Special hypergeometric functions

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.