#import "@preview/physica:0.9.6": *
#import "lib.typ": ccases, fourier

= problem 3

== a)

the truncated fourier series of a function looks like
$
  f_N = #fourier(N: $N$)
$
note: i use $a_0/2$ with a $1/L$ in the definition of $a_0$.

then to calculate the error, we take the difference with the actual function,
which can be expressed as an infinite fourier series
$
  e_N (x) & = f(x) - f_N (x) \
          & = [#fourier()] \
          & - [#fourier(N: $N$)] \
          & = sum_(n=N+1)^oo (a_n cos(n x) + b_n sin(n x))
$
it is the tail end of the infinite sum.

parsevals identity states that
$
  angle.l f, f angle.r =
  1/L integral_(-L)^L abs(f(x))^2 dd(x)
  = a_0^2/2 + sum_(n=1)^oo (a_n^2 + b_n^2)
$


== b)

let $f(x) = e^(-x)$

we first need to find the fourier coefficients

$
  a_0 = integral_(-1)^1 e^(-x) dd(x)
  = [e^x]_(-1)^1 = e - 1/e
$

$
  a_n =
$