 Hello, Myself, MS Bhasargaam, Assistant Professor, Department of Human Design Sciences, Valksandvist of Technology, Solapur. Now, learning outcome. At the end of this session, students will be able to express four-year series of even and odd functions in the interval minus pi to pi. Now, first of all, here we will see about even and odd functions. First one, even function. A function f of x is said to be even if f of minus x equal to f of x. That is, if you replace x by minus x, again we are getting the same function f of x for all x. Then, we can say that the given function is even. Now, we will take some notes f of x contains only even powers of x and may contain only cos x and sec x. Now, secondly, integration from minus a to a f of x dx is equal to twice. Integration from 0 to a f of x dx, when f of x is even, it is a standard theorem of definite integral. Third one, the sum or difference of two even function is even. Fourth one, the product of two even function is even. For example, x square cos x x is to 4 plus cos 2 x plus 2, these are the examples of even functions. Now, about the odd function. A function f of x is said to be odd if f of minus x equal to minus f of x for all x. Now, we will note some points. First one, f of x contains only odd powers of x and may contains only sin x cos x. Second one, integration from minus a to a f of x dx equal to 0, when f of x is odd. The sum or difference of two odd function is odd, odd function. The product of two odd function is even. The product of an odd function and even function is odd. For example, sin x x cube. Now, pause the video for a while and test whether the following functions are even or odd. I hope you have completed. Now, to test even odd function, here first function is f of x is equal to sin 2 x plus x cube. Now, in this one replace x by minus x that is f of minus x equal to sin of 2 minus x plus minus x raise to 3, which is equal to minus sin 2 x minus x cube. Here, you can take minus sin common that is minus sin 2 x plus x cube which is equal to minus f of x. Therefore, f of minus x equal to minus f of x therefore, given function is an odd function. Similarly, second f of x is equal to x raise to 6 plus cos 3 x. Now, replacing x by minus x you get f of minus x equal to minus x raise to 6 plus cos of 3 minus x that is minus x raise to 6 means x raise to 6 and you know cos of minus theta is cos theta that is plus cos 3 x which is again given function f of x. Therefore, f of minus x is equal to f of x therefore, given function is an even function. Now, Fourier series of even odd function let the Fourier series of the function f of x in the interval minus pi to pi b f of x is equal to a naught plus summation of n equal to 1 to infinity a n cos n x plus b n sin n x let us call this as equation number 1 where a naught is equal to 1 by 2 pi integration from minus pi to pi f of x dx let us call this as equation number 2 and a n equal to 1 by pi integration from minus pi to pi f of x cos n x dx let us call this as equation number 3 and b n equal to 1 by pi integration from minus pi to pi f of x sin n x dx let us call this as equation number 4. Now, we will take the case 1. Suppose f of x is an even function in minus pi 2 pi, then b n will be 0. Since in the integrate in 4 in equation number 4 here f of x sin n x is odd is an odd function. Then the 4 here series is given by f of x equal to a naught plus summation of n equal to 1 to infinity a n cos n x, where a naught is equal to 1 by 2 pi integration from minus pi to pi f of x dx, which you can write twice that 2 2 will get cancel 1 by pi integration from 0 to pi f of x dx. And a n is equal to 1 by pi integration from minus pi to pi f of x cos n x dx, which you can write 2 by pi integration from 0 to pi f of x cos n x dx. Now, we will consider the case 2. Suppose f of x is an odd function in minus pi to pi, then a naught and a n are 0, because integrate in 3 is an odd function and also a naught is 0 since f of x is odd. Then 4 here series is given by f of x is equal to summation of n equal to 1 to infinity b n sin n x, where b n is equal to 1 by pi integration from minus pi to pi f of x sin n x dx, which you can modify this is equal to 2 by pi integration from 0 to pi f of x sin n x dx. Now, we will see the example find the 4 here series of f of x equal to pi square by 12 minus x square by 4 in the interval minus pi to pi. And hence deduce that pi square by 12 is equal to 1 by 1 square minus 1 by 2 square plus 1 by 3 square minus so on. Now, here f of x is equal to pi square by 12 minus x square by 4. Now, we will test for even an odd function and for that here we can replace x by minus x that is f of minus x is equal to pi square by 12 minus here replacing x by minus x that is minus x whole square by 4 which is equal to pi square by 12 and minus x whole square is x square that is minus x square by 4 again which is f of x. Hence f of minus x is equal to f of x therefore f of x is equal to pi square by 12 minus x square by 4 is an even function. Hence, here the value of b n is 0. Then the 4-year series of even function is given by f of x is equal to a naught plus summation of n equal to 1 to infinity a n cos n x where a naught is equal to 1 by pi integration from 0 to pi f of x dx which is equal to 1 by pi integration from 0 to pi here f of x is pi square by 12 minus x square by 4 dx which is equal to 1 by pi. Now, integrating with respect to x we get pi square x by 12 minus integration of x square by 4 is x cube by 12 with a limit 0 to pi which is equal to 1 by pi putting the upper limit x equal to pi that we get pi cube by 12 and for low again minus pi cube by 12 and for lower limit putting x equal to 0 both the terms are 0 that is equal to 1 by pi pi cube by 12 minus pi cube by 12 which is equal to 0. Now, a n is equal to 2 by pi integration from 0 to pi f of x cos n x dx which is equal to 2 by pi integration from 0 to pi here f of x is pi square by 12 minus x square by 4 into cos n x dx. Now, again here by using generalized rule of integration by part taking u as a first bracket and v as a cos n x you can write which is equal to 2 by pi keeping pi square by 12 minus x square by 4 as it is an integration of cos n x is sin n x by n minus the derivative of pi square by 12 minus x square by 4 is that is 0 minus 2 x by 4 into integration of sin n x by n is minus cos n x by n square plus the derivative of minus 2 x by 4 is minus 1 by 2 into bracket integration of minus cos n x by n square is minus sin n x by n cube with a limit 0 to pi. Now, put a first upper limit that you replace x by pi here we know sin of n pi 0 therefore, the first term will be 0 minus into bracket putting x equal to pi here you get minus pi by 2 and minus cos of n pi by n square and minus minus plus here sin of n pi is 0. Now, for lower limit putting x equal to 0 the sin 0 is 0 therefore, the first term is 0 and here x is multiplied with cos n x therefore, putting x equal to 0 second term is also 0 and we have again sin 0 is 0 that is for lower limit we get the value 0. Now, by simplifying this which is equal to 2 by pi that is you get minus into minus into minus that is minus pi cos of n pi is minus 1 is to n divided by 2 n square that is 2 2 will get cancel pi pi will get cancel which is equal to minus of minus 1 is to n by n square. Hence f of x is equal to summation of n equal to 1 to infinity now we can write this as a minus 1 is to n plus 1 by n square cos n x. Now, here substituting the value of f of x as a pi square by 12 minus x square by 4 and substituting n equal to 1 to infinity first we will put n equal to 1 that is minus 1 is to 1 plus 1 that is 2 that is 1 by 1 square cos x. Now, putting n equal to 2 we get minus 1 is to 3 means we get minus 1 and divided by n square means minus 1 by 2 square cos of 2 x and putting n equal to 3 we get minus 1 is to 4 that is plus 1 plus 1 upon 3 square cos 3 x and so on. Let us call this equation number 1. Now, putting x equal to 0 in equation number 1 we get the second term will become 0 that is pi square by 12 is equal to as you know cos 0 is 1 that is 1 by 1 square similarly cos 0 is 1 that is minus 1 by 2 square plus 1 by 3 square minus on references higher in the mathematics by Dr. B. S. Graewall. Thank you.