The term "special hypergeometric functions" typically refers to a family of functions that generalize the hypergeometric function, which is a solution to the hypergeometric differential equation.

Zeta functions and L-functions are important concepts in number theory and have applications across various branches of mathematics, particularly in analytic number theory and algebraic geometry. ### Zeta Functions 1.

The **Crenel function**, also known as the rectified function or the rectangular function, is a type of mathematical function that is commonly used in signal processing and analysis. The Crenel function is typically defined as a piecewise constant function that is equal to 1 within a certain interval and equal to 0 outside that interval.

The Goodwin–Staton integral is a specific integral that arises in certain areas of analysis, particularly in relation to the study of functions defined on the real line and their properties. While there is limited detailed information available about this integral in standard texts, it is generally categorized under a class of integrals that may involve special functions or techniques used in advanced mathematical analysis.

The "Hough function" typically refers to the Hough Transform, a technique used in image analysis and computer vision to detect shapes, particularly lines, circles, or other parameterized curves within an image. The Hough Transform is particularly effective for detecting shapes that can be represented as mathematical equations. ### Concept of Hough Transform: 1. **Line Detection**: The basic form of the Hough Transform is used for detecting straight lines in images.

The Kontorovich–Lebedev transform is an integral transform used in mathematics and physics to solve certain types of problems, particularly in the context of integral equations and the theory of special functions. It is named after the mathematicians M. G. Kontorovich and N. N. Lebedev, who developed this transform in the context of mathematical analysis. The transform can be used to relate functions in one domain to functions in another domain, much like the Fourier transform or the Laplace transform.

Legendre form typically refers to a representation of a polynomial or an expression in terms of Legendre polynomials, which are a sequence of orthogonal polynomials that arise in various areas of mathematics, particularly in solving differential equations and problems in physics.

Painlevé transcendents are a class of special functions that arise as solutions to second-order ordinary differential equations known as the Painlevé equations. These equations were first identified by the French mathematician Paul Painlevé in the early 20th century.

The Q-function, or action-value function, is a fundamental concept in reinforcement learning and is used to evaluate the quality of actions taken in a given state. It helps an agent determine the expected return (cumulative future reward) from taking a particular action in a particular state, while following a specific policy thereafter.

The Selberg integral is a notable result in the field of mathematical analysis, particularly in the areas of combinatorics, probability, and number theory. It is named after the mathematician A. Selberg, who introduced it in the context of multivariable integrals.

The Strömgren integral is a concept used in the field of astrophysics, particularly in the study of ionized regions around stars, known as H II regions. It was introduced by the Swedish astronomer Bertil Strömgren in the 1930s. The Strömgren integral refers specifically to the calculation of the ionization balance in a gas that is exposed to a source of ionizing radiation, such as a hot, massive star.

Freiman's theorem is a result in additive combinatorics that provides a structural insight into sets of integers with small sumset sizes. Specifically, it concerns the behavior of subsets of the integers that contain a large number of elements while having a relatively small sumset.

Hall's Marriage Theorem is a result in combinatorial mathematics, specifically in the area of graph theory and bipartite matching. It provides a necessary and sufficient condition for the existence of a perfect matching in a bipartite graph.

Combinatorics of finite geometries is a field of study that explores the properties, structures, and configurations of geometric systems that are finite in nature. It combines principles from combinatorics—the branch of mathematics concerned with counting, arrangement, and combination of objects—with geometric concepts. Here are some key aspects of the combinatorics of finite geometries: 1. **Finite Geometries**: Finite geometries are geometric structures defined over a finite number of points.

An **integrally convex set** refers to a special type of set in the context of integer programming and combinatorial optimization.

Discrete geometry is a branch of geometry that studies geometric objects and properties in a combinatorial or discrete context. It often involves finite sets of points, polygons, polyhedra, and other shapes, and focuses on their combinatorial and topological properties. Theorems in discrete geometry often relate to the arrangement, selection, or structure of these sets in specific ways.

In geometry, triangulation refers to the process of dividing a geometric shape, such as a polygon, into triangles. This is often done to simplify calculations, especially in fields like computer graphics, spatial analysis, and geographic information systems (GIS). **Key points about triangulation in geometry:** 1. **Purpose:** Triangulation allows for easier computation of areas, volumes, and various properties of complex shapes since triangles are the simplest polygons.

Combinatorial Geometry is a branch of mathematics that deals with the study of geometric objects and their combinatorial properties, often in a discrete setting. When we refer specifically to "Combinatorial Geometry in the Plane," we are primarily concerned with planar arrangements of points, lines, polygons, and other geometric figures, and how these arrangements relate to various combinatorial aspects.

A Kakeya set is a set of points in a Euclidean space (typically in two or higher dimensions) that has the property that a needle, or line segment, of unit length can be rotated freely within the set without leaving it. The classic example is the Kakeya set in the plane, which can be thought of as a bounded region that can contain a unit segment that can be rotated to cover all angles.