Definition of the Meijer G-function

A general definition of the Meijer G-function is given by the following line integral in the complex plane (Bateman & Erdélyi 1953, § 5.3-1):

${\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds,}$

 

 

 In mathematics, Spence's function, or dilogarithm, denoted as Li₂(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself::

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,\mathrm {d} u{\text{, }}z\in \mathbb {C} \setminus [1,\infty )}$

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,\mathrm {d} u{\text{, }}z\in \mathbb {C} \setminus [1,\infty )}$

Kummer's function

From Wikipedia, the free encyclopedia

In mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm. Both are named for Ernst Kummer.
Kummer's function is defined by

${\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.}$

${\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.}$

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are solutions of Laguerre's equation:

${\displaystyle xy''+(1-x)y'+ny=0}$

Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:^[9]

${\displaystyle f(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\ \ d\alpha f(\alpha )\ \int _{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ ,}$The super-logarithm, written ${\displaystyle \,\mathrm {slog} _{b}(z)}$, is defined implicitly by
${\displaystyle \,\mathrm {slog} _{b}(b^{z})=\mathrm {slog} _{b}(z)+1}$

For real non zero values of x, the exponential integral Ei(x) is defined as

${\displaystyle \operatorname {Ei} (x)=-\int _{-x}^{\infty }{\frac {e^{-t}}{t}}\,dt.\,}$The piecewise linear function

${\displaystyle x(t)=t-\lfloor t\rfloor =t-\operatorname {floor} (t)}$

or

${\displaystyle x(t)=x(mod1)}$For any real number x the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of the quantity) and is defined as^[6]

${\displaystyle |x|={\begin{cases}x,&{\mbox{if }}x\geq 0\\-x,&{\mbox{if }}x<0.\end{cases}}}$In more precise terms, an algebraic function of degree n in one variable x is a function ${\displaystyle y=f(x)}$ that satisfies a polynomial equation

${\displaystyle a_{n}(x)y^{n}+a_{n-1}(x)y^{n-1}+\cdots +a_{0}(x)=0}$

Definition by Ein

Both ${\displaystyle \mathrm {Ei} }$ and ${\displaystyle E_{1}}$ can be written more simply using the entire function ${\displaystyle \mathrm {Ein} }$^[10] defined as

${\displaystyle \mathrm {Ein} (z)=\int _{0}^{z}(1-e^{-t}){\frac {dt}{t}}=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}z^{k}}{k\;k!}}}$