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.
Fractional Calculus
Calculus is the manipulation of one basic operator: the derivative or . 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
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.
However, functions can also be described in terms of trigonometric functions using the Fourier series Continue reading
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 ( for uniform or
for Gaussian). In order to calculate the probability that any number generated will fall in the range
, one would take the integral
. The probability to get a specific number
would be
which is just 0. Continue reading
On Conservation
Conservation of different properties in nature immensely simplifies calculations to the point where some are impossible without the consideration of them. In some cases, it seems completely intuitive and impossible not to consider. However, not only are there many conservations laws unknown to many but there are also, in some sense, “violations” to these laws. Continue reading
The Shape of a String
Holding a string up with both ends at the same elevation causes the string to form a curve which not many really care to look into more than the first glance. At first, one may just assume that it is a parabola but upon closer look, it actually has a very interesting mathematical shape which is shown below. Continue reading
Strong Force and Nuclear Binding Energies
Forces come in various forms in nature but the nature of the strong force is very peculiar in that it does not create an attraction or repulsion between any two entities. It uses mass energy as a way to create a potential energy well. This is done by the following. Continue reading
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 where
. This however raised a problem when multiplying complex numbers. Continue reading
The Lagrangian
There exists a mathematically different approach to describing classical mechanics than what is usually taught. While it is usually taught using laws like , conservation of energy, and conservation of momentum, there exists a more mathematically elegant and, in some sense, more fundamental way of describing the motion of objects known as Lagrangian mechanics. It is used throughout engineering and physics to solve problems that are too complicated or impractical to solve using classical methods. Continue reading
