Special functions

Special functions are particular mathematical functions that arise frequently in various areas of mathematics, physics, and engineering. These functions have specific properties and often involve solutions to certain types of differential equations or integrals that are encountered in applied mathematics. Some of the most commonly recognized special functions include: 1. **Bessel Functions**: Arise in problems with cylindrical symmetry, such as heat conduction in cylindrical objects.

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.

Elliptic functions are a class of complex functions that are periodic in two directions, making them doubly periodic. This property is essential in many areas of mathematics, including number theory, algebraic geometry, and mathematical physics. Key characteristics of elliptic functions include: 1. **Doubly Periodic**: An elliptic function has two distinct periods, usually denoted as \(\omega_1\) and \(\omega_2\).

Elliptic curves are a specific type of curve defined by a mathematical equation of the form: \[ y^2 = x^3 + ax + b \] where \( a \) and \( b \) are real numbers such that the curve does not have any singular points (i.e., it has no cusps or self-intersections).

Inverse Jacobi elliptic functions are the inverse functions of the Jacobi elliptic functions, which are a set of elliptic functions that generalize the trigonometric and exponential functions.

The inverse lemniscate functions are mathematical functions that are related to the geometrical shape known as the lemniscate, which resembles a figure-eight or an infinity symbol (∞). The most commonly referenced lemniscate is the lemniscate of Bernoulli, which is defined by the equation: \[ (x^2 + y^2)^2 = a^2 (x^2 - y^2) \] for some positive constant \(a\).

Abel elliptic functions, named after the mathematician Niels Henrik Abel, are a specific class of functions that relate to elliptic curves and are used to analyze the properties of elliptic integrals. They arise in the context of the theory of elliptic functions, which are complex functions that are periodic in two directions.

Carlson symmetric form is a mathematical representation used primarily in the context of complex analysis and number theory, particularly in the theory of modular forms and elliptic functions. It is named after the mathematician Borchardt Carlson. In simple terms, the Carlson symmetric form is a way to express certain types of functions that are symmetric in their arguments.

Complex multiplication is a concept from complex number theory that involves multiplying complex numbers. A complex number is expressed in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, \( i \) is the imaginary unit (defined as \( i^2 = -1 \)), \( a \) is the real part, and \( b \) is the imaginary part.

The Dedekind eta function is a complex function that plays a significant role in number theory, modular forms, and the theory of partitions. It is defined for a complex number \( \tau \) in the upper half-plane (i.e.

Dixon elliptic functions are a set of functions that arise in the theory of elliptic functions, which are complex functions that are periodic in two different directions. Specifically, Dixon elliptic functions are a generalization of the classical elliptic functions and are studied primarily in the context of algebraic functions and complex analysis. Named after the mathematician Alfred William Dixon, these functions have particular properties that make them useful in various branches of mathematics, including number theory, algebraic geometry, and mathematical physics.

Elliptic functions are a special class of complex functions that are periodic in two directions. They can be thought of as generalizations of trigonometric functions (which are periodic in one direction) to a two-dimensional lattice. Specifically, an elliptic function is a meromorphic function \( f \) defined on the complex plane that is periodic with respect to two non-collinear periods \( \omega_1 \) and \( \omega_2 \).

An elliptic integral is a type of integral that arises in the calculation of the arc length of an ellipse, as well as in various problems of physics and engineering. Elliptic integrals are generally not expressible in terms of elementary functions, which means that their solutions cannot be represented using basic algebraic operations and standard functions (like polynomials, exponentials, trigonometric functions, etc.).

The term "equianharmonic" generally refers to a relationship in music theory regarding scales, particularly concerning the structure and tuning of musical intervals. Specifically, it is used in the context of musical tuning systems that provide equal temperament relationships between different notes or pitches. One common example relates to the "equianharmonic" concept in the context of different tunings that make different intervals sound similar in terms of harmonic function, even if their pitches differ.

