Hello World! This blog allows posts to be written using Markdown (for which this Markdown cheatsheet is very useful) and HTML/CSS, with support for maths typesetting using LaTeX + MathJax. For example, here is an inline formula formula: $y = ax^2 + bx + c$. We can also present displayed equations in their own (Markdown) section:

Euler’s formula: A complex number $z \in \mathbb{C}$ can be expressed in polar form as $z = r e^{i \theta}$, where

\[e^{i\theta} = \cos(i\theta) + i \sin(i\theta) , \quad \theta \in \mathbb{R} .\]

 

(We just added a blank line by using  , an HTML entity.)

We can even do multi-line equations with the familiar LaTeX environments:

\[\begin{align} (a+b)^2 &= (a+b)(a+b) \\ &= a^2 + 2ab + b^2 . \end{align}\]

Pretty neat! Note that the symbol to be used to for LaTeX commands is $$ ... $$ (or $ ... $ for inline formulas). Furthermore, a blank line needs to be left if a displayed formula is intended.

Note that with LaTeX support, some special characters have to be escaped (within the LaTeX environment, and also sometimes when writing normal text!) For example, a $5 meal has to be written a \$5 meal, since $ is the special character for inline LaTex $ ... $. As an additional note, the escaping character \ might have to be doubled (e.g. \\ is the special new line expression in LaTeX) depending on the content-management systems – we are passing through many layers of parsers! So if something breaks or does not seem to work as intended, this might be the reason.

Back to the features: we can also do matrices!

\[\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}\]

We can also do boxed environments (e.g. for theorems) by using the HTML element <div> and the custom CSS style .Boxed that we have defined, which simply adds a border (with padding, margin , etc.) to the text it surrounds. Note that this approach is very flexible and allows for much more customisation of the final appearance, but requires a fair bit more expertise in HTML/CSS development. For example:

Theorem 1: Let $D(z_0, r)$ be a disk in the complex plane, and $f : D(z_0, r) \to \mathbb{C}$ be a polynomial of degree $n \geq 1$ with $f'(z) \ne 0 $ for all $ z \in D(z_0, r)$. Then $f$ is injective on $D(z_0, \sin \frac{\pi}{n})$.

(This was taken from https://terrytao.wordpress.com/2020/11/28/holomorphic-images-of-disks/ just for illustration!)

Thus, we can rewrite Euler’s formula from above in an alternative, seemingly nicer way:

Euler's formula: A complex number $z \in \mathbb{C}$ can be expressed in polar form as $z = r e^{i \theta}$, where $$ e^{i\theta} = \cos(i\theta) + i \sin(i\theta) , \quad \theta \in \mathbb{R} . $$

Nice! We just have to be careful if we want to say \$10, \$50, \$100 instead (have to escape dollars now, and escape it doubly so with \\, since we are passing through two parsers!)

An example that the HTML element <br> can produce line breaks:

Shopping for apples, and then,
after a break, for bananas.
A picture of a duck
A picture of a duck

Another cool thing is that images and pictures can also be included, using HTML and CSS. For example, this <img> of a duck has been right-aligned by using a custom CSS style .TextWrapRight inside an HTML <figure> ... </figure> element.