Elementary special functions

Elementary special functions are a class of mathematical functions that have important applications across various fields, including mathematics, physics, engineering, and computer science. These functions extend the notion of elementary functions (such as polynomials, exponential functions, logarithmic functions, trigonometric functions, and their inverses) to include a broader set of functions that frequently arise in problems of mathematical analysis.

An exponential function is a mathematical function of the form \( f(x) = a \cdot b^x \), where: - \( a \) is a constant (the initial value), - \( b \) is the base of the exponential function (a positive real number), - \( x \) is the exponent (which can be any real number).

Hyperbolic functions are mathematical functions that are similar to the trigonometric functions but are defined using hyperbolas instead of circles. The two primary hyperbolic functions are the hyperbolic sine (sinh) and the hyperbolic cosine (cosh). ### Definitions: 1. **Hyperbolic Sine**: \[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] 2.

Inverse hyperbolic functions are the inverse functions of the hyperbolic functions, similar to how the inverse trigonometric functions relate to trigonometric functions.

Logarithms are a mathematical concept used to describe the relationship between numbers in terms of their exponents. Specifically, the logarithm of a number is the exponent to which a base must be raised to produce that number.

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. The word "trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure).

The Dirichlet function is a classic example of a function that is used in real analysis to illustrate concepts of continuity and differentiability.

The term "double exponential function" can refer to functions that involve exponentiation of an exponential function. Specifically, a double exponential function is typically of the form: \[ f(x) = a^{(b^x)} \] where \( a \) and \( b \) are constants, and \( a, b > 0 \). This function grows much faster than a regular exponential function due to the "double" exponentiation.

An exponential function is a mathematical function of the form: \[ f(x) = a \cdot b^{x} \] where: - \( f(x) \) is the value of the function at \( x \), - \( a \) is a constant that represents the initial value or coefficient, - \( b \) is the base of the exponential function, a positive real number, - \( x \) is the exponent, which can be any real number.

The Gudermannian function, often denoted as \(\text{gd}(x)\), is a mathematical function that relates the circular functions (sine and cosine) to the hyperbolic functions (sinh and cosh) without explicitly using imaginary numbers. It serves as a bridge between trigonometry and hyperbolic geometry.

The Kronecker delta is a mathematical function that is typically denoted by the symbol \( \delta_{ij} \). It is defined as: \[ \delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} \] In this definition, \( i \) and \( j \) are usually indices that can take integer values.

A logarithm is a mathematical function that helps to determine the power to which a given base must be raised to produce a certain number.

Ptolemy's table of chords is an ancient mathematical construct from Ptolemy's work in the realm of astronomy and trigonometry. In his work "Almagest" (or "Mathematics of the Stars"), Ptolemy compiled a table that lists the lengths of chords in a circle corresponding to various angles. This table served as an early form of trigonometric values before the formal development of trigonometry.

The sigmoid function is a mathematical function that has an "S"-shaped curve (hence the name "sigmoid," derived from the Greek letter sigma). It is often used in statistics, machine learning, and artificial neural networks due to its property of mapping any real-valued input to an output in the range of 0 to 1.

The Soboleva modified hyperbolic tangent function, often represented as \( \tanh_s(x) \), is a mathematical function that is a modification of the standard hyperbolic tangent function. In various domains, including physics and engineering, such modified functions are introduced to better handle specific properties such as asymptotic behavior, smoothness, or to meet certain boundary conditions.