Definition:First Kind
Definition
The term first kind is used to distinguish two or more classes of mathematical object of some common overall type.
Integral Equation of the First Kind
An integral equation of the first kind is an integral equation of the form:
- $\map f x = \lambda \ds \int_{\map a x}^{\map b x} \map K {x, y} \map g y \rd x$
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 First Kind
- $\ds \map K k = \int \limits_0^{\pi / 2} \frac {\d \phi} {\sqrt {1 - k^2 \sin^2 \phi} }$
is the complete elliptic integral of the first kind, and is a function of $k$, defined on the interval $0 < k < 1$.
Fredholm Integral Equation of the First Kind
A Fredholm integral equation of the first kind is an integral equation of the form:
- $\ds \map f x = \lambda \int_a^b \map K {x, y} \map g y \rd y$
where $g$ is an unknown real function.
Volterra Integral Equation of the First Kind
A Volterra integral equation of the first kind is an integral equation of the form:
- $\ds \map f x = \lambda \int_a^x \map K {x, y} \map g y \rd y$
where $g$ is an unknown real function.
Sierpiński Number of the First Kind
The Sierpiński numbers of the first kind are the integers $S_n$ in the form:
- $S_n := n^n + 1$
for all integers $n$.
Single Cusp of the First Kind
A single cusp of the first kind is a single cusp in which both parts of the curve lie on opposite sides of the coincident tangents defining that single cusp.
Double Cusp of the First Kind
A double cusp of the first kind is a double cusp in which both parts of the curve lie on opposite sides of the coincident tangents defining that double cusp.
Bessel Function of the First Kind
A Bessel function of the first kind of order $n$ is a Bessel function which is non-singular at the origin.
It is usually denoted $\map {J_n} x$, where $x$ is the dependent variable of the instance of Bessel's equation to which $\map {J_n} x$ forms a solution.
Cunningham Chain of the First Kind
A Cunningham chain of the first 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.
Thus:
- each term except the last is a Sophie Germain prime
- each term except the first is a safe prime.
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 $n$.
Chebyshev Polynomial of the First Kind
The Chebyshev polynomials of the first kind are defined as polynomials such that:
| \(\ds \map {T_n} x\) | \(=\) | \(\ds \map \cos {n \arccos x}\) |
where $n \in \N$.
Airy Function of the First Kind
An Airy function of the first kind is an Airy function which is of the form:
- $\ds \map {\Ai} x = \dfrac 1 \pi \int_0^\infty \map \cos {\dfrac {t^3} 3 + x t} \rd t$
Discontinuity of the First Kind
$c$ is known as a discontinuity of the first kind of $f$ if and only if:
- $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ exist
where $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ denote the limit from the left and limit from the right at $c$ respectively.
Jacobi Theta Function of the First Type
The Jacobi Theta function of the first type is defined as:
- $\forall z \in \C: \ds \map {\vartheta_1} {z, q} = 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n q^{\paren {n + \frac 1 2}^2} \sin \paren {2 n + 1} z$
where:
- $q = e^{i \pi \tau}$
- $\tau \in \C$ such that $\map \Im \tau > 0$
Associated Legendre Function of the First Kind
Let $m, n \in \Z_{\ge 0}$ be non-negative integers.
The associated Legendre functions of the first kind are the real functions defined and denoted as:
- $\map { {P_n}^m} x = \paren {1 - x^2}^{m / 2} \dfrac {\d^m} {\d x^m} \map {P_n} x$
where $\map {P_n} x$ is the Legendre polynomial of order $n$.
Also known as
Some such objects are referred to as being of the first type rather than the first kind, but the meaning remains the same.
Also see
- Results about first kind can be found here.
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): first kind