"Fundamenta Nova Theoriae Functionum Ellipticarum" is an important work by the mathematician Niels Henrik Abel, published in 1826. The title translates to "New Foundations for the Theory of Elliptic Functions." In this work, Abel laid the groundwork for modern elliptic function theory, providing detailed studies of elliptic integrals and the functions derived from them.

The term "fundamental pair of periods" typically refers to a specific concept in the realm of complex analysis, particularly in the study of elliptic functions and tori. In the context of elliptic functions, a fundamental pair of periods consists of two complex numbers, usually denoted by \(\omega_1\) and \(\omega_2\), which define the lattice in the complex plane that corresponds to an elliptic function. ### Key Points 1.

The half-period ratio, often referred to in the context of periodic functions, is a mathematical concept that describes the relationship between the periods of a function and its symmetry properties. Specifically, for a periodic function, the half-period ratio relates the half-period to the full period of the function. More formally, if \( T \) is the full period of a periodic function, then the half-period, denoted as \( T/2 \), is simply half of that period.

The \( J \)-invariant is an important quantity in the theory of elliptic curves and complex tori. In the context of elliptic curves defined over the field of complex numbers, the \( J \)-invariant is a single complex number that classifies elliptic curves up to isomorphism. Two elliptic curves are isomorphic if and only if their \( J \)-invariants are equal.

Jacobi theta functions are a set of complex functions that play a significant role in various areas of mathematics, including number theory, algebraic geometry, and mathematical physics. They are fundamental in the theory of elliptic functions.

Landen's transformation is a mathematical technique used in the field of elliptic functions and integral calculus. It is primarily applied to transform one elliptic integral into another, typically simplifying the computation or enabling the evaluation of elliptic integrals.

Legendre's relation typically refers to a specific relationship in number theory related to the distribution of primes. It is most commonly associated with Legendre's conjecture, which posits that there is always at least one prime number between any two consecutive perfect squares.

Lemniscate elliptic functions are a class of functions that arise in the study of elliptic curves and are connected to the geometry of the lemniscate, a figure-eight shaped curve.

A **modular lambda function** typically refers to the use of lambda functions within a modular programming context, often in functional programming languages or languages that support functional paradigms, like Python, JavaScript, and Haskell. However, the term isn't standardized and can mean a few things depending on the context. Here are some ways to interpret or use modular lambda functions: 1. **Lambda Functions**: A lambda function is a small anonymous function defined using the `lambda` keyword.

In mathematics, "nome" has a specific meaning related to elliptic functions. A nome is a complex variable often used in the context of elliptic integrals and functions. It is defined in relation to the elliptic modulus \( k \) (or the parameter \( m \), where \( m = k^2 \)).

The Picard–Fuchs equation is a type of differential equation that arises in the context of complex geometry, particularly in the study of algebraic varieties and their deformation theory. It is named after Émile Picard and Richard Fuchs, who contributed to the theory of differential equations and their applications in various mathematical contexts. In simpler terms, the Picard–Fuchs equation typically arises when trying to understand the variation of periods of a family of algebraic varieties or complex manifolds.

The term "quarter period" can refer to a few different contexts depending on the domain in which it is used. Here are a few possible interpretations: 1. **Financial Context**: In finance and business, a quarter period typically refers to a three-month period used by companies to report their financial performance.

The theta function is a special mathematical function often used in various areas of mathematics, including complex analysis, number theory, and mathematical physics. There are several different definitions of theta functions, but the most common ones arise in the context of elliptic functions and modular forms.

The Weierstrass elliptic function is a fundamental object in the theory of elliptic functions, which are special functions that have a periodic nature in two directions. These functions are used extensively in various fields of mathematics, including complex analysis, algebraic geometry, and number theory.

The Weierstrass function is a famous example of a continuous function that is nowhere differentiable. It serves as a significant illustration in real analysis and illustrates properties of functions that may be surprisingly counterintuitive.

Hypergeometric functions are a class of special functions that generalize many series and functions in mathematics, primarily arising in the context of solving differential equations, combinatorics, and mathematical physics.

