MathJax

Fourier Transform

\[ \begin{aligned} f(x) &= \int_{-\infty}^{\infty}F(s)(-1)^{ 2xs}ds \\ F(s) &= \int_{-\infty}^{\infty}f(x)(-1)^{-2xs}dx \end{aligned} \]

The kernel can also be written as \(e^{2i\pi xs}\) which is more frequently used in literature.

Proof that \(e^{ix} = \cos x + i\sin x\) a.k.a Euler's Formula:

\( \begin{aligned} e^x &= \sum_{n=0}^\infty \frac{x^n}{n!} \implies e^{ix} = \sum_{n=0}^\infty \frac{(ix)^n}{n!} \\ \cos x &= \sum_{m=0}^\infty \frac{(-1)^m x^{2m}}{(2m)!} = \sum_{m=0}^\infty \frac{(ix)^{2m}}{(2m)!} \\ \sin x &= \sum_{s=0}^\infty \frac{(-1)^s x^{2s+1}}{(2s+1)!} = \sum_{s=0}^\infty \frac{(ix)^{2s+1}}{i(2s+1)!} \\ \cos x + i\sin x &= \sum_{l=0}^\infty \frac{(ix)^{2l}}{(2l)!} + \sum_{s=0}^\infty \frac{(ix)^{2s+1}}{(2s+1)!} = \sum_{n=0}^\infty \frac{(ix)^{n}}{n!} \\ &= e^{ix} \end{aligned} \)

Pauli Matrices

\[ \begin{aligned} \sigma_x &= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \sigma_y &= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\ \sigma_z &= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \end{aligned} \]