Definition:Periodic Real Function/Period

Definition

Let $f: \R \to \R$ be a periodic real function.

The period of $f$ is the smallest value $L \in \R_{>0}$ such that:

$\forall x \in \R: \map f x = \map f {x + L}$

Also defined as

Some sources define a period of a periodic function $f$ as any $L \in \R_{>0}$ such that $\forall x \in \R: \map f x = \map f {x + L}$, not necessarily the smallest.

Also known as

The period of a periodic function $f$ is also known as the principal period of $f$, particularly by those sources which define a period of $f$ as any $L \in \R_{>0}$ such that:

$\forall x \in \R: \map f x = \map f {x + L}$

Sources

• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $1$. Functions: $1.5$ Trigonometric or Circular Functions: $1.5.2$ Sine Function
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.4$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): period: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): period, periodic