The Appell series is a type of mathematical series that generalizes the concept of power series and is related to certain types of functions known as Appell functions. The series is named after the French mathematician Paul Appell. A typical form of an Appell series can be represented as follows: \[ f(x) = \sum_{n=0}^{\infty} A_n x^n \] where \(A_n\) are the coefficients that depend on certain parameters.

The bilateral hypergeometric series is a generalization of the ordinary hypergeometric series, which allows for the summation of terms indexed by two parameters rather than one.

Dougall's formula is a result in the field of combinatorics and special functions, specifically related to partitions and q-series. It provides an expression for certain types of sums involving binomial coefficients and powers of variables, often used in the study of partitions and generating functions.

The elliptic hypergeometric series is a special class of hypergeometric series that incorporates elliptic functions and is closely related to the theory of elliptic integrals and modular forms. These series generalize the classical hypergeometric series by including parameters that arise from the elliptic functions, which are periodic functions that have two fundamental periods.

The Frobenius solution to the hypergeometric equation refers to the method of finding a series solution near a regular singular point of the hypergeometric differential equation.

The general hypergeometric function, often denoted as \(_pF_q\), is a special function defined by a series expansion that generalizes the concept of hypergeometric functions.

A Horn function is a special type of Boolean function that can be expressed in a specific standard form. More formally, a Boolean function is considered a Horn function if it can be represented as a disjunction (logical OR) of clauses, where each clause has at most one positive literal. In other words, a Horn clause is a disjunction of literals in which at most one literal is positive, while the others are negative.

The Humbert series is a type of mathematical series that arises in the context of certain types of convergent sequences. Specifically, it is often associated with the study of summability methods and can be used in various fields such as number theory and functional analysis. While there isn't a universally accepted definition that is widely recognized under the name "Humbert series," it may refer to specific series associated with Humbert transformations or may arise in particular mathematical contexts or problems.

The hypergeometric function is a special function that generalizes the concept of power series and appears in various areas of mathematics, physics, and statistics. In the context of matrix arguments, the hypergeometric function can be extended to accommodate matrices, leading to the concept of the matrix hypergeometric function.

The Kampé de Fériet function is a special function in the field of mathematical analysis, particularly in relation to hypergeometric functions. It is named after the mathematician Léon Kampé de Fériet. The function generalizes some properties of the hypergeometric functions and is often expressed in terms of series expansions or integrals.

The Lauricella hypergeometric series is a generalization of the classical hypergeometric series and is denoted as \( F_D \). It is a function of several variables and is defined for several complex variables. It generalizes the standard hypergeometric series, which is a function of one variable, to cases with multiple parameters and arguments.

The Legendre functions, often referred to in the context of Legendre polynomials and Legendre functions of the first and second kind, arise in the solution of a variety of problems in physics and engineering, particularly in the fields of potential theory and solving partial differential equations. 1. **Legendre Polynomials**: These are a sequence of orthogonal polynomials defined on the interval \([-1, 1]\) and are denoted as \(P_n(x)\).

The list of hypergeometric identities typically refers to a collection of mathematical equations involving hypergeometric functions, often expressed in terms of the generalized hypergeometric series.

The MacRobert E function, often denoted as \( E(x) \), is a special function in mathematics that is related to the theory of complex variables and is used primarily in the context of mathematical analysis and applied mathematics. It is particularly significant in the studies involving wave equations and stability analysis of certain differential equations. ### Definition The MacRobert E function can be defined in various contexts, including as part of integrals leading to special functions or as solutions to specific types of differential equations.

The Meijer G-function is a special function that generalizes many other special functions, including exponential functions, logarithmic functions, Bessel functions, and hypergeometric functions. It provides a powerful tool for solving a variety of problems in mathematical analysis, physics, engineering, and other fields.

