## Fourier Series in Hindi #3 How to Compute Even Function and Odd Functions of Fourier Series Examples

Even and odd function of Fourier Series Lecture Part 3 |PTU|GNDU|AEM|GTU|Math 3|Hindi| JK ENGINEERING CLASSES (JK SMART CLASSES) introduced a full course of the most difficult subject name is ENGINEERING MATHEMATICS 3(M-3)in the engineering field for all B.TECH TRADE(EE, CE, AE, ME, ECE and CSE, IT) in this video, we discuss the HOW TO COMPUTE EVEN & ODD FUNCTION OF FOURIER SERIES IN HINDI**EVEN FUNCTION OF FOURIER SERIES ****1:15**

A function y = f(x) is said to be even, if f(-x) = f(x). The graph of the even function is always symmetrical about the y-axis.

**ODD FUNCTION OF FOURIER SERIES **4:36

A function y=f(x) is said to be odd, if f(-x) = – f(x). The graph of the odd function is always symmetrical about the origin. For example, the function f(x) = in [-1,1] is even as f(-x) = = f(x) and the function f(x) = x in [-1,1] is odd as f(-x) = -x = -f(x). The graphs of these functions are shown below : 1. If f(x) is even and g(x) is odd, then

• h(x) = f(x) x g(x) is odd

• h(x) = f(x) x f(x) is even

• h(x) = g(x) x g(x) is even

For example,

1. h(x) = x2 cosx is even, since both x2 and cosx are even functions

2. h(x) = xsinx is even, since x and sinx are odd functions

3. h(x) = x2 sinx is odd, since x2 is even and sinx is odd.

4. If f(x) is even, then bn=0 This is FOURIER COSINE SERIES ao and an is present.

accordingly you have to calculate for example, if you are asked what we will do is wherever there is l we will keep ‘pi’ keeping ‘pi’ the Fourier series we will get is this is our Fourier series between -pi to pi we kept ‘pi’ in place of l the value of a0 will be value of an will be value of bn will be s students what we have done are wherever there was l we substituted pi this is the Fourier series we got now we have a concept of odd and even here which you should know if the function is odd then what will happen and if it is even what will happen let me clear you the concept of class 12

if it is an even function then what happens and if it is odd then what happens let me tell you the concept of even and odd as per class 12 the concept of integration in any function if we keep -x in place of x and if it is equal then it is an even function for example cos x is even we denote even function with ‘+’ sign similarly in any function if we keep -x in place of x and there is a negative sign then it is an odd function

For example, sin x sin x is an odd function x is an odd function these all are an odd function you will say how this is an odd function so let me tell you if we keep -x here it is not equal it is equal when the minus sign comes out so this is not equal if it is equal then one is a positive one is negative this is an odd function.

for example to check xcosx is even or odd remember the concept of plus and minus we denote even with + sign and odd with – sign here x is odd and cos x is even function so plus*minus=minus so this is an odd function same if we take xsinx x is odd and sinx is odd but minus*minus=plus so this is an even function now we studied in 12 class

if any function is even from -l to l then, in that case, its value will be equal to so this is the concept of even function but if we have an odd function from -l to l is an odd function then its integration is always 0 in 12 class these are the rules of integration we studied now in Fourier series if the function f(x) is even

if the f(x) is even or if the f(x) is odd if any function is even then what will be our Fourier series and if the function is odd then what will be our Fourier series I am telling you this between -pi to pi so students if the f(x) is even we will denote it with a plus which means it is an even function and if it is even what is the property? then what you will get is the value of a0 will be students,

,

let us talk about an now this is our even function and cos x is an even function so an will be similarly we will get bn as this is our even function but we know sin is odd so + * – = – so the value of bn will be 0 if any function is even we will calculate a0 and an the value of bn will be 0.

remember this concept if you will solve it, it will come 0 but will waste time substituting the value we will get the Fourier series as now we will talk about odd if the function is odd then this f(x) will be –

if this is odd its integration will be 0 so students if the function is odd then our a0 will be 0 this f(x) is odd so minus * plus= minus so our a will also be 0 talking about this function here it is minus, so minus*minus is plus so our Bn will be as this function is even

So our Fourier series will be so students what I told you is this is the formula for our Fourier series between -l to l this is the way we calculate an and bn and we get our answer in the series form if the series is between -pi to pi wherever there is l we will keep pi then I told you what is even and an odd function.

we discussed the difference we denote even function with a plus sign and odd function is denoted by a minus sign if a function is even then there will be a0 and a but bn will be 0 always in the odd function a0 and an is 0 and only we have the value for bn.

so, students, we will discuss two questions one on even and one on odd so here is the question first of all, we have to see the given function is even or odd as you can see that this function is odd and why? if we keep -x in place of x then we will get the negative sign you can see and -x this function is odd and if it is odd then.