Waves-Periodic Function

Periodic function and periodic motion go along with each other. Periodic motion is a repeated motion. The type of motion includes vibratory or oscillatory motion. Examples are the motion of a moon around the sun, the motion of a child sitting on a swing, etc. 

A formula denotes the periodic motion,

f=1/T

where,

T is the time taken by motion to repeat itself. The unit of a time period is in seconds.

f is the frequency. It is the number of times the motion is repeated in a second. It is denoted in-unit Hertz( Hz).

What is Periodic Function?

A function that repeats a pattern of devalues at regular intervals is the Periodic Function. A complete pattern is called a cycle and a period is the horizontal length of one cycle. The Periodic Function is used in science to describe waves, oscillations, and all the other events that exhibit periodicity. 

This Function y=f(x), having a period P, can be represented as f(x+P)= f(x)

P exists as a positive real number.

x belongs to real numbers

So, the least value of the positive real number P is known as the fundamental period.

Example of Periodic Function

The given below are the advanced Periodic function:

Euler’s Formula

This complex number formula is a combination of cosine and sine functions. These are the periodic functions.

Here, both of these are the periodic functions, and the formula given by Euler represents the periodic function and thus has a period of 2/k.

Fourier Series

The Fourier series is considered a superposition of several periodic wave function series. It is to form a complex periodic function. This is a composition of sine and cosine functions. These two wave functions are summated by assigning the respective weight elements to these series.

This series has different applications in the representation of heatwaves, vibration analysis, electrical engineering, image processing, etc.

Jacobi Elliptic Functions

In this function, the graph is elliptical instead of a circle which is usually seen for the trigonometric functions. The reason behind this elliptical shape is the involvement of the two variables together, like the amplitude and the speed of a moving body. It can also be the temperature and viscosity of the substance. Generally, these functions are applied in describing the pendulum’s motion.

Properties of Periodic Function

The points that will be discussed below will help you understand the Periodic Function better:

• The first point includes the graph of a periodic function. It is symmetric, and thus it repeats itself along the vertical axis after a certain interval.
• The period that repeats itself against the period of a periodic function is equal to that of the constant across the whole range of the function.
• The domain in the periodic function contains all the real number values. Also, the range of a Periodic Function is restricted for a fixed interval. 
• If f(x) is considered a periodic function with P as a period, then f(ax+b) is also the same function with a period as P/|a|.
• If f(x) is taken as a periodic function with the period as P, then f(x)+b is also a periodic function along with P.

How Are Periodic Function Generalised?

Bloch-periodic functions

It governs the solution of several periodic differential equations. It appears in a further generalisation of Bloch’s theorem and Floquet theory. It is represented as,

Ψnk=Ψ n(k+p)

f(x+P) = f(x)

Where,

k is a real or a complex number.

Antiperiodic functions

This is one of the common subsets of periodic functions. Antiperiodic function is a function f such that f(x+P)= -f(x) for all x.

Quotient spaces as domain

In signal processing, there is a need to encounter the given problem. The Fourier series represent the periodic functions, so these series satisfy the convolution theorems. Periodic Functions cannot be coiled together with the typical definition as the involved integrals diverge.

The finest way to describe a periodic function is that it is a bounded but periodic domain.

Conclusion

The periodic function is widely used in day-to-day life apart from these mathematical equations. For example, high tides and low tides can be predicted using the periodic functions as scientists use this to determine the height of the day’s water at different intervals of time. Therefore, the Periodic Function definition tells that it is a function that returns to the same value at regular intervals.