Revision #1252 → #1833 · back to history
modifiedGamma function (Euler integral)e252e487f6e6
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| anchors | [{"section":"(Lead)","snippet":"The gamma function can be defined via a convergent improper integral for complex numbers with positive real part"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)=\\int _{0}^{\\infty }t^{z-1}e^{-t}\\,dt,\\ \\qquad \\Re (z)>0.}"}] | — |
modifiedReciprocal gamma is entire5b629d444339
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| anchors | [{"section":"(Lead)","snippet":"Since the gamma function has no zeros, its reciprocal"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)={\\mathcal {M}}\\{e^{-x}\\}(z)\\,.}"}] | — |
modifiedGamma as Mellin transform of exponential decay3f70087f036e
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| anchors | [{"section":"(Lead)","snippet":"the gamma function corresponds to the Mellin transform of the exponential decay"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)={\\mathcal {M}}\\{e^{-x}\\}(z)\\,.}"}] | — |
modifiedFunctional equation interpolating shifted factorial98f1121423ed
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| anchors | [{"section":"Motivation","snippet":"A more restrictive requirement is the functional equation that interpolates the shifted factorial"},{"type":"math_alttext","value":"{\\displaystyle f(x+1)=xf(x)\\ {\\text{ for all }}x>0,\\qquad f(1)=1.}"}] | — |
modifiedEuler integral of the second kinda05de230683f
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| anchors | [{"section":"Main definition","snippet":"converges absolutely , and is known as the Euler integral of the second kind"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)=\\int _{0}^{\\infty }t^{z-1}e^{-t}\\,dt}"}] | — |
modifiedRecurrence via integration by parts3d8a72f8223e
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| anchors | [{"section":"Main definition","snippet":"Integrating by parts , one sees that"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\Gamma (z+1)&=\\int _{0}^{\\infty }t^{z}e^{-t}\\,dt\\\\[6pt]&={\\Bigl [}-t^{z}e^{-t}{\\Bigr ]}_{0}^{\\infty }+\\int _{0}^{\\infty }z\\,t^{z-1}e^{-t}\\,dt.\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\Gamma (z+1)&=z\\int _{0}^{\\infty }t^{z-1}e^{-t}\\,dt\\\\[6pt]&=z\\,\\Gamma (z).\\end{aligned}}}"}] | — |
modifiedEuler's infinite product definitiond320eef18100
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| anchors | [{"section":"Euler's definition as an infinite product","snippet":"This infinite product , which is due to Euler"},{"type":"math_alttext","value":"{\\displaystyle \\lim _{n\\to \\infty }{\\frac {n!\\,\\left(n+1\\right)^{z}}{(n+z)!}}=1\\,.}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}(z-1)!&={\\frac {1}{z}}\\lim _{n\\to \\infty }n!{\\frac {z!}{(n+z)!}}(n+1)^{z}\\\\[6pt]&={\\frac {1}{z}}\\lim _{n\\to \\infty }(1\\cdot 2\\cdots n){\\frac {1}{(1+z)\\cdots (n+z)}}\\left({\\frac {2}{1}}\\cdot {\\frac {3}{2}}\\cdots {\\frac {n+1}{n}}\\right)^{z}\\\\[6pt]&={\\frac {1}{z}}\\prod _{n=1}^{\\infty }\\left[{\\frac {1}{1+{\\frac {z}{n}}}}\\left(1+{\\frac {1}{n}}\\right)^{z}\\right].\\end{aligned}}}"}] | — |
modifiedReciprocal as entire infinite productab2a9eb07db6
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| anchors | [{"section":"Euler's definition as an infinite product","snippet":"The infinite product for the reciprocal"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {1}{\\Gamma (z)}}=z\\prod _{n=1}^{\\infty }\\left[\\left(1+{\\frac {z}{n}}\\right)/{\\left(1+{\\frac {1}{n}}\\right)^{z}}\\right]}"}] | — |
modifiedWeierstrass's definition78c812afa6fa
| Field | From #1252 | To #1833 |
