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# Periodic Function

## Introduction

In order to understand the study of waves and signals, a periodic table is used. In physics, it is observed that the motion is responsible for returning to the same value at a particular interval of time. Based on these specific concepts, all kinds of periodic motions are referred to as periodic functions. Focussing on such concepts this tutorial includes the information related to the periodic function along with its properties of it and the formula is included as well.

## What is Periodic Function?

Over the course, based on a fixed interval of time, the motion that occurs repetitively is generally known as a periodic function. Thus, the periodic function can be defined as a particuler function that comes back to a similar value per unit of time (Zafar et al. 2020). The periodic motion of periodic function can be classified into two different styles oscillatory motion and simple harmonic motion.

**Figure 1: Periodic function**

However, a difference is present between the oscillatory and periodic motion as it is observed that the periodic motion shares a relevance to any motion that shows a reputation over a period of time (Analyzemath.com, 2022).

On the other hand, oscillatory motion refers to such a kind of motion that is executed within two states or about an equilibrium point as represented in the below figure.

**Figure 2: Oscillatory motion**

In order to understand these two different concepts, the motion represented by a pendulum is required to be taken under consideration. Having a particular equilibrium position, the pendulum creates an arc of awing where the motion makes the pendulum move in to and fro motion.

## Properties of Periodic Function

In case of having a deeper understanding of this concept, one needs to know the properties of the periodic function.

- The first attribute that is necessary to be known is that the graph of the periodic “function is symmetric” that repeats itself along the axis that is horizontal (Galileospendulum.org, 2022).
- Next, “the domain of the periodic function includes all” values that are real numbers and “the range of the periodic” motion is recognised at a “fixed interval”. The period, against which the object repeats itself in an equal consistent motion, is considered consistency across the range of the whole function.

## Important periodic Functions

**“Euler's Formula”**

Named after the mathemetician “Leonhard Euler”, it is considered a “mathematical formula” that helps to establish a relationship between exponential functions and trigonometric functions (Galileospendulum.org, 2022). This formula is generally used in order to represent a “particular periodic” function that “has a period of 2π/k”.

**“Jaccobi Elliptic Functions”**

**Figure 3: Jaccobi Elliptic Functions**

The shape of the graphs of this kind of function is elliptical than being a circle, which is used for solving “trigonometric functions” (Semanticscholar.org, 2022). With the “involvement of two variables such as amplitude and speed”, the elliptical shapes are raised. For describing the motion of a pendulum the usage of these kinds of functions is observed.

### Fourier Series

**Figure 4: Fourier Series**

The superposition of different “periodic wave function series” is res[onsible for the formation of the strongest periodic function that is generally referred to by the name of Fourier Series. The composition of such function is generally conducted by sin and cousin functions. In the representation of the heatwaves, quantum mechanics, and vibration analysis the application of the Fourier series is observed.

## The Formula of Periodic Functions

In order to calculate the periodic function, the particular formula that is used is f(x + P) = f(x). In this formula, the f is used in reference to the periodic function where this component is considered as a non-zero constant that is represented by the P for the representation of the values of x. On the other hand, if the function of h is extended to all of R, the equation will be turned into h(t + 2)= h(t) ( Geogebra.org, 2022). This states that the value of the equation depends on certain aspects like P needs to be necessarily a real number. The time travelled between two waves needs to be constant.

## Examples of Periodic Function

In order to find the period of Tan3x + Sin5x/2, the period of Tanx will be considered as π. Therefore, the period of Tanx will be equivalent to π/3. On the contrary, the period of Sinx is 2π. Based on this conceptualisation, the value of Sin5x/2 will be equivalent to 2π/5/2 (Analyzemath.com, 2022).

This formula will be equal to the value of 4π/5. On the basis of the above equations found, “the calculation of the” periodic “function f(x)= tan3x + Sin5x/2” will be as follows:

$$\mathrm{f(x)=\frac{LCM \:of \:π \:and \:4π}{HCF\: of \:3 \:and \:5}}$$The value of this particular calculation will be 4π/1 which will be equivalent to 4π. Therefore, the final value of the period found in the equation provided Tan3x +Sin5x/2 is 4π.

## Conclusion

The tutorial has cast light on the explanation related to the periodic function along with the representation of the formula of the “periodic function”. It has been seen that function “f(x )” is generally recognised to be a “periodic function” if the presence of a real number is observed. The “period of the periodic function” refers to the length of the repeating interval. Besides this, several properties have been stated that include the periodic function needs to the representation of a real number and the graph needs to be symmetric.

## FAQs

**Q1.What is a period in periodic function?**

It has been known that the periodic function must contain a repetitive pattern of the object. The repetitive pattern has a graph representation that is consistent with the period of time that represents the similar kind of interval length of the time between every circle. This length of such repetitive interval is referred to as the period in the periodic function.

**Q2.What is the way to recognize a function as a periodic function?**

In such case identification of a specific range of the reputation that is observed in the function and the presence of a numerical value that is positive helps to determine the equation as a periodic function. As per the mentioned formula of the periodic function, that is f(x + T) = f(x), the value of T needs to be determined as the minimum.

**Q3.What is the phase shift of a periodic function?**

The phase shift of the values is generally determined by the change observed in the values of the range at regular intervals. The instance of phase-shifting includes the positive values.

**Q4.What is the range of a periodic function?**

It has been observed that periodic function includes the ranges that are subject to a limited value. The range is generally represented by a range of positive values. This set of values is subjected to the reputation for achieving different domains in values. The range of periodic functions refers to similar kinds of higher values.

## References

### Journals

Zafar, A., Raheel, M., Ali, K. K., & Razzaq, W. (2020). On optical soliton solutions of new Hamiltonian amplitude equation via Jacobi elliptic functions. The European Physical Journal Plus, 135(8), 1-17. Retrieved from: https://www.researchgate.net

### Websites

Analyzemath.com, (2022). About Periodic Functions. Retrieved from: https://www.analyzemath.com/function/periodic.html [Retrieved on: 7th June 2022]

Galileospendulum.org. (2022). About Physics Quanta: The Pendulum’s Swing. Retrieved from: https://galileospendulum.org/2011/05/24/physics-quanta-the-pendulums-swing/ [Retrieved on: 7th June 2022]

Geogebra.org, (2022). About The Fourier Series. Retrieved from: https://www.geogebra.org/m/eRJ6ygyF [Retrieved on: 7th June 2022]

Semanticscholar.org, (2022). About Jacobi elliptic functions and the complete solution to the bead on the hoop problem. Retrieved from: https://www.semanticscholar.org/paper/Jacobi-elliptic-functions-and-the-complete-solution-Baker-Bill/a4e6feff2f92b68b61bd8c06a4c246ecc8da19f3 [Retrieved on: 7th June 2022]

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