Periodic Function

“Mathematicians have concentrated on the representation of periodic functions using trigonometric series since they are frequently encountered in engineering. We present several simple methods for creating periodic functions in this note, which can be implemented in a variety of computer algebra systems. This makes it easy to grasp how the periodic function’s trigonometric series approaches the periodic function.”

A body is considered to be in periodic motion if the motion it is doing is repeated at regular intervals, such as in a rocking rocker or a swing. The following is the definition of a periodic function:

A function returns to the same value at regular intervals.

Despite the fact that periodic and oscillatory motions sound the same, not all periodic motions are oscillatory motions. The main distinction between a periodic motion and an oscillatory motion is that the former is applicable to any motion that repeats over time, whilst the latter is only applicable to motions that occur around an equilibrium point or between two states. All periodic motions can be defined by a periodic function.

Consider a pendulum bob that is oscillating along with its equilibrium position. If the bob is oscillating, its displacement will also fluctuate from zero to positive and back to zero and negative.

Periodic Function Formula

For some non-zero constant P, a function f is said to be periodic if:

f (x+P)=f(x)

For all possible x values in the domain. A period of the function is a non-zero constant P for which this is the case.

Properties of Periodic Function

The qualities listed below can help you comprehend the idea of a periodic function better.

• A periodic function’s graph is symmetric and repeats down the horizontal axis.
• The range of the periodic function is defined for a fixed interval, and the domain of the periodic function contains all real number values.
• The constant spanning the whole range of a periodic function’s period, against which the period repeats itself, is equal to the period of the function.
• If f(x) has a periodic period of P, then 1/f(x) will have the same fundamental period as f(x).
• If f(x) has a periodic period of P, then f(ax + b) has a periodic period of P/|a|.
• If f(x) has a periodic period of P, then af(x) + b has a periodic period of P as well.

Periodic Functions of Significant Importance

Some sophisticated periodic functions that can be investigated further are listed below.

Euler’s Formula: The cosine and sine functions, which are periodic functions, are used in the complex number formula eikx=Coskx+iSinkx. These two functions are periodic in this case, and the Euler’s formula is a periodic function with a wavelength of 2π/k.

Jacobi Elliptic Functions: In contrast to trigonometric functions, these functions feature an oval graph rather than a circle. These elliptical shapes are the result of the interaction of two parameters, such as the amplitude and speed of a moving body, or the temperature and viscosity of the substance. These functions are widely used to explain the motion of a pendulum.

Fourier Series: The Fourier series is a complex periodic function formed by superimposing several periodic wave function series. It is commonly made up of sine and cosine functions, and the sum of these wave functions is calculated by giving weight components to each of these series. Heatwave representation, vibration analysis, quantum physics, electrical engineering, signal processing, and image processing are all applications of the Fourier series.

What are Periodic Function Used for

• Any phenomena involving regular, recurring behavior is referred to as a periodic function. The most basic example is circular motion with constant velocity, such as the Moon’s orbit around the Earth. Wave motion and behavior, which describes things like sound (a pressure wave in air), water motion, and, most importantly, electromagnetic phenomena, is the most common example (light, electricity, magnetism).
• An analogue clock hangs on my wall. A periodic function with a period of one minute can be used to explain the movements of its second hand.

Conclusion

The sum of sines and cosines can be used to depict a periodic function. The French mathematician and scientist Jean Baptiste Joseph Fourier (1768-1830) discovered this truth, and his work was published in 1822. It is recognised as one of the nineteenth century’s most revolutionary contributions. Peter Gustav Lejeune Dirichlet (1805–59) later confirmed his theorem.

Because the first few terms of the infinite series are usually sufficient for many applications, the fact that all periodic functions may be represented as a sum of sinusoidal components indicates a practical approximation of the function. It also includes an analysis tool for compound waveform decomposition.