Fourier Analysis Introduction

Fourier Sum

f(t) = \frac{a_0}{2} + \sum_{m=1}^{\infty } a_m \cos (m \omega t) + \sum_{m=1}^{\infty } b_m \sin (m \omega t)

a_0 = \frac{2}{T} \int_0^T f(t) dt

a_m = \frac{2}{T} \int_0^T f(t) \cos (m \omega t) dt

b_m = \frac{2}{T} \int_0^T f(t) \sin (m \omega t) dt

f(t) = \sum_{m=-\infty}^{m=\infty} c_n e^{-im\omega t}

c_m = \frac{1}{T} \int_0^T f(t) e^{i m \omega t} dt

f(t) = \int_{-\infty}^{\infty} g(\omega)e^{-i\omega t} d\omega

g(\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty} f(t)e^{i\omega t} dt

If the function

f(\bm r)

is such that

f(\bm r) = f(\bm r + L\bm R)

[For some positive integer

L

and lattice vector

\bm R

. Then/Får något positivt heltal

L

och gittervektor

\bm R

. Då håller att]]

f(\bm r) = \frac{1}{\sqrt{V}} \sum_{\bm k = \frac{\bm G}L} c_{\bm k}\cdot e^{i\bm k\cdot \bm r}

c_{\bm k} = \frac 1{\sqrt{V}}\int_V f(\bm r) \cdot e^{-i\bm k\cdot\bm r}\: d^3r

Where

\bm G

is the reciprocal lattice vector and

V = L^3|\bm a_1\cdot(\bm a_2\times\bm a_3)| = L^3V_a

. The functions

\frac 1{\sqrt{V}}e^{i\bm k\cdot\bm r}

is a complete orthonormal basis in

V

.. If the volume

V

is large, the sum can be replaced by an integral:

\sum_{\bm k} \rightarrow \frac V{(2\pi)^3}\int d^3k

\int_A^Bf(x)\delta(x - x_0)\: dx = \begin{cases} f(x_0) &\textup{if}\; A < x_0 < B \\ 0 &\textup{otherwise} \end{cases}

f(x)

is a "nice" function.

\delta(f(x)) = \sum_{\forall i; f(x_i) = 0, f'(x_i) \neq 0} \frac 1{|f'(x_i)|}\delta(x - x_i)

\delta(x - x_0) = \frac 1{2\pi}\int_{-\infty}^\infty e^{ik(x - x_0)}\: dk

\delta_{n,m} = \frac 1{2\pi}\int_{-\pi}^{\pi} e^{i\phi(n - m)}\: d\phi = \begin{cases} 1 & n = m \\ 0 & n\neq m \end{cases}