1. Let f(x) = egincases 0, -pi le x

1. Let $f(x) = egin{cases} 0, & -pi le x < -frac{pi}{2} \ 2, & -frac{pi}{2} le x < frac{pi}{2} \ 0, & frac{pi}{2} le x < pi end{cases}$ \ (a) Find the Fourier series of $f$. \ (b) Determine if the Fourier series in part (a) converges. If it does converge, what \ are the values to which it converges (on $[-pi, pi])$.

Transcript

00:01 In this question, we are here to find the equation governing or temperature.

00:13 So assuming that a partial differentiation with respect to t of the equation that is r, partial differentiation with respect to r, r, t u by t r.

00:26 So that is r1, r2.

00:31 So a kilogramme temperature, does so.

00:39 Not change in time does not change in time and therefore our partial differentiation with respect to t for 1 is taken as du by t is equal to 0 so that is 0 is equal to k by r by delta r r r delta u by delta t so this is equal to and now we will multiply both sides by r, multiply both side by r.

01:23 So we will get here, d by tr, r -b -u by dr, which is equal to 0.

01:33 So integers both sides, we will integrate here.

01:38 So r -t -u by v -r will be equal to c -1, which is constant of integration.

01:43 Therefore d u by d r is equal to c1 by r so this is a solution here for the equation and now we will start with our first part that is part a so r value for d u by d r which is c1 by r so we will integrate on integrating with respect to r that is d u by d r is equal to c1 by r so we will integrate or integrating with respect to r that is d u by d r is equal to c1 by r.

02:16 So here this is u for r is equal to c1 natural log of r1 added by c2 is equal to t 1 that is our equation 1 first equation and similarly u r2 this was 1 r1 is equal to c1 natural log of r2 added by c2 which is for p2 2 2 2 2 2 2 2 equation equation a and b are here and therefore, when we subtract equation 1, b from 1a, so that will be as c1 anti -n, national log of r1 minus national log of r2, that is equal to t2 minus t1.

03:07 So c -in r1 by r2 is equal to t2 minus t1...

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