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| anchors | [{"section":"Weierstrass's definition","snippet":"The definition for the gamma function due to Weierstrass is also valid for all complex numbers"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)={\\frac {e^{-\\gamma z}}{z}}\\prod _{n=1}^{\\infty }\\left(1+{\\frac {z}{n}}\\right)^{-1}e^{z/n},}"}] | — |
modifiedEuler's reflection formula7c4828441121
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| anchors | [{"section":"General","snippet":"Other important functional equations for the gamma function are Euler's reflection formula"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z+1)=z\\ \\Gamma (z).}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (1-z)\\Gamma (z)={\\frac {\\pi }{\\sin \\pi z}},\\qquad z\\not \\in \\mathbb {Z} ,}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z-n)=(-1)^{n-1}\\;{\\frac {\\Gamma (-z)\\Gamma (1+z)}{\\Gamma (n+1-z)}},\\qquad n\\in \\mathbb {Z} }"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)\\Gamma \\left(z+{\\frac {1}{2}}\\right)=2^{1-2z}\\,{\\sqrt {\\pi }}\\,\\Gamma (2z).}"}] | — |
modifiedLegendre duplication formula90774bd7661a
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| anchors | [{"section":"General","snippet":"and the Legendre duplication formula"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z+1)=z\\ \\Gamma (z).}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (1-z)\\Gamma (z)={\\frac {\\pi }{\\sin \\pi z}},\\qquad z\\not \\in \\mathbb {Z} ,}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z-n)=(-1)^{n-1}\\;{\\frac {\\Gamma (-z)\\Gamma (1+z)}{\\Gamma (n+1-z)}},\\qquad n\\in \\mathbb {Z} }"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)\\Gamma \\left(z+{\\frac {1}{2}}\\right)=2^{1-2z}\\,{\\sqrt {\\pi }}\\,\\Gamma (2z).}"}] | — |
modifiedMultiplication theorem731ee7061aca
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| anchors | [{"section":"General","snippet":"The duplication formula is a special case of the multiplication theorem"},{"type":"math_alttext","value":"{\\displaystyle \\prod _{k=0}^{m-1}\\Gamma \\left(z+{\\frac {k}{m}}\\right)=(2\\pi )^{\\frac {m-1}{2}}\\;m^{{\\frac {1}{2}}-mz}\\;\\Gamma (mz).}"}] | — |
modifiedProperty from limit definitionbf31c197c8a8
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| anchors | [{"section":"General","snippet":"A simple but useful property, which can be seen from the limit definition, is"},{"type":"math_alttext","value":"{\\displaystyle {\\overline {\\Gamma (z)}}=\\Gamma ({\\overline {z}})\\;\\Rightarrow \\;\\Gamma (z)\\Gamma ({\\overline {z}})\\in \\mathbb {R} .}"}] | — |
modifiedClosed form at integer/half-integer real part16e4bf0a0c83
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| anchors | [{"section":"General","snippet":"If the real part is an integer or a half-integer , this can be finitely expressed in closed form"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}|\\Gamma (bi)|^{2}&={\\frac {\\pi }{b\\sinh \\pi b}}\\\\[6pt]\\left|\\Gamma \\left({\\tfrac {1}{2}}+bi\\right)\\right|^{2}&={\\frac {\\pi }{\\cosh \\pi b}}\\\\[6pt]\\left|\\Gamma \\left(1+bi\\right)\\right|^{2}&={\\frac {\\pi b}{\\sinh \\pi b}}\\\\[6pt]\\left|\\Gamma \\left(1+n+bi\\right)\\right|^{2}&={\\frac {\\pi b}{\\sinh \\pi b}}\\prod _{k=1}^{n}\\left(k^{2}+b^{2}\\right),\\quad n\\in \\mathbb {N} \\\\[6pt]\\left|\\Gamma \\left(-n+bi\\right)\\right|^{2}&={\\frac {\\pi }{b\\sinh \\pi b}}\\prod _{k=1}^{n}\\left(k^{2}+b^{2}\\right)^{-1},\\quad n\\in \\mathbb {N} \\\\[6pt]\\left|\\Gamma \\left({\\tfrac {1}{2}}\\pm n+bi\\right)\\right|^{2}&={\\frac {\\pi }{\\cosh \\pi b}}\\prod _{k=1}^{n}\\left(\\left(k-{\\tfrac {1}{2}}\\right)^{2}+b^{2}\\right)^{\\pm 1},\\quad n\\in \\mathbb {N} \\\\[-1ex]&\\end{aligned}}}"}] | — |
modifiedReflection applied to specific argumentf0b796a92b03
