OpenMath Content Dictionary: gammaFnsPlus

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gammaFnsPlus.ocd
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gammaFnsPlus.omcd
Defines:
Beta, Digamma, Gamma, IncompleteBeta, IncompleteBetaQuotient, IncompleteGamma, IncompleteGammaComplement, IncompleteGammaQuotient, IncompleteGammaStar, PolyGamma
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This content dictionary contains symbols which describe the Gamma function and related functions.


Gamma

The symbol Gamma represents the Euler Gamma function, it is denoted by $\Gamma(z)$ and is defined as $\Gamma(z)=\int_{0}^{\infty} t^{z-1}e^{-t} dt$. $Gamma(z)$ is a single valued and analytic over the entire complex plane. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.1.1]

Commented Mathematical property (CMP):
\Gamma : real $\rightarrow$ real
Commented Mathematical property (CMP):
\Gamma : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
\Gamma : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\Gamma(n)=1 \cdot 2 \cdot 3 \dots (n-1)n=n!$, integer value
Commented Mathematical property (CMP):
$\Gamma(z+1)=z \Gamma(z)=z!$, recurrence relation
Commented Mathematical property (CMP):
$\Gamma(n+z)=(n-1+z)(n-2+z)\dots(1+z)\Gamma(1+z)$, recurrence relation
Commented Mathematical property (CMP):
$\Gamma(z) \Gamma(1-z)=-z \Gamma(-z) \Gamma(z)$, reflection relation
Commented Mathematical property (CMP):
$\Gamma(\frac{1}{2})=(-\frac{1}{2})!$, fractional value
Commented Mathematical property (CMP):
$\Gamma(\frac{3}{2})=(\frac{1}{2})!$, fractional value
Signatures:
sts


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Digamma

The symbol Digamma represents the Digamma function, it is denoted by $\psi(z)$ and is defined as $\psi(z)=\frac{d[ln\Gamma(z)]}{dz}=\frac{\Gamma(z)^{\prime}}{\Gamma(z)}$. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.3.1]

Commented Mathematical property (CMP):
\psi : real $\rightarrow$ real
Commented Mathematical property (CMP):
\psi : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
\psi : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\psi(1)=-\gamma$, integer value
Commented Mathematical property (CMP):
$\psi(n)=-\gamma+\sum_{k=1}^{n-1}k^{-1}$, $n \geq 2$, integer value
Commented Mathematical property (CMP):
$\psi(z+1)=\psi(z)+\frac{1}{z}$, recurrence relation
Commented Mathematical property (CMP):
$\psi(1-z)=\psi(z)+\pi cot\pi z$, reflection relation
Signatures:
sts


[Next: PolyGamma] [Previous: Gamma] [Top]

PolyGamma

The symbol PolyGamma represents the Polygamma function, it is denoted by $\psi^{(n)}(z)$ and is defined as $\psi^{(n)}(z) =\frac{d^n}{dz^n}\psi(z)=\frac{d^{n+1}}{dz^{n+1}}ln\Gamma(z) =(-1)^{n+1}\int_{0}^{\infty}\frac{t^ne^{-zt}}{1-e^{-t}} dt$ where $n \ge 0$. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.4.1]

Commented Mathematical property (CMP):
\psi : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
\psi : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
\psi : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\psi^{(n)}(1)=(-1)^{n+1}n! \zeta (n+1)$, $n>0$, integer value
Commented Mathematical property (CMP):
$\psi^{(n)}(\frac{1}{2})=(-1)^{n+1}n!(2^{n+1}-1) \zeta(n+1)$, $n=1, 2, \dots$, fractional value
Commented Mathematical property (CMP):
$\psi^{(n)}(z+1)=\psi^{(n)}(z)+(-1)^nn!z^{-n-1}$, recurrence relation
Commented Mathematical property (CMP):
$\psi^{(n)}(z)=(-1)^{n+1}n!\sum_{k=0}^{\infty}(z+k)^{-n-1}$ , $z \neq 0. -1. -2. \dots$, series expansion
Signatures:
sts


[Next: IncompleteGammaQuotient] [Previous: Digamma] [Top]

IncompleteGammaQuotient

The symbol IncompleteGammaQuotient represents the Incomplete Gamma Quotient function, it is denoted by $P(a,x)$ and is defined by, the integral $P(a,x)=\frac{1}{\Gama (a)}\int_{0}^{x} e^{-t}t^{a-1}dt$. It has a branch cut along the negative real axis in the complex $a$ plane. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.5.1]

Commented Mathematical property (CMP):
P : (real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
P : (complex, real) $\rightarrow$ complex
Commented Mathematical property (CMP):
P : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$P(n,x)=1-e^{-x} \sum_{j=0}^{n-1} \frac{x^j}{j!}$ Special values
Commented Mathematical property (CMP):
$P(a+1,x)=P(a,x)-\frac{x^ae^{-x}}{\Gamma(a+1)}$ Recurrence Formulas
Signatures:
sts