Riemann's differential equation typically refers to a type of linear partial differential equation associated with Riemann surfaces and complex analysis. However, there isn't a single, universally recognized differential equation directly defined as "Riemann's differential equation." One prominent equation related to Riemann surfaces is the Riemann-Hilbert problem, which is a type of boundary value problem for holomorphic functions, involving a piecewise constant function defined on contours in the complex plane.

Schwarz's list is a classification of certain interesting or notable groups of mathematical objects, specifically in the context of algebraic topology and complex geometry. It is named after the mathematician Hermann Schwarz. In algebraic topology, Schwarz's list typically refers to specific examples or types of manifolds that exhibit particular properties or behaviors, often with an emphasis on those that are closely related to the study of Riemann surfaces, complex manifolds, or other geometric structures.

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.

Bessel functions are a family of solutions to Bessel's differential equation, which arises in various problems in mathematical physics, particularly in wave propagation, heat conduction, and static potentials. The equation is typically expressed as: \[ x^2 y'' + x y' + (x^2 - n^2) y = 0 \] where \( n \) is a constant, and \( y \) is the function of \( x \).

The Bessel–Clifford function is a type of special function that arises in the solution of certain boundary value problems, particularly in cylindrical coordinates. It is closely related to Bessel functions, which are a family of solutions to Bessel's differential equation. The Bessel–Clifford function is often used in contexts where the problems have cylindrical symmetry, and along with the Bessel functions, it can represent wave propagation, heat conduction, and other phenomena in cylindrical domains.

The Coulomb wave functions are solutions to the Schrödinger equation for a particle subject to a Coulomb potential, which is the potential energy associated with the interaction between charged particles. This potential is typically represented as \( V(r) = -\frac{Ze^2}{r} \), where \( Z \) is the atomic number (or effective charge), \( e \) is the elementary charge, and \( r \) is the distance from the charge.

The Cunningham function, often denoted as \( C_n \), is a sequence of numbers defined as follows: - \( C_0 = 1 \) - \( C_1 = 1 \) - For \( n \geq 2 \), \( C_n = 2 \cdot C_{n-1} + C_{n-2} \) This recurrence relation means that each term is generated by taking twice the previous term and adding the term before that.

The error function, often denoted as \(\text{erf}(x)\), is a mathematical function used in probability, statistics, and partial differential equations, particularly in the context of the normal distribution and heat diffusion problems.

Incomplete Bessel functions are special functions that arise in various areas of mathematics, physics, and engineering, particularly in problems involving cylindrical symmetry or wave phenomena. Specifically, they are related to Bessel functions, which are solutions to Bessel's differential equation. The incomplete Bessel functions can be thought of as Bessel functions that are defined only over a finite range or with a truncated domain.

Kelvin functions, also known as cylindrical harmonics or modified Bessel functions of complex order, are special functions that arise in various problems in mathematical physics, particularly in wave propagation, heat conduction, and other areas where cylindrical symmetry is present. They are denoted as \( K_{\nu}(z) \) and \( I_{\nu}(z) \) for the Kelvin functions of the first kind and second kind, respectively.

Lentz's algorithm is a numerical method used for computing the value of certain types of functions, particularly those that can be expressed in the form of an infinite series or continued fractions. This algorithm is particularly useful for evaluating functions that are difficult to calculate directly due to issues such as convergence or numerical instability.

The logarithmic integral function, denoted as \( \mathrm{Li}(x) \), is a special function that is defined as follows: \[ \mathrm{Li}(x) = \int_2^x \frac{dt}{\log(t)} \] for \( x > 1 \). The function is often used in number theory, particularly in relation to the distribution of prime numbers.

Solid harmonics are mathematical functions that are used in various fields such as physics, engineering, and applied mathematics to describe functions on the surface of a sphere and in three-dimensional space. They are a generalization of spherical harmonics, which are typically defined on the surface of a sphere. In essence, solid harmonics can be thought of as a set of basis functions for representing scalar fields in three-dimensional space.

