Definition:Third Kind

Definition

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

Integral Equation of the Third Kind

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

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

where:

$\map u x$, $\map f x$ 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 Third Kind

$\ds \map \Pi {k, n} = \int \limits_0^{\pi / 2} \frac {\d \phi} {\paren {1 + n \sin^2 \phi} \sqrt {1 - k^2 \sin^2 \phi} }$

is the complete elliptic integral of the third kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$

$n \in \Z$

Bessel Function of the Third Kind

Another term for Hankel function:

A Hankel function is the sum of Bessel functions in either of the following two ways:

Hankel Function of the First Kind

The Hankel function of the first kind is defined as:

$\map {H_n^{\paren 1} } z = \map {J_n} z + i \map {Y_n} z$

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 $n$.

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 $n$.

Also see

• Definition:First Kind
• Definition:Second Kind

• Results about third kind can be found here.