Finding a period of an expression

### 8.2.2 Finding a period of an expression

The period command finds a period of a given periodic expression.

- period takes one mandatory argument and one optional argument:
  - expr, an expression f(x) where x∈ℝ.
  - Optionally, x, a real variable (by default x).
- period(expr ⟨,x ⟩) returns a (not necessarily the smallest) period of f or +∞ if f is not periodic. Note that +∞ is also returned if period is unable to find a period. For periodic functions with arbitrary small periods, such as constant functions, zero is returned.
- period is able to detect periodicity of functions built from the basic periodic functions ga(x)=eiax (which covers for the standard trigonometric functions by Euler’s formula) and/or functions returned by the periodic command (see Section 8.2.1).

##### Remark.

If periodic returns zero for an expression f(x), it does not always mean that f is constant. For example, f(x)=⌊ x−⌊ x−1⌋⌋ is always equal to 1 because 1≤ x−⌊ x−1⌋<2, but g(x)=⌈ x−⌊ x−1⌋⌉ is not constant by the same argument (for x∈ℤ, g(x) is equal to 1, and for x∉ℤ its value is 2). However, periodic returns zero for both functions, as it is unable to detect changes in isolated points.

#### Examples

To define and display a square wave, you can enter:

sw:=sign(sin(pi*x)):; plot(sw,x)

You observe that the period is equal to 2. Indeed:

period(sw)

2 |  |  |  |  |  |  |  |  |  |

Any rational function with periodic variables is periodic if the variable periods are commeasurable (i.e. if the quotient of any two periods is a rational number). Also, if f is periodic, then g∘ f is periodic for any function g. For example, define two periodic functions f and g with periods 2 and 3, respectively, and then the function h(x)=f(x)g(x)/ln(1+f(x)2+g(x)4):

f:=periodic(x^2,x=0..1):; g:=periodic(x^3,x=-1..1):; h:=f*g/ln(1+f^2+g^4):; plot(h)

The above graph indicates that h is possibly periodic; indeed, its period is equal to 6, which is the least common multiple of the periods of f and g.

period(h)

6 |  |  |  |  |  |  |  |  |  |

A function built from periodic functions with periods which are not commeasurable is not periodic. For example, f(x)=sin(x)+sin(π x) is not periodic:

period(sin(x)+sin(pi*x),x)

+∞ |  |  |  |  |  |  |  |  |  |

Often (but generally not), the smallest period is returned, such as for the function f(x)=sin2 x, for instance.

period(sin(x)^2,x)

π |  |  |  |  |  |  |  |  |  |