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% Formulas for Complex Variables
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\title{Complex Analysis}
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\begin{tabular}[t]{cc}

\begin{minipage}{0.5\textwidth}
{De Moivre}
\begin{eqnarray*}
z & = & x + iy = e^{i\theta} = \cos\theta + i\sin\theta \\
\bar{z} & = \dfrac{1}{z} = & x - iy = e^{-i\theta} = \cos\theta - i\sin\theta \\
e^{i\frac{\theta}{n}} &=& \cos\frac{\theta + 2\pi{}k}{n} + i\sin\frac{\theta + 2\pi{}k}{n}
\end{eqnarray*}
\end{minipage}

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{Powers}
\begin{eqnarray*}
(\cos\theta + i\sin\theta)^n &= & \cos{}n\theta + i\sin{}n\theta \\
(\cos\theta + i\sin\theta)^{-n} &= & \cos{}n\theta - i\sin{}\theta \\
2\cos{}n\theta &=& z^n + z^{-n} \\
2i\sin{}n\theta &=& z^n - z^{-n}
\end{eqnarray*}
\end{minipage}

\end{tabular}



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\begin{tabular}[t]{ccc}

\begin{minipage}{0.5\textwidth}
{Trigonometric}
\begin{eqnarray*}
\Im z = \sin\theta &= & \dfrac{e^{i\theta} - e^{-i\theta}}{2i} = \dfrac{1}{2i} \left( z - \dfrac{1}{z} \right)  \\
\Re z = \cos\theta &= & \dfrac{e^{i\theta} + e^{-i\theta}}{2}  = \dfrac{1}{2} \left( z + \dfrac{1}{z} \right) \\
\tan\theta &= &\dfrac{\sin\theta}{\cos\theta}
\end{eqnarray*}
\end{minipage}

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{Hyperbolic}
\begin{eqnarray*}
\sinh\theta &= \dfrac{e^{\theta} - e^{-\theta}}{2} \\
\cosh\theta &= \dfrac{e^{\theta} + e^{-\theta}}{2} \\
\tanh\theta &= \dfrac{\sinh\theta}{\cosh\theta}
\end{eqnarray*}
\end{minipage}

\end{tabular}



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\begin{tabular}[t]{cc}

\begin{minipage}{0.5\textwidth}
{Complex Equivalences}
\begin{eqnarray*}
\sin{}iz &= i\sinh{}z \\
\sinh{}iz &= i\sin{}z
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.5\textwidth}
\begin{eqnarray*}
\cos{}iz &= \cosh{}z \\
\cosh{}iz &= \cos{}z \\
\cosh x &= k \quad\rightarrow\quad x = \ln (k \pm \sqrt{k^2 - 1}), {\textrm{where} (k > 1)}
\end{eqnarray*}
\end{minipage}

\end{tabular}



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\begin{tabular}[t]{ccc}

\begin{minipage}{0.5\textwidth}
{Hyperbolic Identities}
\begin{eqnarray*}
\cosh(a+b) &= \cosh{}a \cosh{}b + \sinh{}a \sinh{}b \\
\cosh{}z &= \cosh{}x \cos{}y + i\sinh{}x \sin{}y \\
\end{eqnarray*}
\end{minipage}

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{}
\begin{eqnarray*}
\end{eqnarray*}
\end{minipage}

\end{tabular}

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\begin{minipage}{0.5\textwidth}
{Complex Roots}
\begin{eqnarray*}
\omega & = & z^{\frac{1}{n}} = r^{\frac{1}{n}}(\cos\theta + i\sin\theta)^{\frac{1}{n}} = r^{\frac{1}{n}}e^{i\frac{\theta}{n}} \\
& = & r^{\frac{1}{n}} \left(\cos\dfrac{\theta + 2\pi{k}}{n} + i\sin\dfrac{\theta + 2\pi{k}}{n}\right) \\
\end{eqnarray*}
\end{minipage}

\begin{minipage}{0.5\textwidth}
{Roots of Unity}
\begin{eqnarray*}
1 = \cos{2\pi{}k} + i\sin{2\pi{}k} = e^{i2\pi{}k} \\
1, \alpha, \alpha^2, \dots, \alpha^{n-1}  \quad\textrm{with}\, \alpha = \exp\left(\dfrac{2\pi{}i}{n}\right)
\end{eqnarray*}
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\end{tabular}






{\vfill\hfill{\tiny Author: Martin Blais, 2009.
This work is licensed under the Creative Commons
``Attribution - Non-Commercial - Share-Alike'' license.}}
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