, ÷)$ and continuation through removable singularities. Therefore elementary functions are analytic, but can be multi-valued. According to this definition, elementary functions include all algebraic functions, and by allowing these functions to be complex valued, trigonometric functions and their inverses become included in the elementary functions. For example: By taking branch cut of $\log$ to be negative imaginary axis and $-\frac{\pi}{2}\lt \arg(z)\lt \frac{3\pi}{2},$ we have real elemenrary functions

  • $\sqrt{x}=e^{1/2\log(x)}$ for all $x\ge 0$
  • $x^2=e^{2\log(x)}$ and similarly all polynomials
  • $\vert x\vert=\sqrt{x^2}$
  • $\sin(x)=\dfrac{1}{2i}(e^{ix}-e^{-ix})$
  • $\arctan x =\dfrac{1}{2i}\log\left(\dfrac{1+ix}{1-ix}\right)$

There are other definitions with bit more subtleties as well, see here. But there is no requirement that elementary functions includes "inverse" functions, in general. For example, local inverses of $f(z)=ze^z,$ known as (branches of) Lambert $W$-function are not elementary. Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of elementary functions can have an inverse which is an elementary function. Also, elementary functions do not required to closed under limits, infinite sums and integration, but they are closed under differentiation as a consequence of chain rule. This excludes many nice classes of functions from being elementary, such as elliptic functions, Bessel functions and hyper-geometric functions.

Observe that, the roots of polynomial equations are the implicitly defined functions of its constant coefficients. For polynomials of degree four and smaller there are explicit formulae for the roots (the formulae are elementary functions), but not for degrees five and higher. For example the unique real root of the polynomial $p(x)=x^5+x+a,$ called the Bring radical $\operatorname{BR}(a),$ is not elementary in the usual sense. But it is an elementary function in the following sense.

Liouvillian Elementary Functions

Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions known as Differential algebra was started by Joseph Fels Ritt in the 1930s. This generalization of "elementary functions" and their differential calculus allow us to solve two main problems:

  • What antiderivatives (of elementary functions) that can be expressed as elementary functions? See Liouville's theorem.

  • Solutions of linear differential equations in one variable (This subject is also known as the Picard–Vessiot theory/ Differential Galois theory, analogue of familiar Galois theory of polynomial equations aims to understand solutions of differential equations).

Let $(\mathbb{F},\partial)$ be a differential field with constants $\operatorname{const}(\mathbb{F}).$ A differential field extension $\mathbb{E}/\mathbb{F}$ is an elementary extension if there exists a tower of differential fields $$\mathbb{F}=\mathbb{F}_0\subseteq \mathbb{F}_1\subseteq\cdots \subseteq\mathbb{F}_n=\mathbb{E}$$

over the same field of constants such that each extension $\mathbb{F}_{j+1}/\mathbb{F}_j$ is either algebraic, exponential or logarithmic. See here for more details.

When $\operatorname{const}(\mathbb{F})=\mathbb{C},$ "Liouvillian elementary functions" $\mathcal{E}$ is the collection of all complex valued functions which lie in some elementary extension of $\mathbb{C}(x)$ (formal rational functions in a single complex variable) equip with usual derivative.

As a corollary to (above linked) Liouville's theorem, given two elementary functions $f, g$ the integral $$\displaystyle\int f e^g$$ is elementary if and only if $f=h'+hg'$ for some elementary function $h.$ Therefore logarithmic integral, dilogarithm, error function are not elementary. When this integral has an elementary solution one can use Risch algorithm to explicitly find it.

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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…
RghtHndSd
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What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, logarithmic, exponential, and $n$th roots, and solving…
user23784
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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}},\ \ \…
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Nice expression for minimum of three variables?

As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function. $\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$ There's even a nice intuitive explanation to go along with this: If we go to the…
Oscar Cunningham
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What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the connections between these ways are not clarified mostly…
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Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. The problem I got from this is what is an…
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Why do all elementary functions have an elementary derivative?

Considering many elementary functions have an antiderivative which is not elementary, why does this type of thing not also happen in differential calculus?
Ian Krasnow
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Approximating the error function erf by analytical functions

The Error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ shows up in many contexts, but can't be represented using elementary functions. I compared it with another function $f$ which also starts linearly, has $f(0)=0$ and…
Nikolaj-K
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Is there an elementary expression for every real sequence?

By elementary expression for the sequence $\{a_n\}_{n=0}^\infty$, I mean an elementary function $f : X \to \mathbb C$, where $\mathbb N \subset X \subset \mathbb R$, such that $f(n)=a_n$ for all $n$. The set of elementary functions, is the smallest…
Trebor
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Algebraic numbers that cannot be expressed using integers and elementary functions

Can we give an explicit${^*}$ example of a real algebraic number that provably cannot be represented as an expression built from integers and elementary${^{**}}$ functions only? ${^*}$ explicit means we can write down a polynomial equation with…
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Fastest way to calculate $e^x$ up to arbitrary number of decimals?

What are other faster methods of calculating $e^x$ up to any number of decimals other than using the Taylor series formula?
Pushpak Dagade
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A closed form for $\int_0^\infty\frac{\sin(x)\ \operatorname{erfi}\left(\sqrt{x}\right)\ e^{-x\sqrt{2}}}{x}dx$

Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the integral $$I=\int_0^\infty\frac{\sin(x)\ \operatorname{erfi}\left(\sqrt{x}\right)\…
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What kind of functions cannot be described by the Taylor series? Why is this?

It's true that I'm not familiar with too many exotic functions, but I don't understand why there exist functions that cannot be described by a Taylor series? What makes it okay to describe any particular functions with such a series? Is there any…
smaude
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What is a good way to explain why the graph of polynomials do not exhibit ripples, even in an arbitrarily small interval?

I was showing someone the graph of $0.1x^9+0.6x^5+0.5x^2 + x$ on Wolframalpha (for this question, any real valued polynomial will do) Someone asked me why the graphs of polynomials are smooth no matter what interval on $\mathbb{R}$ we look. More…
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When is ${\large\int}\frac{dx}{\left(1+x^a\right)^a}$ an elementary function?

Consider the following indefinite integral: $$F_a(x)=\int\frac{dx}{\left(1+x^a\right)^a}.$$ Here $a\in\mathbb R$ is a parameter, and $x>0$ is a variable. For what values of the parameter $a$ the function $F_a(x)$ is an elementary function of the…
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