Periodic Functions: Definition, Examples, Properties

The periodic functions are such functions whose values are repeated at regular intervals or periods. The period of a periodic function is its main characteristic. In this article, we study the periodic functions with their examples and properties.

Periodic Function Definition

A function f(x) is called a periodic function if f(x) is defined for all real x, except possible at some points, and if there is a positive number p such that

f(x+p) = f(p) for all x.

The number p is called the period of the function.

For example, f(x)=tanx is a periodic function with period π. This is because tan(π+x)=tanx. Note that tanx is not defined for x = ±π/2, ±3π/2, ±5π/2, …

Periodic Function Example

The following are examples of periodic functions.

• sinx is a periodic function with period 2π.
• cosx is a periodic function with period 2π.
• tanx with period π.
• cotx with period π.

Note that the functions f(x) = x, x², x³, e^x, sinhx, coshx, lnx are not periodic in ℝ.

Properties of Periodic Functions

The properties of a periodic function is listed below.

1. Fundamental Period:

Let f(x) be a periodic function with period p, that is, f(x+p) = f(x) ∀ x.

Now,

f(x+2p) = f(x+p+p) = f(x+p) = f(x). f(x+3p) = f(x+2p+p) = f(x+2p) = f(x).

Continuing in this fashion, we deduce that

f(x+np) = f(x) for any integer n = 1, 2, 3, …

Thus, np is also a period of f(x) for any positive integer n.

The smallest positive period of a function is called the fundamental period. For example, cosx, sinx has fundamental period 2π, and that of tanx, cotx is π.

2. Linearity:

Let f(x), g(x) be two periodic functions with period p. Then for any constants a and b, af(x)+bg(x) is also a periodic function with period p. This is because af(x+p)+bg(x+p) = af(x)+bg(x).

3. Integral of a Periodic Function:

For a periodic function f(x) with period p, we have

$\boxed{\int_a^{a+p} f(x)~dx = \int_0^{p} f(x)~dx}$

Related Articles:

• Even and Odd Functions
• Orthogonal and Orthonormal Functions

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