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| anchors | [{"section":"General","snippet":"First, consider the reflection formula applied to"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (bi)\\Gamma (1-bi)={\\frac {\\pi }{\\sin \\pi bi}}}"},{"type":"math_alttext","value":"{\\displaystyle -bi\\cdot \\Gamma (bi)\\Gamma (-bi)={\\frac {\\pi }{\\sin \\pi bi}}}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (bi)\\Gamma (-bi)={\\frac {\\pi }{-bi\\sin \\pi bi}}={\\frac {\\pi }{b\\sinh \\pi b}}}"}] | — |
modifiedValue of Gamma(1/2)5c7b12d633d5
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| anchors | [{"section":"General","snippet":"Perhaps the best-known value of the gamma function at a non-integer argument is"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma \\left({\\tfrac {1}{2}}\\right)={\\sqrt {\\pi }},}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\Gamma \\left({\\frac {1}{2}}+n\\right)&={(2n)! \\over 4^{n}n!}{\\sqrt {\\pi }}={\\frac {(2n-1)!!}{2^{n}}}{\\sqrt {\\pi }}={\\binom {n-{\\frac {1}{2}}}{n}}\\,n!\\,{\\sqrt {\\pi }}\\\\[6pt]\\Gamma \\left({\\frac {1}{2}}-n\\right)&={(-4)^{n}n! \\over (2n)!}{\\sqrt {\\pi }}={\\frac {(-2)^{n}}{(2n-1)!!}}{\\sqrt {\\pi }}={\\frac {\\sqrt {\\pi }}{{\\binom {-1/2}{n}}\\,n!}}\\end{aligned}}}"}] | — |
modifiedDerivatives via polygamma function8a67e54e4543
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| anchors | [{"section":"General","snippet":"The derivatives of the gamma function are described in terms of the polygamma function"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma '(z)=\\Gamma (z)\\psi ^{(0)}(z).}"}] | — |
modifiedDerivative at positive integersf5c7b9005386
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| anchors | [{"section":"General","snippet":"For a positive integer"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma '(z)=\\Gamma (z)\\psi ^{(0)}(z).}"}] | — |
modifiednth derivative formula41c182474e00
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| anchors | [{"section":"General","snippet":"th derivative of the gamma function is"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {d^{n}}{dz^{n}}}\\Gamma (z)=\\int _{0}^{\\infty }t^{z-1}e^{-t}(\\log t)^{n}\\,dt.}"}] | — |
modifiedLaurent series expansion633a88d931c4
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| anchors | [{"section":"General","snippet":"we have in particular the Laurent series expansion of the gamma function"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma ^{(n)}(1)=(-1)^{n}B_{n}(\\gamma ,1!\\zeta (2),\\ldots ,(n-1)!\\,\\zeta (n)),}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)={\\frac {1}{z}}-\\gamma +{\\frac {1}{2}}\\left(\\gamma ^{2}+{\\frac {\\pi ^{2}}{6}}\\right)z-{\\frac {1}{6}}\\left(\\gamma ^{3}+{\\frac {\\gamma \\pi ^{2}}{2}}+2\\zeta (3)\\right)z^{2}+O(z^{3}).}"}] | — |
modifiedJensen-type inequality from log-convexity22650e588f51
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| anchors | [{"section":"Inequalities","snippet":"Logarithmic convexity and Jensen's inequality together imply"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma \\left({\\frac {a_{1}x_{1}+\\cdots +a_{n}x_{n}}{a_{1}+\\cdots +a_{n}}}\\right)\\leq {\\bigl (}\\Gamma (x_{1})^{a_{1}}\\cdots \\Gamma (x_{n})^{a_{n}}{\\bigr )}^{\\frac {1}{a_{1}+\\cdots +a_{n}}}.}"}] | — |
modifiedGautschi's inequality09a656a50b8a
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| anchors | [{"section":"Inequalities","snippet":"The best-known is Gautschi's inequality"},{"type":"math_alttext","value":"{\\displaystyle x^{1-s}<{\\frac {\\Gamma (x+1)}{\\Gamma (x+s)}}<\\left(x+1\\right)^{1-s}.}"}] | — |
modifiedStirling's formula092576203921
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| anchors | [{"section":"Stirling's formula","snippet":"is given by Stirling's formula"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (x+1)\\sim {\\sqrt {2\\pi x}}\\left({\\frac {x}{e}}\\right)^{x},}"}] | — |
