Elementary function

In mathematics, elementary functions are those functions that result from recurring, basic functions such as B. form polynomials or the logarithm using the basic arithmetic operations and concatenation . The exact list of permitted functions from which elementary functions may be assembled sometimes varies from author to author.

The elementary functions often result as solutions to a simple differential or functional equation and are therefore - even more than the special functions - fundamental for many natural sciences such as physics or chemistry, because they appear again and again in a wide variety of contexts.

It is usually quite difficult to show that a given function is not elementary. Important non-elementary functions, such as the error integral or the integral sine , are antiderivatives that cannot be integrated in an elementary manner. One speaks of elementary integrable functions when the antiderivative of an elementary function is elementary itself. This way of speaking is also not exact.

Elementary functions were introduced by Joseph Liouville in a series of articles from 1833 to 1841.

definition

Most of the time, a function is called elementary when it appears in the following list:

constant functions : etc. ${\ displaystyle 2, \ \ pi, \ e,}$
Power functions : etc. ${\ displaystyle x, \ x ^ {2}, \ x ^ {3},}$
Root functions : etc. ${\ displaystyle {\ sqrt {x}}, \ {\ sqrt [{3}] {x}},}$
natural exponential function : ${\ displaystyle e ^ {x}}$
natural logarithm : ${\ displaystyle \ ln x}$
Trigonometric functions : etc. ${\ displaystyle \ sin x, \ \ cos x, \ \ tan x,}$
Inverse trigonometric functions : etc. ${\ displaystyle \ arcsin x, \ \ arccos x, \ arctan x}$
Hyperbolic functions : etc. ${\ displaystyle \ sinh x, \ \ cosh x,}$
Inverse hyperbolic functions : etc. ${\ displaystyle \ operatorname {arsinh} x, \ \ operatorname {arcosh} x,}$

or can be generated from functions in this list in a finite number of steps by addition, subtraction, multiplication, division or concatenation.

Note that the secondary condition “in a finite number of steps” is important so that, for example, not all power series are elementary.

Examples

From the above definition it follows directly that the following functions are all elementary:

Addition, e.g. B. ( x +1)
Multiplication, e.g. B. (2 x )
Polynomial functions , e.g. B. ${\ displaystyle 7x ^ {5} + 4x ^ {4} -9x ^ {2} +1}$
Rational functions , e.g. B. ${\ displaystyle {\ frac {7x ^ {5} + 4x ^ {4} -9x ^ {2} +1} {x ^ {4} +9}}}$
${\ displaystyle {\ dfrac {e ^ {\ tan x}} {1 + x ^ {2}}} \ sin \ left ({\ sqrt {1 + (\ ln x) ^ {2}}} \ right) }$

Counterexamples

An example of a non-elementary function is the error function :

${\ displaystyle \ mathrm {erf} (x) = {\ frac {2} {\ sqrt {\ pi}}} \ int _ {0} ^ {x} e ^ {- t ^ {2}} \, dt .}$

It is not at all obvious that this function is not elementary, but it can be shown with the Risch algorithm .

The following functions are all elementary, but have no elementary antiderivative:

${\ displaystyle \ exp (x ^ {2})}$
${\ displaystyle {\ sqrt {\ log (x)}}}$
${\ displaystyle {\ frac {\ sin (x)} {x}}}$
${\ displaystyle \ exp (\ exp (x))}$
${\ displaystyle \ log (\ log (x))}$

Properties of the class of elementary functions

It follows directly from the definition that the class of elementary functions is closed under addition, subtraction, product and quotient formation, as well as concatenation. With the help of the product rule , quotient rule and chain rule , you can quickly see that the derivative of an elementary function is always elementary (provided that the function is differentiable).

Antiderivatives of elementary functions are often not elementary, such as B. the above mentioned error function.

Web links

Elementary functions at Encyclopedia of Mathematics
Eric W. Weisstein : Elementary function . In: MathWorld (English).

literature

JH Davenport: What Might "Understand a Function" Mean. In: M. Kauers, M. Kerber, R. Miner, W. Windsteiger: Towards Mechanized Mathematical Assistants. Springer, Berlin / Heidelberg 2007, pp. 55–65. (semanticscholar.org)
Maxwell Rosenlicht: Liouville's Theorem on Functions with Elementary Integrals. In: Pacific Journal of Mathematics. 24, No. 1, 1968, pp. 153-161.
Maxwell Rosenlicht: Integration in Finite Terms. In: The American Mathematical Monthly. 79, 1972, pp. 963-972.

Individual evidence

^ Ordinary Differential Equations . Dover, 1985, ISBN 0-486-64940-7 , pp. 17 ( online ).
↑ Elena Anne Marchisotto, Gholam-Ali Zakeri: An Invitation to Integration in Finite Terms

[1] Ordinary Differential Equations . Dover, 1985, ISBN 0-486-64940-7 , pp. 17 ( online ).

[2] Elena Anne Marchisotto, Gholam-Ali Zakeri: An Invitation to Integration in Finite Terms