Periodic Function - Definition, Examples, Formula, Equations

A periodic function is a function that repeats its values at regular intervals or periods. Periodic functions are fundamental in various branches of mathematics, physics, and engineering, especially in the study of waves, oscillations, and signal processing.

In this article, we will cover the concept of periodic function. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

Periodic function

A function $f(x)$ is called a periodic function, if there exists a +ve real number $T$ such that $f(x+T)=f(x) \forall x$ belongs Domain of $f(x)$.

Here, $T$ is called the period of $f(x)$, where $T$ is the least $+v e$ value.

Graphically: if the graph repeats at a fixed interval, the function is said to be periodic and its period is the width of that interval.

Eg

Graph of $sin(x)$ is repeated at an interval of $2π$

Some standard results of periodic function

Properties of the periodic function

i) if $f(x)$ is periodic with period $T$, then
1. $\operatorname{cf}(x)$ is periodic with period $T$
2. $f(x+c)$ is periodic with period $T$
3. $f(x) \pm c$ is periodic with period $T$, where $c$ is any constant.

ii) if $f(x)$ is periodic with period $T$, then $k f(c x+d)$ has period $\frac{T}{|c|}$ i.e. period is only affected by the coefficient of $x$,
iii) if $f_1(x), f_2(x)$ are periodic functions with periods $T_1, T_2$ respectively, then $h(x)=f_1(x)+f_2(x)$ has a period
a). LCM of $\left\{T_1, T_2\right\}$, if $\mathrm{h}(\mathrm{x})$ is not an even function.
or
b) 0.5 LCM of $\left\{T_1, T_{2\} \text {, if }} f_1(x)\right.$ and $f_2(x)$ are complementary pairwise comparable functions.

Some Important Periodic Functions

The following are some of the advanced periodic functions, which can be explored further.

Euler's Formula: The periodic functions sine and cosine make up the complex number formula $e^{i x}=\cos k x+i \sin k x$ In this case, the two functions are periodic, and the periodic function represented by the euler's formula has a period of $2π/k$.

Jacobi Elliptic Functions: Unlike trigonometric functions, which typically have a circle-shaped graph, these functions have an elliptical graph. These elliptical forms result from the simultaneous involvement of two variables, such as the temperature and viscosity of the material or the amplitude and speed of a moving body. These functions are frequently employed to explain a pendulum's motion.

Fourier Series: The Fourier series is a complex periodic function that is created by superimposing different periodic wave function series. It is often made up of sine and cosine functions, and the sum of these wave functions is calculated by giving each series the appropriate weight component. Applications of the Fourier series include vibration analysis, electrical engineering, signal processing, quantum mechanics, heatwave representation, and image processing.

How to Determine the Period

1. Inspection:

   • For standard trigonometric functions like $sin⁡(x), cos⁡(x)$, and $tan⁡(x)$, the periods are well-known: $2π$ for sine and cosine, $π$ for tangent.

2. Algebraic Method:

   • For a function $f(x)$ defined by a formula, set up the equation $f(x+T)=f(x)$ and solve for $T$.

3. Graphical Method:

   • Plot the function and visually inspect the distance between repeating patterns to identify the period.

Solved Examples Based On the Periodic Functions

Example 1: What is the period of $f(x)=\cos (\cos x)+\cos (\sin x)$ ?
Solution:
The period of $\cos (\cos x)$ is $\pi$ and it is an even function.
The period of $\cos (\sin x)$ is $\pi$ and it is an even function
Thus, the period of $f(x)$ is $1 / 2($ LCM of $\pi, \pi)=\pi / 2$
Hence, the answer is $\pi / 2$.

Example 2: What is the period of tan(5x)?

Solution:

As we have learned in Properties of the Periodic function

If $f(x)$ is periodic with period $T$, then $k f(c x+d)$ has a period $\frac{T}{|c|}$

Now,

As the period of $tan(x)$ is $π$, the period of $tan(5x)$ is $π/5$.

Hence, the answer is $π/5$.

Example 3: Let $f(x)$ be a differentiable function for all $x \in R$. The derivative of $f(x)$ is an even function. If the period of $f(2 x)$ is 1 , then the value of $f(2)+f(4)-f(6)-f(8)$ is
Solution:

$\begin{aligned}
& f(2 x)=f(2 x+1) \\
& f(x)=f\left(x+\frac{1}{2}\right)
\end{aligned}$

So period of $f(x)_{\text {is }} \frac{1}{2}$

$\begin{aligned}
& f(4)=f\left(2+\frac{1}{2} \times 4\right) \\
& f(6)=f\left(2+\frac{1}{2} \times 8\right) \\
& f(8)=f\left(2+\frac{1}{2} \times 12\right)
\end{aligned}$

So $f(2)=f(4)=f(6)=f(8)$
So value of $f(2)+f(4)-f(6)=f(8)=0$

Hence the answer is $0$.

Example 4: The function $f(x)=|\sin 4 x|+|\cos 2 x|$, is a periodic function with a period:

Solution:

As we have learned,
- If period of $f(x)$ is $T$, then period of $f(c x+d)$ is $T /|c|$
- If period of $f(x)$ is $T1$ and period of $g(x)$ is T2, then period for $f(x)+$ $\mathrm{g}(\mathrm{x})$ is the lcm of $T1$ and $T2$
- Period of $|\sin (\mathrm{x})|$ and $|\cos (\mathrm{x})|$ is $\pi$

Now,
As period of $|\sin (x)|$ is $\pi$, so period of $|\sin (4 x)|$ is $\pi / 4$, and
As period of $|\cos (\mathrm{x})|$ is $\pi$, period of $|\cos (2 \mathrm{x})|$ is $\pi / 2$
So period of $|\sin (4 x)|+|\cos (2 x)|$ is the $\mid c m$ of $\pi / 4$ and $\pi / 2$, which is $\pi / 2$.
Hence, the answer is $\pi / 2$.

Example 5: What is the period of $\sin x / 2+\cos 2 x$ ?
Solution:

As we have learned
Period of a Trigonometric Ratio -
Period of $\sin x$ and $\cos x$ is $2 \pi$
- wherein

$\sin (2 n \pi+x)=\sin x$
$\cos (2 n \pi+x)=\cos x$
period of $\sin x / 2$ is $4 \pi$
period of $\cos 2 x$ is $2 \pi / 2=\pi$
Thus period $=4 \pi$
Hence, the answer is $4 \pi$.

Summary

Periodic functions repeat their values at regular intervals and are characterized by their fundamental period T. Examples include trigonometric functions like sine, cosine, and tangent, as well as other oscillatory functions. Understanding periodic functions is crucial in many scientific and engineering disciplines, where they model repeating phenomena and signals. Identifying the period of a function can be done through inspection, algebraic methods, or graphical analysis.