find a fourier series for the function x+xsqr in the interval -pi<x<pi?

Question

and hence show that (pi)sqr/6= 1+1/4+1/9+.............now please tell me what is the basic concept to solve the second part of such type of questions.

Answer 1

You want the Fourier series for x+x^2 in the interval [-pi, pi] and show that (pi^2)/6 is the sum of the series 1/n^2 where 1<=n<=infinity.


For the second part you probably need to use the Parseval identity that relates the L^2 norm of the function and the sum of the square of the modulus of its Fourier coefficients.

Then you'll need some theorical argument to show that the series converges puntually. In the series something very like 1/n^2 will show up for some x in [-pi, pi] and you'll finally use some elementar algebra to conclude.

Answer 2

For section a million you look to have a really undemanding function that would not want a Fourier series. Are you particular that that's properly written in the question? For section 2a, the function is defined for C - (0,-i). this implies the finished complicated plane except the only aspect (0,-i) no longer the portion of the imaginary plane between this and the muse. (you should any way have suggested that portion of the imaginary axis from 0 to - a million, you probably did not favor the i because you've already suggested alongside that axis.) i imagine that i visit also grant you with some help with some thing else of section 2. you should discover that the first spinoff may be rearranged to i*f(z)/(z + i)^2. (this received't be the case as i have finished it particularly right away.) if so, this makes it more beneficial accessible to discover the 2d spinoff. a similar issue may ensue back yet i have not lengthy gone that a procedures.