00:01
Hello everyone, the question is given that a fourier function and we need to find the fourier integral representation where the function is given is like that fx equals to 0 when x less than 0, cos x when x belongs to closed interval 0, pi by 2 and 0 when x greater than pi by 2.
00:23
So, for this we need to find some basic information that is fourier integral transformation.
00:28
Fourier integral transformation is nothing but fx equals to integration 0 to infinity au cos ux plus bu sin ux dx where au equal to 1 upon pi integration minus infinity to infinity ft cos ut dt bu equal to 1 upon pi integration minus infinity to infinity ft sin ut dt.
00:51
Now moving to the calculation part that is first au, au equals to 1 upon pi integration minus infinity to infinity ft cos ut dt which is equals to 1 upon pi we are going to break this integration part into interval.
01:11
So, first interval will be minus infinity to 0, second will be 0 to pi by 2 and third will be pi by 2 to infinity as in the question mentioned in that three particular interval for this we need to break this interval in into three parts and now we are going to put the values in those intervals where ft is 0 in this interval, ft is cos t in this interval and ft is again 0 in pi by 2 to infinity.
01:42
Now this is the simple integration so and again we know that sin pi by 2 plus minus x equals to cos x for this we can write that 1 upon 2 pi cos u pi by 2 upon 1 plus u plus sin u pi by 2 upon 1 minus u.
02:04
So, au equal to this is cos u pi by 2 upon 2 pi this is taking common this will be 1 upon 1 plus u plus 1 upon 1 minus u therefore the final value will be cos u pi by 2 upon pi into 1 minus u square this is the value of au...