Elementary function

An elementary function is formed from constants and the identity function by finitely many arithmetic operations, composition, roots, exponentials, logarithms, and trigonometric functions.

Arithmetic operation: An operation that combines numbers using addition, subtraction, multiplication, or division. Arithmetic operations are among the permitted steps for constructing elementary functions.
Non-elementary function: A function that cannot be expressed using finitely many elementary operations and functions. It names the functions outside the class defined by these construction rules.
Differential algebra: The study of algebraic structures equipped with a derivation, especially fields of differential functions. Differential algebra formalizes how elementary functions and their derivatives extend differential fields.
Elementary antiderivative: An antiderivative expressible using elementary functions and the variable of integration. Many elementary integrands still lack antiderivatives within the same function class.
Function composition: An operation that applies one function to the output of another. Composition lets elementary expressions nest functions such as logarithms and trigonometric functions.
Error function: A special function proportional to the integral of exp(−t²) from zero to its argument. Its derivative is elementary, but the function itself has no elementary expression.
Liouville's theorem: A result characterizing when an elementary antiderivative can exist within differential field extensions. It supplies a theoretical test for whether an elementary antiderivative is possible.
Differential equation: An equation relating an unknown function to one or more of its derivatives. Solutions to differential equations often require functions outside the elementary class.
Exponential function: A function of the form f(x)=a^x, with fixed positive base a. Exponentials are one of the defining function families in the construction.
Gamma function: A function extending the factorial to complex arguments through an integral or analytic continuation. The gamma function is a standard special function beyond elementary formulas.

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Differentiation rulesNarrower topic: The rules cover derivatives of these functions and their combinations.