Period of a Function

Question 1

What is the period of a function? (Definition)

Answer

The period $t$ of a periodic function $f(x)$ is the smallest value $t$ such that $f(x+t) = f(x)$

Graphically, its curve is repeated over the interval of each period. The function is equal to itself for every cycle of length $t$ (it presents a pattern/graph that is repeated by translation).

The value of the period $t$ is also called the periodicity of the function or fundamental period.

Question 2

How to find the period of a function?

Answer

To find the period $t$ of a signal or a function $f(x)$ , demonstrate that $f(x+t)=f(x)$

Example: The trigonometric function $\sin(x + 2\pi) = \sin(x)$ so $\sin(x)$ is periodic of period $2\pi$

Trigonometric/sinusoidal functions are usually periodic, with a period $2\pi$ , to guess the period, try multiples of pi for value $t$ .

If the period is equal to 0, then the function is not periodic.

Question 3

How to find the value f(x) of a periodic function?

Answer

Any periodic function of period $t$ repeats every $t$ values. To predict the cycle value of a periodic function, for a value $x$ calculate $x_t = x \mod t$ (modulo t) and find the known value of $f(x_t) = f(x)$

Example: The function $f(x) = \cos(x)$ has a period of $2\pi$ , the value for $x = 9 \pi$ is the same as for $x \equiv 9 \pi \mod 2 \pi \equiv \pi \mod 2 \pi$ and therefore $\cos(9 \pi) = \cos(\pi) = -1$

Question 4

How to find the amplitude of a periodic function?

Answer

The amplitude is the absolute value of the non-periodic part of the function.

Example: $a \sin(x)$ has for amplitude $| a |$

Question 5

How to prove the periodicity of a function?

Answer

The demonstration of the existence of a period $t$ for a function $f$ consists in calculating if the equation $f(x+t)=f(x)$ is true.

Question 6

How to prove that a function is not periodic?

Answer

If $f$ is periodic, then it exists a real not null such as $f(x+t)=f(x)$

Demonstration consists in proving that it is impossible. For example with a reductio ad absurdum or performing a calculation that leads to a contradiction.

Question 7

What are usual periodic functions?

Answer

The most common periodic functions are trigonometric functions based on sine and cosine functions (which have a period of 2 Pi).

Function	Period
Sine $\sin(x)$	$2\pi$
Cosine $\cos(x)$	$2\pi$
Tangent $\tan(x)$	$\pi$

Period of a Function