Periodic Function

The motion of a swing or a rocking chair repeats at regular intervals, indicating periodic motion. A periodic function is like a pattern that repeats after a fixed interval. Even though periodic motion and oscillatory motion may seem similar, they are not the same, periodic motion refers to any motion that repeats at regular intervals, while oscillatory motion is a specific type of periodic motion that moves back and forth around an equilibrium point. A periodic function is a function that repeats its values at regular intervals. 
 

The properties of periodic functions help us understand periodic functions better. Here are some of the properties of periodic functions:

• The graph of a periodic function repeats its pattern at regular intervals along the horizontal axis. 
• A periodic function is defined for all real numbers as its domain, while its range depends on the values the function takes within one period. 
• The period of a periodic function is the constant interval after which its values repeat.
• If f(x) is a periodic function with period P, then 1f(x) is also a periodic with the same period P.
• If f(x) has a period P, then f (ax + b) becomes a periodic with a new period 1|a|.
• If f(x) has a period P, then af(x) + b will still be periodic with the same period P. 
   

The period of a function is found using the steps added below:

Step 1: The periodic function is a function that repeats its value at regular intervals.

Step 2: The periodic function is represented as f(x + p) = f(x), where p represents the period of the function, and it is a real number, p ∈ R.

Step 3: The period is the time between two consecutive repetitions of the wave. 
 

In trigonometry, we have three fundamental functions, namely, sine (sin), cosine (cos), and tangent (tan). The periods of sin, cos, and tan are 2, 2, and . The starting point of the graph of any trigonometric function is taken as x = 0. Trigonometric functions are periodic functions, and the period of the trigonometric functions is given below:

Period of Sin x and Cos x is 2
I.e. sin(x + 2) = sin x and cos(x + 2) = cos x

Period of Tan x and Cot x is 
tan(x + ) = tan x and cot(x + ) = cot x

Period of Sec x and Cosec x is 2
sec(x + 2) = sec x and cosec (x + 2) = cosec x
 

If p is the period of the periodic function f(x),  then 1/f(x) is also a periodic function and will have the same fundamental period of p as f(x).
If f(x + p) = f(x)
F(x) = 1/f(x), then F(x + p) = F(x)

If p is the period of the periodic function f(x), then f(ax + b), a > 0 is also a periodic function with a period of p/|A|.
Period of Sin(ax + b) and Cos(ax + b) is 2/|A|.
Period of Tan(ax + b) and Cot(ax + b) is /|A|.
Period of Sec(ax + b) and Cosec(ax + b) is 2/|A|.

If p is the period of the periodic function f(x), then af(x) + b, a > 0 is also a periodic function with a period of p. 
Period of [a Sin x + b] and [a Cos x + b] is 2.
Period of [a Tan x + b] and [a Cot x + b] is .
Period of [a Sec x + b] and [a Cosec x + b] is 2.
 

Some periodic functions are more complex and are used in advanced mathematics and science. They help in describing patterns and motions that cannot be explained by simple sine or cosine functions:

Euler's Formula: The formula, eix = cos x + i sin x, combines both sine and cosine, which are periodic. It repeats its values every 2.

Jacobi Elliptic Functions: These functions create oval-shaped graphs instead of circular ones like sine and cosine. They are used to explain things like pendulum motion or the relation between speed and position. 

Fourier Series: It is like combining many sine and cosine waves to form a more complex repeating wave. It is used in things like studying heat, vibrations, signals, and even images.
 

Periodic functions are used in many real-life situations where a cycle or pattern repeats over time. Periodic functions help us understand and predict regular changes, making them useful in science, engineering, and daily life.

• Electrical Engineering: Periodic functions are used to design and study electric circuits. Alternating Current (AC), which powers our homes, changes direction in a repeating pattern like a sine wave.
• Music and Sound Waves: Sound is produced by vibrations that repeat regularly. Musical notes come from these repeating sound waves, and periodic functions are used for tuning instruments and creating digital music. 
• Astronomy: Scientists use periodic functions to predict events such as eclipses, phases of the moon, and the movement of planets and stars, as their motions often repeat in cycles. 
• Medical Field: Doctors use periodic patterns in ECG and EEG tests to check heart and brain health. Brain waves and heartbeats are examples of periodic biological signals.
• Weather and Climate: Weather changes, tides and seasons follow repeating patterns. Meteorologists use periodic functions to study and forecast these trends.