Chapter 1

HTML:

$$\begin{gathered} e^{i\theta }=\cos \theta +i\sin \theta \\ \Rightarrow x+iy=r{e}^{i\theta } \end{gathered}$$

Markdown (requires mdbook-katex):

$${\oint }_{C}f(x,y)\mathrm{d}A$$

Inspect element and use Sources tab (under Debugger on Firefox) to check that all CSS and fonts are properly loaded from GitHub pages instead of external CDN.

Proof that $e^{ix}=\cos x+i\sin x$

$$\begin{array}{rl}{e}^x& =\sum _{n=0}^{\infty }\frac{x^n}{n!}⟹e^{ix}=\sum _{n=0}^{\infty }\frac{(ix{)}^n}{n!}\\ \cos x& =\sum _{m=0}^{\infty }\frac{(-1{)}^m{x}^{2m}}{(2m)!}=\sum _{m=0}^{\infty }\frac{(ix{)}^{2m}}{(2m)!}\\ \sin x& =\sum _{s=0}^{\infty }\frac{(-1{)}^s{x}^{2s+1}}{(2s+1)!}=\sum _{s=0}^{\infty }\frac{(ix{)}^{2s+1}}{i(2s+1)!}\\ \cos x+i\sin x& =\sum _{l=0}^{\infty }\frac{(ix{)}^{2l}}{(2l)!}+\sum _{s=0}^{\infty }\frac{(ix{)}^{2s+1}}{(2s+1)!}=\sum _{n=0}^{\infty }\frac{(ix{)}^n}{n!}\\ & =e^{ix}\end{array}$$

Fourier Transform:

$$\begin{array}{rl}f(t)& ={\int }_{-\infty }^{\infty }F(\omega )i^{4t\omega }d\omega \\ F(\omega )& ={\int }_{-\infty }^{\infty }f(t)i^{-4t\omega }dt\end{array}$$

Pauli Matrices:

$$\begin{array}{rl}{\sigma }_1& =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\\ {\sigma }_2& =\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\\ {\sigma }_3& =\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\end{array}$$