Definition:Elementary Function

Definition

An elementary function is one of the following:

• The constant function: $\map {f_c} x = c$ where $c \in \R$

• Powers of $x$: $\map f x = x^y$, where $y \in \R$

• Exponentials: $\map f x = e^x$

• Natural logarithms: $\map f x = \ln x$

• Trigonometric functions: $\map f x = \sin x$, $\map f x = \cos x$

• Inverse trigonometric functions: $\map f x = \arcsin x$, $\map f x = \arccos x$

• All functions that are compositions of the above, for example $\map f x = \ln \sin x$, $\map f x = e^{\cos x}$

• All functions obtained by adding, subtracting, multiplying and dividing any of the above types any finite number of times.

Examples

Examples of elementary functions include:

$2 x - \ln x + \dfrac {e^{\sin x} } x$

$\paren {\arccos x}^2 + \dfrac {e^{x^2 + 2x + 1} } {\sqrt {3 x} } - \map \ln {\ln 4 x}$

Also see

• Definition:Transcendental Function

• Results about the elementary functions can be found here.

Sources

• 1946: F.E. Relton: Applied Bessel Functions ... (previous) ... (next): Chapter $\text {I}$: The Error Function; Beta and Gamma Functions: $1 \cdot 1$. The study of functions
• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text E$: The Elementary Functions
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $10$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): elementary: 2.
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): elementary function
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): elementary function