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IncompleteGamma

The symbol IncompleteGamma represents the Incomplete Gamma function, it is denoted by $\gamma(a,x)$ and is defined by the integral $\gamma(a,x)=\int_{0}^{x} e^{-t}t^{a-1}dt$. It has a branch cut along the negative real axis in the complex $a$ plane. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.5.2]

Commented Mathematical property (CMP):
\gamma : (real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
\gamma : (complex, real) $\rightarrow$ complex
Commented Mathematical property (CMP):
\gamma : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\gamma(\frac12,x^2)=2\int_0^xe^{-t^2}dt=\sqrt\pi erf(x)$ Special Values
Commented Mathematical property (CMP):
$\gamma(a+1,x)=a\gamma(a,x)-x^ae^{-x}$ Reccurence Formula
Commented Mathematical property (CMP):
$\gamma(a,x+y)-\gamma(a,x) = e^{-x}x^{a-1}\sum_{n=0}^\infty \frac{(a-1)(a-2)\cdots(a-n)}{x^n}\left [1-e^{-y}e_n(y)\right ] (|y|< |x|)$ Series Development
Signatures:
sts


[Next: IncompleteGammaComplement] [Previous: IncompleteGammaQuotient] [Top]

IncompleteGammaComplement

The symbol IncompleteGammaComplement represents the Incomplete Gamma Complement function, it is denoted by $\Gamma(a,x)$ and is defined by the integral $\Gamma(a,x)=\int_{x}^{\infty} e^{-t}t^{a-1}dt$. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.5.3]

Commented Mathematical property (CMP):
\Gamma : (real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
\Gamma : (complex, real) $\rightarrow$ complex
Commented Mathematical property (CMP):
\Gamma : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\Gamma(0,x)=\int_x^\infty e^{-t}t^{-1} dt$ Special values
Commented Mathematical property (CMP):
$\Gamma(a,x)=e^{-x}x^a\left (\frac{1}{x+}~\frac{1-a}{1+}~\frac{1}{x+} ~\frac{2-a}{1+}~ \frac{2}{x+} \cdots\right )~~(x>0,|a|\lt\infty)$ Continued Fraction
Signatures:
sts


[Next: IncompleteGammaStar] [Previous: IncompleteGamma] [Top]

IncompleteGammaStar

The symbol IncompleteGammaStar represents the Incomplete Gamma star function, it is denoted by $\gamma^*(a,x)$ and is defined by $\gamma^*(a,x)=\frac{x^{-a}}{\Gamma(a)}\gamma(a,x)$. $\gamma^*$ is a single valued analytic function of $a$ and $x$ possessing no finite singularities. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.5.4] and [6.5.29]

Commented Mathematical property (CMP):
\gamma^* : (real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
\gamma^* : (complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
\gamma^* : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$\gamma^*(-n,x)=x^n$ Special values
Commented Mathematical property (CMP):
$\gamma^*(a,z)=e^{-z}\sum_{n=0}^\infty\frac{z^n}{\Gamma(a+n+1)}= \frac{1}{\Gamma(a)}\sum_{n=0}^\infty \frac{{(-z)}^n}{(a+n)n!}$ Series Development
Signatures:
sts


[Next: Beta] [Previous: IncompleteGammaComplement] [Top]

Beta

The symbol Beta represents the Euler Beta function, it is denoted by $B(z,w)$ and is defined as $B(z,w)=\int_{0}^{1} t^{z-1} (1-t)^{w-1} dt$. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.2.1]

Commented Mathematical property (CMP):
B : (complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
B : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$B(z, w)=\frac{\Gamma(z)(w)}{\Gamma(z+w)}=B(w, z)$
Signatures:
sts


[Next: IncompleteBeta] [Previous: IncompleteGammaStar] [Top]

IncompleteBeta

The symbol IncompleteBeta represents the Incomplete Euler Beta function, it is denoted by $B_{x}(a,b)$ and is defined as, $B_{x}(a,b)=\int_{0}^{x} t^{a-1} (1-t)^{b-1} dt$. It has a branch cut along the negative real axis in the complex $z$ plane. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.6.1]

Commented Mathematical property (CMP):
B : (real, real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
B : (real, complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
B : (symbolic, symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$I_{x}(a, b)=B_{x}(a, b)/B(a, b)$
Commented Mathematical property (CMP):
$B_{z}(a, b)=a^{-1}z^aF(a, 1-b;a+1;z)$, relation to hypergeometric function
Signatures:
sts


[Next: IncompleteBetaQuotient] [Previous: Beta] [Top]

IncompleteBetaQuotient

The symbol IncompleteBetaQuotient represents the Incomplete Beta Quotient function, it is denoted by $I_{x}(a,b)$ and is defined as, $I_{x}(a,b)=B_{x}(a, b)/B(a, b)$. It has a branch cut along the negative real axis in the complex $a,b$ plane. It is defined as in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.6.2]

Commented Mathematical property (CMP):
I : (real, real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
I : (real, complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
I : (symbolic, symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$I_{x}(a, b)=B_{x}(a, b)/B(a, b)$
Commented Mathematical property (CMP):
$I_{x}(a, b)=1-I_{1-x}(b, a)$, symmetry
Signatures:
sts


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