The Sonine formula, also known as Sonine's theorem, is a mathematical expression that describes the tails of certain probability distributions, particularly in the context of the normal distribution. It is used in statistical theory to approximate the cumulative distribution function (CDF) of a normal random variable for values far from the mean, specifically in the tails of the distribution.

Spherical harmonics are a set of mathematical functions that are defined on the surface of a sphere and are used in a variety of fields, including physics, engineering, computer graphics, and geophysics. They can be viewed as the multidimensional analogs of Fourier series and are particularly useful in solving problems that have spherical symmetry.

A table of spherical harmonics typically provides a set of orthogonal functions defined on the surface of a sphere, which are used in various fields such as physics, engineering, and computer graphics. Spherical harmonics depend on two parameters: the degree \( l \) and the order \( m \).

The term "Toronto function" does not refer to a well-known concept or standard term in mathematics, computer science, or any other widely recognized field up to my last knowledge update in October 2023. It is possible that it could refer to something specific within a niche context or a recent development that has emerged since then.

Zonal spherical harmonics are a specific class of spherical harmonics that depend only on the polar angle (colatitude) and are independent of the azimuthal angle (longitude). They are used in various applications such as geophysics, astronomy, and climate science, often to represent functions on the surface of a sphere.

Theta functions are a special class of functions that arise in various areas of mathematics, including complex analysis, number theory, and algebraic geometry. They are particularly significant in the study of elliptic functions and modular forms.

Jacobi forms are a class of functions that arise in the context of several areas in mathematics, including number theory, algebraic geometry, and the theory of modular forms. They are particular kinds of quasi-modular forms that exhibit specific transformation properties under the action of certain groups.

The metaplectic group is a significant concept in the fields of mathematics, particularly in representation theory and the theory of symplectic geometry. It is a double cover of the symplectic group, which means that it serves as a sort of "two-fold" representation of the symplectic group, capturing additional structure that cannot be represented by the symplectic group alone.

The Schottky problem, often referred to in the context of number theory and algebraic geometry, is named after the mathematician Friedrich Schottky. It addresses questions related to the representation of certain algebraic structures, particularly in connection with the theory of abelian varieties and modular forms. In more specific terms, the Schottky problem can be framed as follows: it concerns the characterization of Jacobians of algebraic curves.

In the context of mathematics and specifically in the field of number theory, the term "Theta characteristic" often refers to a certain type of characteristic of a Riemann surface or algebraic curve that arises in the study of Abelian functions, Jacobi varieties, and the theory of divisors. 1. **Theta Functions**: Theta characteristics are closely related to theta functions, which are special functions used in various areas of mathematics, including complex analysis and algebraic geometry.

In mathematics, particularly in the theory of abelian varieties and algebraic geometry, a *Theta divisor* is a specific kind of divisor associated with a principally polarized abelian variety (PPAV). More formally, if \( A \) is an abelian variety and \( \Theta \) is a quasi-projective variety corresponding to a certain polarization, then the theta divisor \( \theta \) is defined as the zero locus of a section of a line bundle on \( A \).

The theta function of a lattice is a special type of mathematical function that arises in the context of complex analysis, number theory, and mathematical physics. Specifically, it is related to the theory of elliptic functions, modular forms, and can be used in various applications including statistical mechanics and string theory. A lattice in this context is typically defined as a discrete subgroup of the complex plane generated by two linearly independent complex numbers \( \omega_1 \) and \( \omega_2 \).

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 Airy zeta function is a mathematical function that is related to the solutions of the Airy differential equation. The Airy functions, denoted as \( \text{Ai}(x) \) and \( \text{Bi}(x) \), are special functions that arise in various physical problems, particularly in quantum mechanics and wave phenomena, where they describe the behavior of a particle in a linear potential.

Apéry's theorem is a result in number theory that concerns the value of the Riemann zeta function at positive integer values. Specifically, the theorem states that the value \(\zeta(3)\), the Riemann zeta function evaluated at 3, is not a rational number. The theorem was proven by Roger Apéry in 1979 and is significant because it was one of the first results to demonstrate that certain values of the zeta function are irrational.

