Residue Theorem: Let

be analytic in the region

except for the isolated singularities $a_1,a_2,\dots,a_m$ . If $\gamma$ is a closed rectifiable curve in

which does not pass through any of the points a_k

and if $\gamma\approx 0$ in

, then

$\frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{\mathbf{N}\in\mathbb{C}^{N\times 10}}\Bigr) = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.$

Maximum Modulus: Let

be a bounded open set in $\mathbb{C}$ and suppose that

is a continuous function on G^-

which is analytic in

. Then

$\max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.$

$\iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z \leq \oint_{\partial Q} f'\Biggl(\max\Biggl\{ \frac{\Vert w\Vert}{\vert w^2+x^2\vert}; \frac{\Vert z\Vert}{\vert y^2+z^2\vert}; \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert} \Biggr\}\Biggr)\,.$

$\begin{tikzpicture} % horizontal axis \draw[->] (0,0) -- (6,0) node[anchor=north] {$f/f_N$}; % labels \draw (0,0) node[anchor=north] {0} (2,0) node[anchor=north] {1} (4,0) node[anchor=north] {2}; % ranges \draw (1,3.5) node{{\scriptsize Constant flux}} (4,3.5) node{{\scriptsize Field weakening}}; % vertical axis \draw[->] (0,0) -- (0,4) node[anchor=east] {$U_s,\varPsi_s$}; % nominal speed \draw[dotted] (2,0) -- (2,4); % Us \draw[thick] (0,0) -- (2,2) -- (6,2); \draw (1,1.5) node {$U_s$}; %label % Psis \draw[thick,dashed] (0,3) -- (2,3) parabola[bend at end] (6,1); \draw (2.5,3) node {$\varPsi_s$}; %label \end{tikzpicture}$

$\begin{algorithm} \caption{An algorithm with caption}\label{alg:cap} \begin{algorithmic} \Require $n \geq 0$ \Ensure $y = x^n$ \State $y \gets 1$ \State $X \gets x$ \State $N \gets n$ \While{$N \neq 0$} \If{$N$ is even} \State $X \gets X \times X$ \State $N \gets \frac{N}{2}$ \Comment{This is a comment} \ElsIf{$N$ is odd} \State $y \gets y \times X$ \State $N \gets N - 1$ \EndIf \EndWhile \end{algorithmic} \end{algorithm}$