Question

Fourier complex

Answer 1

\[1+x, -\pi<x<\pi\]

Answer 2

\[c_o=\frac{ 1 }{ 2\pi } \int\limits_{-\pi}^{\pi} (1+x) dx= 1\]

Answer 3

\[c_n= \frac{ 1 }{ 2\pi }\int\limits_{-\pi}^{\pi} (1+x)e^{-inx} dx\]

Answer 4

how to evaluate c_n more efficiently?

Answer 5

@AlexandervonHumboldt2

Answer 6

\[\frac{ 1 }{ 2\pi }\int\limits_{-\pi}^{\pi} e^{-inx} dx+ \frac{ 1 }{ 2\pi }\int\limits_{-\pi}^{\pi} xe^{-inx} dx\]

Answer 7

part 1: \[\frac{ n i}{ 2\pi } (e^{-i n \pi} -e^{i n \pi})= \frac{ n }{ \pi }(\sin(n \pi))\]

Answer 8

part 2: \[\frac{ -1}{ 2 i n}(e^{-i n \pi}+ e^{i n \pi})+ \frac{1}{2 \pi n^2}(e^{-i n \pi}-e^{i n \pi})\]

Answer 9

\[\sum_{n=1}^{\infty}\frac{ (-1)^{n-1}}{ n i }+ 0\]

Answer 10

we have part 1 goes to 0 and so the final coef= ^