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