Updated documentation.

EXAMPLES.md

@@ -1,13 +1,13 @@
-# MathRender Examples
+# SwiftMath Examples
 
 ## Square of sums
 ```LaTeX
 (a_1 + a_2)^2 = a_1^2 + 2a_1a_2 + a_2^2
 ```
 
-As rendered by MathJax: $(a_1 + a_2)^2 = a_1^2 + 2a_1a_2 + a_2^2$
+MathJax: $(a_1 + a_2)^2 = a_1^2 + 2a_1a_2 + a_2^2$
 
-MathRender: 
+SwiftMath: 
 
 ![Square Formula](img/square.png)
 
@@ -16,12 +16,21 @@ MathRender:
 x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
 ```
 
+MathJax: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
+
+SwiftMath: 
+
 ![Quadratic Formula](img/quadratic.png)
 
 ## Standard Deviation
 ```LaTeX
 \sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2}
 ```
+
+MathJax: $\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2}$
+
+SwiftMath: 
+
 ![Standard Deviation](img/standard.png)
 
 ## De Morgan's laws
@@ -29,6 +38,10 @@ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
 \neg(P\land Q) \iff (\neg P)\lor(\neg Q)
 ```
 
+MathJax: $\neg(P\land Q) \iff (\neg P)\lor(\neg Q)$
+
+SwiftMath: 
+
 ![De Morgan](img/demorgan.png)
 
 ## Log Change of Base
@@ -36,6 +49,10 @@ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
 \log_b(x) = \frac{\log_a(x)}{\log_a(b)}
 ```
 
+MathJax: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$
+
+SwiftMath: 
+
 ![Log Base Change](img/log.png)
 
 ## Cosine addition
@@ -43,6 +60,10 @@ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
 \cos(\theta + \varphi) = \cos(\theta)\cos(\varphi) - \sin(\theta)\sin(\varphi)
 ```
 
+MathJax: $\cos(\theta + \varphi) = \cos(\theta)\cos(\varphi) - \sin(\theta)\sin(\varphi)$
+
+SwiftMath: 
+
 ![Cos Sum](img/trig.png)
 
 ## Limit e^k
@@ -50,6 +71,10 @@ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
 \lim_{x\to\infty}\left(1 + \frac{k}{x}\right)^x = e^k
 ```
 
+MathJax: $\lim_{x\to\infty}\left(1 + \frac{k}{x}\right)^x = e^k$
+
+SwiftMath: 
+
 ![Limit](img/limit.png)
 
 ## Calculus
@@ -57,6 +82,10 @@ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
 f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\xi
 ```
 
+MathJax: $f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\xi$
+
+SwiftMath: 
+
 ![Calculus](img/calculus.png)
 
 ## Stirling Numbers of the Second Kind
@@ -64,6 +93,10 @@ f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\
 {n \brace k} = \frac{1}{k!}\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}(k-j)^n
 ```
 
+MathJax: ${n \brace k} = \frac{1}{k!}\sum_{j=0}^k (-1)^{k-j}\binom{k}{j}(k-j)^n$
+
+SwiftMath: 
+
 ![Stirling Numbers](img/stirling.png)
 
 ## Gaussian Integral
@@ -71,6 +104,10 @@ f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\
 \int_{-\infty}^{\infty} \! e^{-x^2} dx = \sqrt{\pi}
 ```
 
+MathJax: $\int_{-\infty}^{\infty} \! e^{-x^2} dx = \sqrt{\pi}$
+
+SwiftMath: 
+
 ![Gauss Integral](img/gaussintegral.png)
 
 ## Arithmetic mean, geometric mean inequality
@@ -78,12 +115,21 @@ f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\
 \frac{1}{n}\sum_{i=1}^{n}x_i \geq \sqrt[n]{\prod_{i=1}^{n}x_i}
 ```
 
+MathJax: $\frac{1}{n}\sum_{i=1}^{n}x_i \geq \sqrt[n]{\prod_{i=1}^{n}x_i}$
+
+SwiftMath: 
+
 ![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)
 ```
+
+MathJax: $\left(\sum_{k=1}^n a_k b_k \right)^2 \le \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right)$
+
+SwiftMath: 
+
 ![Cauchy Schwarz](img/cauchyschwarz.png)
 
 ## Cauchy integral formula
@@ -91,13 +137,23 @@ f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\
 f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_\gamma\frac{f(z)}{(z-z_0)^{n+1}}dz
 ```
 
+MathJax: $f^{(n)}(z_0) = \frac{n!}{2\pi i}\oint_\gamma\frac{f(z)}{(z-z_0)^{n+1}}dz$
+
+SwiftMath: 
+
 ![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)
 ```
 
+MathJax: $i\hbar\frac{\partial}{\partial t}\mathbf\Psi(\mathbf{x},t) = -\frac{\hbar}{2m}\nabla^2\mathbf\Psi(\mathbf{x},t)
++ V(\mathbf{x})\mathbf\Psi(\mathbf{x},t)$
+
+SwiftMath: 
+
 ![Schroedinger](img/schroedinger.png)
 
