There are formulas: \begin{align*} &e^{a i \omega}+e^{b i \omega} &=2 \cos \left(\frac{a-b}{2} \omega\right) e^{\frac{a+b}{2} i \omega} \\ e^{a i \omega}-e^{b i \omega} &=2 i \sin \left(\frac{a-b}{2} \omega\right) e^{\frac{a+b}{2} i \omega} .\end{align*} Why this is useful: you can reduce a sum of two exponentials to a complex scalar times a real trig function, e.g. when computing a residue to get a real number. Why this is true: for the right choice of \(\ell\), \begin{align*} e^{aiw} + e^{biw} = e^{\ell iw} \qty{ e^{(l-a)iw} + e^{(\ell - b)iw} } = e^{\ell i w} \qty{ e^{kiw} + e^{-kiw}} = e^{\ell i w}\cdot 2\cos(kw) .\end{align*} To make this hold, choose

• \(\ell \coloneqq{a+b\over 2}\)
• Then \(\ell - a = {b-a \over 2} \coloneqq k\)
• \(\ell -b = {a-b\over 2} = -k\)

An example: \begin{align*} e^{-i\pi \over 2}+ e^{-3i\pi \over 2} &\coloneqq e^{-iw} + e^{-3iw} \\ &= e^{-2iw} \qty{e^{iw} + e^{-iw}}\\ &= e^{-2iw}\cdot 2\cos(w) \\ &= e^{-2i\cdot {\pi \over 2}}\cdot 2\cos\qty{\pi \over 2} \\ &= -i\cdot 0 = 0 .\end{align*}

\begin{align*} z + \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu &= 2\Re(z) && z - \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu = 2i\Im(z) \\ z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu &= {\left\lvert {z} \right\rvert}^2 && \operatorname{Arg}(z/w) = \operatorname{Arg}(z) - \operatorname{Arg}(w) \\ \Re(z) &= { z + \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu \over 2} && \Im(z) = {z - \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu \over 2i} \\ {1\over i} &= -i = i^3 && {1\over i^3} = i .\end{align*}

Exponential forms of cosine and sine, where it’s sometimes useful to set \(w\coloneqq e^{iz}\): \begin{align*} \cos(z) &= \frac 1 2 \qty{e^{iz} + e^{-iz}} = {1\over 2}(w+ w^{-1})\\ \sin(z) &= \frac{1}{2i}\qty{e^{iz} - e^{-iz}} = {1\over 2i}(w-w^{-1}) .\end{align*}

Exponential forms of hyperbolic cosine and sin: \begin{align*} \cosh(z) &= \cos(iz) = {1\over 2}\qty{e^z + e^{-z}} \\ \sinh(z) &= -i \sin(iz) = {1\over 2}\qty{e^z - e^{-z}} .\end{align*}

Some other useful facts about the hyperbolic exponentials:

• They are periodic with period \(2\pi i\).
• \({\frac{\partial }{\partial z}\,}\cosh(z) = \sinh(z)\) and \({\frac{\partial }{\partial z}\,}\sinh(z) = \cosh(z)\).
• \(\sinh\) is odd and \(\cosh\) is even.
• \(\cosh(z + i\pi) = -\cosh(z)\) and \(\sinh(z + i\pi) = -\sinh(z)\).
• \(\cosh\) has zeros at \(\left\{{i\pi\qty{2k+1\over 2}}\right\} = \left\{{i \qty{\pi/2 + k\pi}}\right\}\), i.e. \(\cdots, -\pi/2, \pi/2, 3\pi/2,\cdots\), the half-integers.
• \(\sinh\) has zeros at \(\left\{{i\pi k}\right\}\), i.e. the integers.
• \(\sinh(z)\) has a particularly nice formula for derivatives: \begin{align*} f(z) &\coloneqq\cosh(z) \implies f^{(n)}(z) = {e^z + (-1)^n e^{-z}\over 2} \\ f(z) &\coloneqq\sinh(z) \implies f^{(n)}(z) = {e^z - (-1)^{n} e^{-z}\over 2} .\end{align*}

Some computations that come up frequently: \begin{align*} {\left\lvert {z \pm w} \right\rvert}^2 &= {\left\lvert {z} \right\rvert}^2 + {\left\lvert {w} \right\rvert}^z + 2\Re(\mkern 1.5mu\overline{\mkern-1.5muw\mkern-1.5mu}\mkern 1.5muz) \\ (a+bi)(c+di) &= (ac - bd) + (ad + bc) \\ {1\over {\left\lvert {a+b} \right\rvert}} &\leq {1 \over {{\left\lvert {a} \right\rvert} - {\left\lvert {b} \right\rvert}}} && {\left\lvert {e^{z}} \right\rvert} = e^{\Re(z)}, \quad \arg(e^z) = \Im(z) .\end{align*}

The inversion \(z\to 1/z\) is easiest to understand in polar coordinates, \(Re^{i\theta}\mapsto {1\over R}e^{-i\theta}\):