# Tutorial – Read Me

This forum is supported with MathJax, that is, it is $$\LaTeX$$ enabled. Underneath is a small introduction to the syntax (Credit: Mario Laux). The commands for symbols that are not introduced below can (probably) be found e.g. here.

### Typesetting basics

Formulas inside square brackets $...$ (each leading with a backslash!) are shown big and centered in their own paragraph (“display mode”). Formulas inside round brackets $$...$$ are embedded in continuous text and, if possible, are properly adjusted to row hight (“inline mode”).

The source code of each equation can usually be displayed by right-hand clicking on the formula itself.

For clarity reasons, most of the here presented formulas are in display mode $...$, however, each command works fine in inline mode.

#### Fractions

$\frac{3}{2} = \frac{\frac{11-5}{2}}{\frac{6}{4-1}}$
$\frac{3}{2} = \frac{\frac{11-5}{2}}{\frac{6}{4-1}}$

#### Brackets

There is a difference between normal brackets (...) and scalable brackets \left(...\right); the latter adjust to their content.

$f(x) := \left(1+\frac{x}{2}\right) \left(1-\frac{x}{2}\right)$
$f(x) := \left(1+\frac{x}{2}\right) \left(1-\frac{x}{2}\right)$

#### Sub- and Superscript

$­­(x_1 + x_2)^2 = x_1^2 + 2 x_1 x_2 + x_2^2$
$(x_1 + x_2)^2 = x_1^2 + 2 x_1 x_2 + x_2^2$

#### Grouping

To avoid ambiguities, connected arguments have to be grouped together using braces {...}.

$­­2^{3x} \neq 2^3x$
$2^{3x} \neq 2^3x$

#### Font style

Elementary functions are generated by \sin, \cos, \ln,.... The leading backslash forces upright instead of italic letters to distinguish them from variables.

$­­\sin^2(x) + \cos^2(x) = 1$
$\sin^2(x) + \cos^2(x) = 1$

Generally, mathematical operators, that are no variables, can be written upright using \mathrm{...}.

$­­\mathrm{e}^{\mathrm{i} x} = \cos(x) + \mathrm{i} \sin(x)$
$\mathrm{e}^{\mathrm{i} x} = \cos(x) + \mathrm{i} \sin(x)$

#### Sums

$­­\mathrm{e}^x = \sum_{n=0}^\infty \frac{x^n}{n!}$
$\mathrm{e}^x = \sum_{n=0}^\infty \frac{x^n}{n!}$

#### Integrals

$­­\int_a^b f(g(x)) g'(x) \mathrm{d}x = \int_{g(a)}^{g(b)} f(u) \mathrm{d}u$
$\int_a^b f(g(x)) g'(x) \mathrm{d}x = \int_{g(a)}^{g(b)} f(u) \mathrm{d}u$

#### Limes, Supremum, Infimum

$­­\limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{k\geq n} a_k$
$\limsup_{n \to \infty} a_n = \lim_{n \to \infty} \sup_{k\geq n} a_k$

### Additional commands

#### Blackboard Bold

$­­1 \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$
$1 \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

#### Text

$­­\frac{\text{Zähler}}{\text{Nenner}}$
$\frac{\text{numerator}}{\text{denominator}}$

#### Complex conjugate

$­­\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$
$\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}$

#### Spacing

Multiple spaces or line breaks have no effect in inline or display mode. Small space can be added using \, or \;, commands as \quad or \qquad generate bigger spaces.

$­­\int_0^\pi \mathrm{e}^{-R \sin(x)} \, \mathrm{d}x \leq \frac{\pi}{R} \quad \text{for all } R>0$
$\int_0^\pi \mathrm{e}^{-R \sin(x)} \, \mathrm{d}x \leq \frac{\pi}{R} \quad \text{for all } R>0$

#### Comments

$­­f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even function}} + \underbrace{\frac{f(x) - f(-x)}{2}}_{\text{odd function}}$
$f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even function}} + \underbrace{\frac{f(x) – f(-x)}{2}}_{\text{odd function}}$

#### Braces

Besides square and round brackets, braces are also possible, however, they need a leading backslash like \{...\} or \left{...\right}.

$­­\sqrt{2} := \sup \left\{x \in \mathbb{R}: \; x^2 < 2 \right\}$
$\sqrt{2} := \sup \left\{x \in \mathbb{R}: \; x^2 < 2 \right\}$

#### Absolute value, Norm

$­­\left| \int_a^b f(x) \, \mathrm{d}x \right| \leq |b-a| \cdot \|f\|_\infty$
$\left| \int_a^b f(x) \, \mathrm{d}x \right| \leq |b-a| \cdot \|f\|_\infty$

#### Matrices

$­­\mathrm{id}_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
$\mathrm{id}_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

#### Case differentiation

$­­|x| = \begin{cases} x &\text{if } x \geq 0 \\ -x &\text{if } x < 0 \end{cases}$
$|x| = \begin{cases} x &\text{if } x \geq 0 \\ -x &\text{if } x < 0 \end{cases}$

### Examples

#### Binomial Theorem

­For all $$x,y \in \mathbb{R}$$ and all $$n \in \mathbb{N}_0$$ it holds $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}.$ 
For all $$x,y \in \mathbb{R}$$ and all $$n \in \mathbb{N}_0$$ it holds
$(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}.$

#### Fourier series

­Let $$f:\mathbb{R} \to \mathbb{C}$$ be a $$1$$-periodic, two times continuously differentiable function. Then for all $$x \in \mathbb{R}$$ it holds $f(x) = \sum_{k=-\infty}^\infty c_k(f) \, \mathrm{e}^{2\pi\mathrm{i} k x}$ where the convergence is uniformly. The Fourier coefficients are give by $c_k(f) = \int_0^1 f(x) \, \mathrm{e}^{-2\pi \mathrm{i}k x} \, \mathrm{d}x.$ 
Let $$f:\mathbb{R} \to \mathbb{C}$$ be a $$1$$-periodic, two times continuously differentiable function. Then for all $$x \in \mathbb{R}$$ it holds
$f(x) = \sum_{k=-\infty}^\infty c_k(f) \, \mathrm{e}^{2\pi\mathrm{i} k x}$
and the convergence is uniformly. The Fourier coefficients are given by
$c_k(f) = \int_0^1 f(x) \, \mathrm{e}^{-2\pi \mathrm{i}k x} \, \mathrm{d}x.$