Definition:Second Kind

Definition

The term second kind is used to distinguish two or more classes of mathematical object of some common overall type.

Integral Equation of the Second Kind

An integral equation of the second kind is an integral equation of the form:

$\map g x = \map f x + \lambda \ds \int_{\map a x}^{\map b x} \map K {x, y} \map g y \rd x$

where:

$\map f x$ and $\map K {x, y}$ are known functions

$\map a x$ and $\map b x$ are known functions of $x$, or constant

$\map g x$ is an unknown function.

Elliptic Integral of the Second Kind

$\ds \map E k = \int \limits_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \rd \phi$

is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.

Fredholm Integral Equation of the Second Kind

A Fredholm integral equation of the second kind is an integral equation of the form:

$\ds \map g x = \map f x + \lambda \int_a^b \map K {x, y} \map g y \rd y$

Volterra Integral Equation of the Second Kind

A Volterra integral equation of the second kind is an integral equation of the form:

$\ds \map g x = \map f x + \lambda \int_a^x \map K {x, y} \map g y \rd y$

where $g$ is an unknown real function.

Sierpiński Number of the Second Kind

A Sierpiński number of the second kind is an odd positive integer $k$ such that integers of the form $k2^n + 1$ are composite for all positive integers $n$.

That is, when $k$ is a Sierpiński number of the second kind, all elements of the set:

$\set {k 2^n + 1}$

are composite.

Cusp of the Second Kind

A cusp of the second kind is a cusp in which both parts of the curve lie on the same side of the coincident tangents defining that cusp.

Double Cusp of the Second Kind

A double cusp of the second kind is a double cusp in which both parts of the curve lie on the same side of the coincident tangents defining that double cusp.

Bessel Function of the Second Kind

A Bessel function of the second kind of order $n$ is a Bessel function which is singular at the origin.

It is usually denoted $\map {Y_n} x$, where $x$ is the dependent variable of the instance of Bessel's equation to which $\map {Y_n} x$ forms a solution.

Cunningham Chain of the Second Kind

A Cunningham chain of the second kind is a (finite) sequence $\tuple {p_1, p_2, \ldots, p_n}$ such that:

$(1): \quad \forall i \in \set {1, 2, \ldots, n - 1}: p_{i + 1} = 2 p_i - 1$

$(2): \quad p_i$ is prime for all $i \in \set {1, 2, \ldots, n - 1}$

$(3): \quad n$ is not prime such that $2 n - 1 = p_1$

$(4): \quad 2 p_n - 1$ is not prime.

Hankel Function of the Second Kind

The Hankel function of the second kind is defined as:

$\map {H_n^{\paren 2} } z = \map {J_n} z - i \map {Y_n} z$

where:

$\map {J_n} z$ denotes the Bessel function of the first kind of order $n$

$\map {Y_n} z$ denotes the Bessel function of the second kind of order $n$.

Chebyshev Polynomials of the Second Kind

The Chebyshev polynomials of the second kind are defined as polynomials such that:

\(\ds \map {U_n} {\cos \theta} \sin \theta\)

\(=\)

\(\ds \map \sin {\paren {n + 1} \theta}\)

Also see

• Definition:First Kind
• Definition:Third Kind

• Results about second kind can be found here.