modifiedAsymptotic limit for Gammacdc87ff6191a
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| anchors | [{"section":"Stirling's formula","snippet":"Another useful limit for asymptotic approximations for"},{"type":"math_alttext","value":"{\\displaystyle {\\Gamma (x+\\alpha )}\\sim {\\Gamma (x)x^{\\alpha }},\\qquad \\alpha \\in \\mathbb {C} .}"}] | — |
modifiedStirling product definitiondc92c7d64fbc
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| anchors | [{"section":"Stirling's formula","snippet":"When writing the error term as an infinite product, Stirling's formula can be used to define the gamma function"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (x)={\\sqrt {\\frac {2\\pi }{x}}}\\left({\\frac {x}{e}}\\right)^{x}\\prod _{n=0}^{\\infty }\\left[{\\frac {1}{e}}\\left(1+{\\frac {1}{x+n}}\\right)^{x+n+{\\frac {1}{2}}}\\right].}"}] | — |
modifiedExtension to negative non-integer valuese356e449d81b
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| anchors | [{"section":"Extension to negative, non-integer values","snippet":"its domain can be extended with analytic continuation"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (-x)={\\frac {1}{\\Gamma (x+1)}}{\\frac {\\pi }{\\sin {\\big (}\\pi (x+1){\\big )}}},}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (-x):={\\frac {\\,1}{-x}}\\,\\Gamma (-x+1),}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma \\!\\left(\\!-{\\frac {1}{2}}\\right)=-2\\,\\Gamma \\!\\left({\\frac {1}{2}}\\right).}"}] | — |
modifiedMeromorphic with simple poles at non-positive integers1f907e843c56
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| anchors | [{"section":"Residues","snippet":"it is a meromorphic function with simple poles at the non-positive integers"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)={\\frac {\\Gamma (z+n+1)}{z(z+1)\\cdots (z+n)}},}"}] | — |
modifiedResidue at a simple pole69f2b3a916c4
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| anchors | [{"section":"Residues","snippet":"at a simple pole"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {Res} (f,c)=\\lim _{z\\to c}(z-c)f(z).}"}] | — |
modifiedResidues of the gamma function5da19777d508
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| anchors | [{"section":"Residues","snippet":"So the residues of the gamma function at those points are"},{"type":"math_alttext","value":"{\\displaystyle (z+n)\\Gamma (z)={\\frac {\\Gamma (z+n+1)}{z(z+1)\\cdots (z+n-1)}}.}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z+n+1)=\\Gamma (1)=1}"},{"type":"math_alttext","value":"{\\displaystyle z(z+1)\\cdots (z+n-1)=-n(1-n)\\cdots (n-1-n)=(-1)^{n}n!.}"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {Res} (\\Gamma ,-n)={\\frac {(-1)^{n}}{n!}}.}"}] | — |
modifiedReciprocal gamma is entire with specified zeros63e6b68f16ef
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| anchors | [{"section":"Residues","snippet":"and hence the reciprocal gamma function"},{"type":"math_alttext","value":"{\\displaystyle (z+n)\\Gamma (z)={\\frac {\\Gamma (z+n+1)}{z(z+1)\\cdots (z+n-1)}}.}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z+n+1)=\\Gamma (1)=1}"},{"type":"math_alttext","value":"{\\displaystyle z(z+1)\\cdots (z+n-1)=-n(1-n)\\cdots (n-1-n)=(-1)^{n}n!.}"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {Res} (\\Gamma ,-n)={\\frac {(-1)^{n}}{n!}}.}"}] | — |
modifiedAlternative integral representationsef53e2e08a89
