complex analysis
De Moivre's Theorem
$$\small (\cos x+ i \sin x)^n=\cos(nx)+i\sin(nx)$$
Hyperbolic Functions
$$\sinh(x) = \frac{e^x - e^{-x}}{2}$$
Cauchy's Integral Theorem
$$\oint_C f(z)dz = 0$$
Euler's Identity/Formula
$$e^{i\pi} = -1$$
Trig functions in terms of e
$$\small{\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}, \cos(x) = \frac{e^{ix} + e^{-ix}}{2}}$$