Calculus Definitions > The standard form of any line in a Cartesian plane is Ax + By = C where: A, B, and C are placeholders for constants (at least one of A and B must be nonzero), x and…

Sequence and Series > An asymptotic series (also called an asymptotic expansion or Poincaré expansion) is a way of investigating the general behavior of a function as the independent variable (e.g. x) becomes very large. The procedure, which truncates a…

Types of Functions > Orthonormal functions are orthogonal and normalized. They are often used to find approximations for other difficult to compute functions. You can construct any well behaved function f using a complete set of orthonormal functions (Errede, 2008):…

Sequence and Series > The “classical” Stirling series is defined as (Dominic, 2008): Where the numbers Bk are the Bernoulli numbers. The series was formulated by the French mathematician Abraham DeMoivre (1667-1994), based on work by Scottish mathematician James Stirling…

Calculus Definitions > A unit square has one unit for each side. The unit is problem specific. For example, if you’re working with cm, the unit square would have sides of 1 cm and would contain one square cm—an area…

Calculus Definitions > Einstein Summation, introduced by Einstein in 1916 in his generalized theory of relativity paper, is a shorthand way to avoid the tedium of writing summations. It is stated as: “If a suffix occurs twice in a term,…

Types of Functions > Contents: Orthogonal Functions Orthogonal Series Orthogonal Functions Orthogonal functions are two functions with an inner product of zero. Orthogonal functions are particularly useful for finding solutions to partial differential equations like Schrodinger’s equation and Maxwell’s equations.…

Types of Functions > In general terms, a prediction function is a mathematical function that tells you something about a future event, based on past events. There are many different kinds of functions that might be classified as prediction functions,…

Sequence and Series > A Cauchy sequence is a sequence of real numbers with terms that eventually cluster together—if the difference between terms eventually gets closer to zero. Whether or not a sequence is Cauchy is determined only by its…

Sequence and Series > Trigonometric Series & Polynomial A trigonometric series is a series of cosines and sines of multiple angles. This powerful way to represent and study functions has the form (Kechris, 2020): Where: an ∈ ℂ (the complex…