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
\[
c_k(f) = \int_0^1 f(x) \, \mathrm{e}^{-2\pi \mathrm{i}k x} \, \mathrm{d}x.
\]