## Complex Exponentials

Complex exponentials are used immensely in math and as a result, in many fields of science. It is also used in abundance throughout this site so it is important to understand what they are for future reference. They show the relationship between exponentials and trigonometry on a fundamental level. The following is the relationship.

$e^{ix}=\textup{cos}(x)+i\textup{sin}(x)$ Continue reading Complex Exponentials

## Fractional Calculus

Calculus is the manipulation of one basic operator: the derivative or $\inline&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}$. This operator operates on functions and by repeatedly applying it, you can get higher order derivatives. It’s inverse operator is known as the integral. Similar to matrix operators which have eigenvalues and eigenvectors, this operator also has eigenvalues and eigenfunctions. The eigenfunction is the function which only goes through some scalar change when acted on by the operator. This scalar that the function is scaled by is called the eigenvalue of the eigenfunction. For the derivative operator, Continue reading Fractional Calculus

## Space Transforms

Functions have the ability to be described in terms of the infinite summation of other functions with a common example being polynomials using Taylor series as shown below.

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$

However, functions can also be described in terms of trigonometric functions using the Fourier series Continue reading Space Transforms

## Probability Distribution Transformations

Random number generators generate random numbers that tend to follow some kind of distribution. Uniform number generators generate a random number between 0 and 1 with equal likely hood everywhere in the range, Gaussian generators generate numbers that have higher probability to be near zero, etc. These distributions are given functions ($\inline&space;p(x)=1$ for uniform or $\inline&space;p(x)=e^{-x^2/2}$ for Gaussian). In order to calculate the probability that any number generated will fall in the range $\inline&space;[a,b]$, one would take the integral $\inline&space;\int_{a}^{b}p(x)\textup{d}x$. The probability to get a specific number $\inline&space;x$ would be $\inline&space;p(x)\textup{d}x$ which is just 0. Continue reading Probability Distribution Transformations

## Quaternions

Many are familiar with the idea of imaginary/complex numbers but in 1843, Hamilton invented hypercomplex numbers. Initially, he created three components such that they had the form $\inline&space;a+bi+cj$ where $\inline&space;j^2=-1$. This however raised a problem when multiplying complex numbers. Continue reading Quaternions