00:01
Hello students, to find the fourier series expansion of the piecewise function f of x over the interval minus where x lies between minus pi to pi, we will need to compute the fourier coefficient for both parts of the function separately and then combine them.
00:14
The fourier series for a function is f of x defined on the interval that is minus l to x lies between minus l to l is given by f of x is equal to a0 by 2 plus summation a n into cos n pi x by l plus b n into sin n pi x by l where n ranges from 1 to infinity.
00:51
Let's find the fourier coefficient for the two parts of the function f of x over the interval x lies between minus pi to 0 and x lies between 0 to pi separately.
01:02
So for first one for x lying between minus pi to 0 f of x is equal to 1 minus x.
01:11
The function is odd over the interval that means f minus x is equal to 1 minus of minus x that is equal to 1 plus x.
01:22
So the only coefficient we need to find is b1.
01:25
So b1 is equal to 1 by pi integration from minus pi to 0 1 minus x to the power of x into sin x into dx.
01:36
Solving this we will find 1 by pi into 1 by pi.
01:42
1 by pi minus pi to 0 sin x minus x sin x dx.
01:46
So sin x will got cos x and x cos x minus x sin x will got x cos x from minus pi to 0.
01:52
So solving this we will get 1 by pi 1 by pi.
01:59
So 1 by pi plus 1 by pi into 1 plus pi that is equal to 1 plus pi by pi that is equal to 1 plus 1 by pi.
02:15
Now for x lying between 0 to pi f of x is equal to x minus 2...