2 Complex Mathematical Equations

This chapter demonstrates various mathematical equations and their formatting across different output formats. We’ll explore both inline and display mathematics, using examples from various fields.

2.1 Basic Equations

Let’s begin with some fundamental equations. The quadratic formula states that for an equation ax² + bx + c = 0, the solution is:

2.2 Complex Analysis

In complex analysis, we often encounter Euler’s formula:

A special case of this is Euler’s Identity, one of the most beautiful equations in mathematics Dunham1999:

2.3 Linear Algebra

In linear algebra, we work with matrices and vectors. Consider the eigenvalue equation:

Where A is a square matrix, v is an eigenvector, and λ is an eigenvalue Strang2006.

2.4 Differential Equations

The ordinary differential equation for simple harmonic motion is:

2.5 Statistical Formulas

In statistics, we frequently use the normal distribution, whose probability density function is:

This formula is essential in many statistical applications Ross2014.