 ## Lorentz Equations
@@ -111,6 +167,14 @@ equations.
 \end{gather}
 ```
 
+MathJax: $\begin{gather}
+\dot{x} = \sigma(y-x) \\
+\dot{y} = \rho x - y - xz \\
+\dot{z} = -\beta z + xy"
+\end{gather}$
+
+SwiftMath: 
+
 ![Lorentz](img/lorentz.png)
 
 ## Cross product
@@ -122,6 +186,14 @@ equations.
 \end{vmatrix}
 ```
 
+MathJax: $\vec \bf V_1 \times \vec \bf V_2 =  \begin{vmatrix}
+\hat \imath &\hat \jmath &\hat k \\
+\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
+\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
+\end{vmatrix}$
+
+SwiftMath: 
+
 ![Cross Product](img/cross.png)
 
 ## Maxwell's Equations
@@ -136,6 +208,15 @@ multiple equations.
 \end{eqalign}
 ```
 
+MathJax: $begin{eqalign}
+\nabla \cdot \vec{\bf E} & = \frac {\rho} {\varepsilon_0} \\
+\nabla \cdot \vec{\bf B} & = 0 \\
+\nabla \times \vec{\bf E} &= - \frac{\partial\vec{\bf B}}{\partial t} \\
+\nabla \times \vec{\bf B} & = \mu_0\vec{\bf J} + \mu_0\varepsilon_0 \frac{\partial\vec{\bf E}}{\partial t}
+\end{eqalign}$
+
+SwiftMath: 
+
 ![Maxwell's Equations](img/maxwell.png)
 
 ## Matrix multiplication
@@ -154,6 +235,19 @@ c\alpha + d\gamma & c\beta + d \delta
 \end{pmatrix}
 ```
 
+MathJax: $\begin{pmatrix}
+a & b\\ c & d
+\end{pmatrix}
+\begin{pmatrix}
+\alpha & \beta \\ \gamma & \delta
+\end{pmatrix} = 
+\begin{pmatrix}
+a\alpha + b\gamma & a\beta + b \delta \\
+c\alpha + d\gamma & c\beta + d \delta 
+\end{pmatrix}$
+
+SwiftMath: 
+
 ![Matrix Multiplication](img/matrixmult.png)
 
 ## Cases
@@ -164,6 +258,13 @@ f(x) = \begin{cases}
 \end{cases}
 ```
 
+MathJax: $f(x) = \begin{cases}
+\frac{e^x}{2} & x \geq 0 \\
+1 & x < 0
+\end{cases}$
+
+SwiftMath: 
+
 ![Cases](img/cases.png)
 
 ## Splitting long equations
@@ -174,4 +275,11 @@ f(x) = \begin{cases}
 \left( o_t - \hat{\mu}_m^{(s)} \right) ^T \cal C_m^{(s)-1} \right) 
 ```
 
+MathJax: $\frak Q(\lambda,\hat{\lambda}) =
+-\frac{1}{2} \mathbb P(O \mid \lambda ) \sum_s \sum_m \sum_t \gamma_m^{(s)} (t) +\\
+\quad \left( \log(2 \pi ) + \log \left| \cal C_m^{(s)} \right| +
+\left( o_t - \hat{\mu}_m^{(s)} \right) ^T \cal C_m^{(s)-1} \right)$
+
+SwiftMath: 
+
 ![Long equation](img/long.png)

README.md

@@ -4,6 +4,10 @@
 for displaying beautifully rendered math equations in iOS and MacOS applications. It typesets formulae written 
 using LaTeX in a `UILabel` equivalent class. It uses the same typesetting rules as LaTeX and
 so the equations are rendered exactly as LaTeX would render them. 
+`
+SwiftMath` is a Swift translation of the latest `iosMath` v0.9.5 release but includes bug fixes
+and enhancements like a new \lbar (lambda bar) character and cyrillic alphabet support.
+Please let me know of any bugs or bug fixes that you find. 
 
 `SwiftMath` prepackages everything needed for direct access via the Swift Package Manager.
 No need for complicated alien pods that never seem to work quite right.
@@ -16,12 +20,29 @@ importantly, it is significantly faster than using a `UIWebView`.
 ## Examples
 Here are screenshots of some formulae that were rendered with this library:
 
+```LaTeX
+x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
+```
+
 ![Quadratic Formula](img/quadratic.png) 
 
+```LaTeX
+f(x) = \int\limits_{-\infty}^\infty\!\hat f(\xi)\,e^{2 \pi i \xi x}\,\mathrm{d}\xi
+```
+
 ![Calculus](img/calculus.png)
 
+```LaTeX
+\frac{1}{n}\sum_{i=1}^{n}x_i \geq \sqrt[n]{\prod_{i=1}^{n}x_i}
+```
+
 ![AM-GM](img/amgm.png)
 
+```LaTex
+    \frac{1}{\left(\sqrt{\phi \sqrt{5}}-\phi\\right) e^{\frac25 \pi}}
+    = 1+\frac{e^{-2\pi}} {1 +\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }
+```
+
 ![Ramanujan Identity](img/ramanujan.png)
 
 More examples are included in [EXAMPLES](EXAMPLES.md)