Extension of the factorial function
Hadamard's gamma function plotted over part of the real axis. Unlike the classical gamma function, it is holomorphic; there are no poles. In mathematics , Hadamard's gamma function , named after Jacques Hadamard , is an extension of the factorial function , different from the classical gamma function (it is an instance of a pseudogamma function ). This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex numbers in a different way than Euler's gamma function. It is defined as:
H
(
x
)
=
1
Γ
(
1
−
x
)
d
d
x
{
ln
(
Γ
(
1
2
−
x
2
)
Γ
(
1
−
x
2
)
)
}
,
{\displaystyle H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma (1-{\frac {x}{2}})}}\right)\right\},}
where Γ(x ) denotes the classical gamma function. If n
H
(
n
)
=
Γ
(
n
)
=
(
n
−
1
)
!
{\displaystyle H(n)=\Gamma (n)=(n-1)!}
Properties
Unlike the classical gamma function, Hadamard's gamma function H (x )entire function , i.e. it has no poles in its domain. It satisfies the functional equation
H
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x
+
1
)
=
x
H
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x
)
+
1
Γ
(
1
−
x
)
,
{\displaystyle H(x+1)=xH(x)+{\frac {1}{\Gamma (1-x)}},}
with the understanding that
1
Γ
(
1
−
x
)
{\displaystyle {\tfrac {1}{\Gamma (1-x)}}}
0 for positive integer values of x .
Representations
Hadamard's gamma can also be expressed as
H
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x
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=
ψ
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1
−
x
2
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−
ψ
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1
2
−
x
2
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2
Γ
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1
−
x
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=
Φ
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−
1
,
1
,
−
x
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Γ
(
−
x
)
{\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\right)-\psi \left({\frac {1}{2}}-{\frac {x}{2}}\right)}{2\Gamma (1-x)}}={\frac {\Phi \left(-1,1,-x\right)}{\Gamma (-x)}}}
where
Φ
{\displaystyle \Phi }
Lerch zeta function , and as
H
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x
)
=
Γ
(
x
)
[
1
+
sin
(
π
x
)
2
π
{
ψ
(
x
2
)
−
ψ
(
x
+
1
2
)
}
]
,
{\displaystyle H(x)=\Gamma (x)\left[1+{\frac {\sin(\pi x)}{2\pi }}\left\{\psi \left({\dfrac {x}{2}}\right)-\psi \left({\dfrac {x+1}{2}}\right)\right\}\right],}
where ψ (x )digamma function .
See also
References
![{\displaystyle H(x)=\Gamma (x)\left[1+{\frac {\sin(\pi x)}{2\pi }}\left\{\psi \left({\dfrac {x}{2}}\right)-\psi \left({\dfrac {x+1}{2}}\right)\right\}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9881a5068931a0b8b16c27f16787824d4dadb1c)