# iosMath Examples ## Square of sums LaTeX (a_1 + a_2)^2 = a_1^2 + 2a_1a_2 + a_2^2  ![Square Formula](img/square.png) ## Quadratic Formula LaTeX x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}  ![Quadratic Formula](img/quadratic.png) ## Standard Deviation LaTeX \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2}  ![Standard Deviation](img/standard.png) ## De Morgan's laws LaTeX \neg(P\land Q) \iff (\neg P)\lor(\neg Q)  ![De Morgan](img/demorgan.png) ## Log Change of Base LaTeX \log_b(x) = \frac{\log_a(x)}{\log_a(b)}  ![Log Base Change](img/log.png) ## Cosine addition LaTeX \cos(\theta + \varphi) = \cos(\theta)\cos(\varphi) - \sin(\theta)\sin(\varphi)  ![Cos Sum](img/trig.png) ## Limit e^k LaTeX \lim_{x\to\infty}\left(1 + \frac{k}{x}\right)^x = e^k  ![Limit](img/limit.png) ## Calculus LaTeX f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\xi  ![Calculus](img/calculus.png) ## Stirling Numbers of the Second Kind LaTeX {n \brace k} = \frac{1}{k!}\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}(k-j)^n  ![Stirling Numbers](img/stirling.png) ## Gaussian Integral LaTeX \int_{-\infty}^{\infty} \! e^{-x^2} dx = \sqrt{\pi}  ![Gauss Integral](img/gaussintegral.png) ## Arithmetic mean, geometric mean inequality LaTeX \frac{1}{n}\sum_{i=1}^{n}x_i \geq \sqrt[n]{\prod_{i=1}^{n}x_i}  ![AM-GM](img/amgm.png) ## Cauchy-Schwarz inequality LaTeX \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 Schwarz](img/cauchyschwarz.png) ## Cauchy integral formula LaTeX f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_\gamma\frac{f(z)}{(z-z_0)^{n+1}}dz  ![Cauchy Integral](img/cauchyintegral.png) ## Schroedinger's Equation LaTeX i\hbar\frac{\partial}{\partial t}\mathbf\Psi(\mathbf{x},t) = -\frac{\hbar}{2m}\nabla^2\mathbf\Psi(\mathbf{x},t) + V(\mathbf{x})\mathbf\Psi(\mathbf{x},t)  ![Schroedinger](img/schroedinger.png) ## Lorentz Equations Use the gather or displaylines environments to center multiple equations. LaTeX \begin{gather} \dot{x} = \sigma(y-x) \\ \dot{y} = \rho x - y - xz \\ \dot{z} = -\beta z + xy" \end{gather}  ![Lorentz](img/lorentz.png) ## Cross product LaTeX \vec \bf V_1 \times \vec \bf V_2 = \begin{vmatrix} \hat \imath &\hat \jmath &\hat k \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \end{vmatrix}  ![Cross Product](img/cross.png) ## Maxwell's Equations Use the aligned, eqalign or split environments to align multiple equations. LaTeX \begin{eqalign} \nabla \cdot \vec{\bf E} & = \frac {\rho} {\varepsilon_0} \\ \nabla \cdot \vec{\bf B} & = 0 \\ \nabla \times \vec{\bf E} &= - \frac{\partial\vec{\bf B}}{\partial t} \\ \nabla \times \vec{\bf B} & = \mu_0\vec{\bf J} + \mu_0\varepsilon_0 \frac{\partial\vec{\bf E}}{\partial t} \end{eqalign}  ![Maxwell's Equations](img/maxwell.png) ## Matrix multiplication Supported matrix environments: matrix, pmatrix, bmatrix, Bmatrix, vmatrix, Vmatrix. LaTeX \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} = \begin{pmatrix} a\alpha + b\gamma & a\beta + b \delta \\ c\alpha + d\gamma & c\beta + d \delta \end{pmatrix}  ![Matrix Multiplication](img/matrixmult.png) ## Cases LaTeX f(x) = \begin{cases} \frac{e^x}{2} & x \geq 0 \\ 1 & x < 0 \end{cases}  ![Cases](img/cases.png) ## Splitting long equations LaTeX \frak Q(\lambda,\hat{\lambda}) = -\frac{1}{2} \mathbb P(O \mid \lambda ) \sum_s \sum_m \sum_t \gamma_m^{(s)} (t) +\\ \quad \left( \log(2 \pi ) + \log \left| \cal C_m^{(s)} \right| + \left( o_t - \hat{\mu}_m^{(s)} \right) ^T \cal C_m^{(s)-1} \right)  ![Long equation](img/long.png)