Periodic Function

It is a periodic function when a function repeats itself every time at regular intervals. The period of a function is an important aspect of periodic functions, and it aids in the definition of a function by providing a reference point. An exponentially periodic function y = f(x) with a period P can be expressed as f(X + P) = f(X).

Let’s study more about the periodic function, its features, and some instances of periodic functions in this section.

Periodic Function

It is considered to be a periodic function if there is an integer P such that the function f(x + P) = f(x), when all of the inputs to the function are real integers, and this is true for all x is real. The fundamental period of a function is defined as the value of the positive real number P that has the least value. This fundamental period of a function is also referred to as the period of the function, because it is the period during which the function repeats.

f(x + P) = f(x)

Because the sine function has a period of 2π, it is a periodic function. Sin(2π + x) equals Sinx.

Some Important Periodic Functions and Their Periods

In order to determine the interval after which the range of a periodic function repeats itself, we need to know how long the function has been in existence. When a periodic function f(x) is defined, its domain encompasses all possible real number values of x; however, the range of a periodic function is defined as a set of values inside an interval. The period of a periodic function is the length of this repeating interval, or the interval after which the range of the function repeats itself, and it is defined as the interval between two consecutive repetitions of the function.

These are some of the most important periodic functions and their periods are as follows:

• The period of Cosx and Sinx  is 2π.

• The period of Cotx and  Tanx  is π.

• The period of Cosecx and  Secx  is 2π.

Periodic Functions Have Certain Characteristics

The following characteristics of a periodic function are useful in developing a more in-depth understanding of the ideas of periodic functions.

• The graph of a periodic function is symmetric and repeats itself along the horizontal axis, indicating that the function is periodic.

• The periodic function’s domain comprises all of the real number values, and the periodic function’s range is determined for a fixed interval of time.

• Across the entire range of a periodic function’s range, the period against which the period repeats itself equals the constant of the function.

• Assuming that f(x) is a periodic function with a period of P, it follows that 1/f(x) is also a periodic function with the same fundamental period of P.

• Assuming that f(x) has a periodic period of P and that f(ax + b) has a period of P/|a|, we can conclude that f(ax + b) has a period of P/|a|.

• Then if f(x) is a periodic function with a period of P, then af(x)+b is a periodic function with a period of P, and so on.

Some of the most important periodic functions are as follows

The following are some of the more complex periodic functions that can be investigated in further depth.

• Euler’s Formula is as follows: The complex number formula eix = Coskx + iSinkx is made up of the cosine and sine functions, both of which are periodic functions, as well as the square root of the square root of the sine function. Both of these functions are periodic, and the euler’s formula represents a periodic function with a period of 2π/k, as seen in the following equation.

• In contrast to the graphs of trigonometric functions, the Jacobi Elliptic Functions have an ellipse shape rather than a circle, as is common for these functions. Elliptical forms are formed when two variables are combined, such as the amplitude and speed of a moving body, or the temperature and viscosity of a liquid or a solid. In the description of the motion of a pendulum, these functions are frequently used. 

• Fourier Series: The Fourier series is a superposition of various periodic wave function series that results in a complex periodic function. Fourier Series: It is often formed of sine and cosine functions, and the summation of these wave functions is obtained by giving relevant weight components to each series in the series of wave functions. The Fourier series has applications in the modelling of heatwaves, vibration analysis, quantum physics, electrical engineering, signal processing, and image processing, to name a few areas of research.

Conclusion

It is a periodic function when a function repeats itself every time at regular intervals. The period of a function is an important aspect of periodic functions, and it aids in the definition of a function by providing a reference point. An exponentially periodic function y = f(x) with a period P can be expressed as f(X + P) = f(X).In order to determine the interval after which the range of a periodic function repeats itself, we need to know how long the function has been in existence.The graph of a periodic function is symmetric and repeats itself along the horizontal axis, indicating that the function is periodic.The periodic function’s domain comprises all of the real number values, and the periodic function’s range is determined for a fixed interval of time.