iosMath Examples Square of sums (a_1 + a_2)^2 = a_1^2 + 2a_1a_2 + a_2^2 Quadratic Formula x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} Standard Deviation \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2} De Morgan's laws \neg(P\land Q) \iff (\neg P)\lor(\neg Q) Log Change of Base \log_b(x) = \frac{\log_a(x)}{\log_a(b)} Cosine addition \cos(\theta + \varphi) = \cos(\theta)\cos(\varphi) - \sin(\theta)\sin(\varphi) Limit e^k \lim_{x\to\infty}\left(1 + \frac{k}{x}\right)^x = e^k Stirling Numbers of the Second Kind {n \brace k} = \frac{1}{k!}\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}(k-j)^n Gaussian Integral \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} Arithmetic mean, geometric mean inequality \frac{1}{n}\sum_{i=1}^{n}x_i \geq \sqrt[n]{\prod_{i=1}^{n}x_i} Cauchy-Schwarz inequality \left(\sum_{k=1}^n a_k b_k \right)^2 \le \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right) Cauchy integral formula f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_\gamma\frac{f(z)}{(z-z_0)^{n+1}}dz Schroedinger's Equation i\hbar\frac{\partial}{\partial t}\Psi(x,t) = -\frac{\hbar}{2m}\nabla^2\Psi(x,t) + V(x)\Psi(x,t)
