FunctionPeriod

The fundamental period of f in x, or 0 if f is provably not periodic.

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FunctionPeriod(f, x)the smallest positive p with f(x+p) = f(x) for every x, or 0 when f is not periodic.

Typed signatures

Type	Package	Applies
`(expression, symbol) -> expression`	`analytic`	always

Domain: Elementary functions

Details

A nonconstant polynomial, a genuine rational function, Sqrt, Ln, and Exp of a nonconstant affine argument are all provably non-periodic (each is monotonic or unbounded), so this answers 0 for them rather than declining.
Sin/Cos(a x + b) has period $2\pi/|a|$ ; Tan(a x + b) has period $\pi/|a|$ .
Does not yet combine periods across a sum of trig terms with different (even commensurate) frequencies, such as $\sin x + \cos 2x$ — that composition is outside the single-shape classifier this file is built on, and is declined.

Examples

1 more kept as data, checked against the oracles

Basic #

Sine's period is 2*pi

\mathrm{FunctionPeriod}(\sin(x), x)

Out

2π

Known value · Wolfram 15.0.0, hand cross-checked

Tangent's period is pi, half of sine's

\mathrm{FunctionPeriod}(\tan(x), x)

Out

π

Known value · Wolfram 15.0.0, hand cross-checked

Doubling the argument halves the period

\mathrm{FunctionPeriod}(\sin(2x), x)

Out

π

Known value · Wolfram 15.0.0, hand cross-checked

Exp is monotonic, hence not periodic

\mathrm{FunctionPeriod}(e^{x}, x)

Out

0

Known value · Wolfram 15.0.0, hand cross-checked

Scope #

With a symbolic frequency the period is left unevaluated even under $a>0$

\mathrm{Assuming}(a\gt0, \mathrm{FunctionPeriod}(\sin(ax), x))

Out

\mathrm{FunctionPeriod}(\sin(ax), x)

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