Elementary function

The elementary functions (of x) include:

• Constant functions: ${\displaystyle 2,\ \pi ,\ e,}$ etc.
• Powers of ${\displaystyle x}$: ${\displaystyle x,\ x^{2},\ x^{3},}$ etc.
• Roots of ${\displaystyle x{\text{: }}{\sqrt {x}},\ {\sqrt[{3}]{x}},}$ etc.
• Exponential functions: ${\displaystyle e^{x}}$
• Logarithms: ${\displaystyle \log x}$
• 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 any of the previous functions[6]
• All functions obtained by composing previously listed functions

Some elementary functions, such as roots, logarithms, or inverse trigonometric functions, are not entire functions and may be multivalued.

Examples of elementary functions include:

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

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

All monomials, polynomials and rational functions are elementary. Also, 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 ${\displaystyle x}$: ${\displaystyle |x|={\sqrt {x^{2}}}}$.

An example of a function that is not elementary is 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.

• See also the examples in Liouvillian function and Nonelementary integral.