The Arakawa–Kaneko zeta function is a mathematical construct that arises in the study of dynamical systems, particularly in the context of the study of lattice models and statistical mechanics. Specifically, it is related to the treatment of certain integrable systems and is connected to concepts like partition functions and statistical weights. In general, the Arakawa–Kaneko zeta function is defined in the context of a two-dimensional lattice and is associated with a discrete set of variables.

The arithmetic zeta function, often associated with number theory, is a generalization of the Riemann zeta function, which traditionally sums over integers. The arithmetic zeta function, denoted by \( \zeta(s) \), is defined in various ways depending on the context, typically involving sums or products over prime numbers or algebraic structures. One prominent example of an arithmetic zeta function is the **Dedekind zeta function** associated with a number field.

The Artin L-function is a generalization of the classical Riemann zeta function and is an important object in number theory and arithmetic geometry, particularly in the context of class field theory and algebraic number theory. It is associated with a representations of a Galois group, collections of characters, and the study of L-functions in the context of number fields. ### Definition 1.

The Artin conductor is a concept from algebraic number theory, specifically in the study of Galois representations and local fields. It is a tool used to measure the ramification of a prime ideal in the extension of fields, particularly in the context of class field theory.

The Artin–Mazur zeta function is a function associated with a dynamical system, particularly in the context of number theory and arithmetic geometry. It is primarily used in the study of iterative processes and can also be applied to understand the behavior of various types of mathematical objects, such as algebraic varieties and their associated functions over finite fields.

In number theory and representation theory, an automorphic L-function is a type of complex analytic function that encodes significant arithmetic information about automorphic forms, which are certain types of functions defined on algebraic groups over global fields (like the rational numbers) that exhibit certain symmetries and transformation properties. ### Key Concepts: 1. **Automorphic Forms**: These are generalizations of modular forms, defined on the quotient of a group (often the general linear group) over a number field.

The Barnes zeta function is an extension of the classical Riemann zeta function and is defined in the context of number theory and special functions. It is primarily associated with the theory of multiple zeta values and has connections to various areas of mathematics, including algebra, topology, and mathematical physics. The Barnes zeta function, denoted as \( \zeta_B(s, a) \), depends on two parameters: \( s \) and \( a \).

The Basel problem is a famous problem in the field of mathematics, specifically in the study of series. It asks for the exact sum of the reciprocals of the squares of the natural numbers. Formally, it is expressed as: \[ \sum_{n=1}^{\infty} \frac{1}{n^2} \] The solution to the Basel problem was famously found by the Swiss mathematician Leonhard Euler in 1734.

The Beurling zeta function is a mathematical object related to number theory, specifically in the study of prime numbers. It is named after the Swedish mathematician Arne Magnus Beurling, who introduced it in the 1930s. The Beurling zeta function generalizes the classical Riemann zeta function and is used in the context of "pseudo-primes" or "generalized prime numbers.

The Birch and Swinnerton-Dyer (BSD) conjecture is a fundamental hypothesis in number theory that relates the number of rational points on an elliptic curve to the behavior of an associated L-function. Specifically, it concerns the properties of elliptic curves defined over the rational numbers \(\mathbb{Q}\).

The Brumer-Stark conjecture is a significant hypothesis in number theory that relates to the structure of abelian extensions of number fields and their class groups. It plays a crucial role in the study of L-functions and their special values, specifically in the context of p-adic L-functions and the behavior of class numbers. The conjecture can be understood in relation to certain aspects of class field theory.

The Chowla–Mordell theorem is a result in number theory related to the properties of rational numbers and algebraic equations. Specifically, it deals with the existence of rational points on certain types of algebraic curves.

The Clausen function, denoted as \( \text{Cl}_{2}(x) \), is a special function that is related to the integration of the sine function.

The Dedekind zeta function is an important invariant in algebraic number theory associated with a number field.