Testing Mathematical Formulas in Markdown

This document serves as a test for rendering mathematical formulas in Markdown using $$ delimiters.

Basic Algebra

Let’s start with some fundamental algebraic expressions.

The quadratic formula is given by:

x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

A simple linear equation:

y = mx + c

Expansion of a binomial square:

(a+b)^2 = a^2 + 2ab + b^2

Here are some common expressions from calculus.

The limit definition of a derivative:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

A definite integral:

\int_{a}^{b} f(x) dx

The Taylor series expansion of $e^x$ around $x=0$ :

e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots

Some basic trigonometric identities.

Pythagorean identity:

\sin^2\theta + \cos^2\theta = 1

Angle addition formula for sine:

\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta

Euler’s formula:

e^{i\theta} = \cos\theta + i\sin\theta

Formulas commonly used in statistics and probability.

The formula for the mean ( $\mu$ ) of a set of $n$ numbers $x_1, x_2, \ldots, x_n$ :

\mu = \frac{1}{n} \sum_{i=1}^{n} x_i

The probability density function of a normal distribution:

f(x | \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Bayes’ theorem:

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Examples from linear algebra.

A 2x2 matrix:

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

The determinant of a 2x2 matrix:

\det(A) = ad - bc

Matrix multiplication of two matrices A and B:

C = AB

A couple of well-known physics equations.

Einstein’s mass-energy equivalence:

E = mc^2

Newton’s second law of motion:

F = ma

This should provide a good test of how various mathematical formulas are rendered.

Thanks for reading!

周二 5月 31 2022 LaTeX

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