Questions tagged [elementary-functions]

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.