Elementary function

Elementary functions of a single variable x include:

• Constant functions: ${\displaystyle 2,\ \pi ,\ e,}$ etc.
• Rational powers of x: ${\displaystyle x,\ x^{2},\ {\sqrt {x}}\ (x^{\frac {1}{2}}),\ x^{\frac {2}{3}},}$ etc.
• Exponential functions: ${\displaystyle e^{x},\ a^{x}}$
• Logarithms: ${\displaystyle \log x,\ \log _{a}x}$
• Trigonometric functions: ${\displaystyle \sin x,\ \cos x,\ \tan x,}$ etc.
• Inverse trigonometric functions: ${\displaystyle \arcsin x,\ \arccos x,}$ etc.
• Hyperbolic functions: ${\displaystyle \sinh x,\ \cosh x,}$ etc.
• Inverse hyperbolic functions: ${\displaystyle \operatorname {arsinh} x,\ \operatorname {arcosh} x,}$ etc.
• All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions[6]
• All functions obtained by root extraction of a polynomial with coefficients in elementary functions[7]
• All functions obtained by composing a finite number of any of the previously listed functions

Certain elementary functions of a single complex variable z, such as ${\displaystyle {\sqrt {z}}}$ and ${\displaystyle \log z}$, may be multivalued. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function ${\displaystyle e^{z}}$ composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with ${\displaystyle z^{i}}$ instead provides the trigonometric functions.

Examples of elementary functions include:

• Addition, e.g. (x+1)
• Multiplication, e.g. (2x)
• Polynomial functions
• ${\displaystyle {\frac {e^{\tan x}}{1+x^{2}}}\sin \left({\sqrt {1+(\log x)^{2}}}\right)}$
• ${\displaystyle -i\log \left(x+i{\sqrt {1-x^{2}}}\right)}$

The last function is equal to ${\displaystyle \arccos x}$, the inverse cosine, in the entire complex plane.

All monomials, polynomials, rational functions and algebraic functions are elementary. The absolute value function, for real ${\displaystyle x}$, is also elementary as it can be expressed as the composition of a power and root of ${\displaystyle x}$: ${\textstyle |x|={\sqrt {x^{2}}}}$.

Some examples of functions that are not elementary:

• tetration
• non-elementary Liouvillian functions, including
  • the exponential (Ei), logarithmic (Li or li) and Fresnel (S and C) integrals.
  • the error function, ${\displaystyle \mathrm {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt,}$ a fact that may not be immediately obvious, but can be proven using the Risch algorithm.
• other Nonelementary integrals, including the Dirichlet integral and elliptic integral.