While working on **Fourier Series** or some other **Mathematical Problem**, you might sometime have to work with **Periodic Functions**.

Periodic Functions are those that give the same value after a particular period.

So we will use this definition to define a periodic function in **PYTHON**.

Let’s say that there is a function * f(x)* which is defined in the interval

**[li,lf]**and is periodic with a period of

*.*

**T=lf-li**Then the function should have the same value at:** f(x), f(x+T), f(x+2*T), ….**

i.e.* f(x)=f(x+T)=f(x+2*T)=*……. since period=

**T**.

But I said that the function is defined only in the interval **[li,lf]**. So how is the computer supposed to calculate it’s value at **x>lf**?

That’s easy. Since the value of the function at * f(x+T)* is simply

*, therefore we can generalize that whenever*

**f(x)***: then,*

**x>lf**

**f(x)=f(x-T).****Note**: We have to keep taking

*back by*

**x***until it lies within*

**T i.e****(x-T)****[li,lf]**where the function is well-defined.

Similarly what about the value of function at * x* less than

*cause the function is not defined for values less than*

**(li)****(li)**?

Again, this time we use:

*. Note: We keep translating*

**f(x)=f(x+T)***forward by*

**x***until it lies within*

**T i.e****(x+T)****[li,lf]**where the function is well-defined.

Using the above two arguments we can create a function which will make any given function defined within** [li,lf]** and with a period **T** a periodic function.

**Here is the code for that:**

# Function that will convert any given function 'f' defined in a given range '[li,lf]' to a periodic function of period 'lf-li' def periodicf(li,lf,f,x): if x>=li and x<=lf : return f(x) elif x>lf: x_new=x-(lf-li) return periodicf(li,lf,f,x_new) elif x<(li): x_new=x+(lf-li) return periodicf(li,lf,f,x_new)

In the above code I have created a function ‘periodicf’ which takes as arguments the limits [li,lf] for which the function ‘f’ (third argument) is originally defined and the fourth argument is the value of x at which I want the value of ‘f’.

The above code assumes the function to be defined within [li,lf] therefore the function’s starting point is (li).

However, if you want to create a function that is defined in a different manner then you will have to define ‘f’ correspondingly.

Following is a program which you can use to define various periodic functions such as sawtooth wave, cycloids, square wave, and triangular wave.

import numpy as np import matplotlib.pyplot as plt from matplotlib.pyplot import * fig = figure(figsize=(8, 8), dpi=120) # Function that will convert any given function 'f' defined in a given range '[li,lf]' to a periodic function of period 'lf-li' def periodicf(li,lf,f,x): if x>=li and x<=lf : return f(x) elif x>lf: x_new=x-(lf-li) return periodicf(li,lf,f,x_new) elif x<(li): x_new=x+(lf-li) return periodicf(li,lf,f,x_new) # The periodic version of sawtooth function def sawtoothP(li,lf,x): return periodicf(li,lf,sawtooth,x) # Non-periodic sawtooth function defined for a range [-l,l] def sawtooth(x): return x # The periodic version of square function def squareP(li,lf,x): return periodicf(li,lf,square,x) # Non-periodic square wave function defined for a range [-l,l] def square(x): if x>0: return 5 else: return 0 # The periodic version of triangle function def triangleP(li,lf,x): return periodicf(li,lf,triangle,x) # Non-periodic triangle wave function defined for a range [-l,l] def triangle(x): if x>0: return x else: return -x # The periodic version of cycloid function def cycloidP(li,lf,x): return periodicf(li,lf,cycloid,x) # Non-periodic cycloid wave function defined for a range [-l,l] def cycloid(x): return np.sqrt(5**2-x**2) if __name__ == "__main__": # plt.style.use('dark_background') plt.style.use('seaborn') plt.title('Periodic functions\nCycloid') li = -5 lf = 5 step_size = 0.05 x_l = -20 x_u = 50 x = np.arange(x_l,x_u,step_size) y1 = [sawtoothP(li,lf,xi) for xi in x] y2 = [squareP(li,lf,xi) for xi in x] y3 = [triangleP(li,lf,xi) for xi in x] y4 = [cycloidP(li,lf,xi) for xi in x] x_plot =[] y_plot1 = [] y_plot2 = [] y_plot3 = [] y_plot4 = [] x_l_plot = x_l - 15 x_u_plot = x_l_plot + 20 plt.xlim(x_l_plot,x_u_plot) plt.ylim(-6,7) for i in range(x.size): x_plot.append(x[i]) y_plot1.append(y1[i]) y_plot2.append(y2[i]) y_plot3.append(y3[i]) y_plot4.append(y4[i]) #Sawtooth plt.plot(x_plot,y_plot1,c='darkkhaki') #Square plt.plot(x_plot,y_plot2,c='tomato') #Triangular plt.plot(x_plot,y_plot3,c='orange') #Cycloid plt.plot(x_plot,y_plot4,c='slateblue') x_l_plot = x_l_plot + step_size x_u_plot = x_u_plot + step_size plt.xlim(x_l_plot,x_u_plot) plt.pause(0.1) plt.show()

### OUTPUT