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| anchors | [{"section":"Integral representations","snippet":"There are many formulas, besides the Euler integral of the second kind, that express the gamma function as an integral"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)=\\int _{-\\infty }^{\\infty }e^{zt-e^{t}}\\,dt}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)=\\int _{0}^{1}\\left(\\log {\\frac {1}{t}}\\right)^{z-1}\\,dt,}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)=2c^{z}\\int _{0}^{\\infty }t^{2z-1}e^{-ct^{2}}\\,dt\\,,\\;c>0}"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (1/2)=2\\int _{0}^{\\infty }e^{-t^{2}}\\,dt={\\sqrt {\\pi }}\\;.}"}] | — |
modifiedBinet's first integral formula9d9510b39d47
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| anchors | [{"section":"Integral representations","snippet":"Binet's first integral formula for the gamma function states that"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)=\\left(z-{\\frac {1}{2}}\\right)\\log z-z+{\\frac {1}{2}}\\log(2\\pi )+\\int _{0}^{\\infty }\\left({\\frac {1}{2}}-{\\frac {1}{t}}+{\\frac {1}{e^{t}-1}}\\right){\\frac {e^{-tz}}{t}}\\,dt.}"},{"type":"math_alttext","value":"{\\displaystyle \\log \\left(\\Gamma (z)\\left({\\frac {e}{z}}\\right)^{z}{\\sqrt {\\frac {z}{2\\pi }}}\\right)={\\mathcal {L}}\\left({\\frac {1}{2t}}-{\\frac {1}{t^{2}}}+{\\frac {1}{t(e^{t}-1)}}\\right)(z).}"}] | — |
modifiedBinet's second integral formula0eb82a860b4b
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| anchors | [{"section":"Integral representations","snippet":"Binet's second integral formula states that"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)=\\left(z-{\\frac {1}{2}}\\right)\\log z-z+{\\frac {1}{2}}\\log(2\\pi )+2\\int _{0}^{\\infty }{\\frac {\\arctan(t/z)}{e^{2\\pi t}-1}}\\,dt.}"}] | — |
modifiedHankel's formulaa93bc219d9d9
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| anchors | [{"section":"Integral representations","snippet":"then Hankel's formula for the gamma function is"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)=-{\\frac {1}{2i\\sin \\pi z}}\\int _{C}(-t)^{z-1}e^{-t}\\,dt,}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {1}{\\Gamma (z)}}={\\frac {i}{2\\pi }}\\int _{C}(-t)^{-z}e^{-t}\\,dt,}"}] | — |
modifiedContinued fraction representationc81bcc0166b6
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| anchors | [{"section":"Continued fraction representation","snippet":"The gamma function can also be represented by a sum of two continued fractions"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\Gamma (z)&={\\cfrac {e^{-1}}{2+0-z+1{\\cfrac {z-1}{2+2-z+2{\\cfrac {z-2}{2+4-z+3{\\cfrac {z-3}{2+6-z+4{\\cfrac {z-4}{2+8-z+5{\\cfrac {z-5}{2+10-z+\\ddots }}}}}}}}}}}}\\\\&+\\ {\\cfrac {e^{-1}}{z+0-{\\cfrac {z+0}{z+1+{\\cfrac {1}{z+2-{\\cfrac {z+1}{z+3+{\\cfrac {2}{z+4-{\\cfrac {z+2}{z+5+{\\cfrac {3}{z+6-\\ddots }}}}}}}}}}}}}}\\end{aligned}}}"}] | — |
modifiedKummer/Malmsten Fourier series for log Gammafd3ae36ca7fe
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| anchors | [{"section":"Fourier series expansion","snippet":"The logarithm of the gamma function has the following Fourier series expansion"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)=\\left({\\frac {1}{2}}-z\\right)(\\gamma +\\log 2)+(1-z)\\log \\pi -{\\frac {1}{2}}\\log \\sin(\\pi z)+{\\frac {1}{\\pi }}\\sum _{n=1}^{\\infty }{\\frac {\\log n}{n}}\\sin(2\\pi nz),}"}] | — |
modifiedRaabe's formula183c8f4316dd
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| anchors | [{"section":"Raabe's formula","snippet":"In 1840 Joseph Ludwig Raabe proved that"},{"type":"math_alttext","value":"{\\displaystyle \\int _{a}^{a+1}\\log \\Gamma (z)\\,dz={\\tfrac {1}{2}}\\log 2\\pi +a\\log a-a,\\quad a>0.}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{1}\\log \\Gamma (z)\\,dz={\\frac {1}{2}}\\log(2\\pi ).}"}] | — |
modifiedGauss's Pi function93d50ae360bf
| Field | From #1252 | To #1833 |
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| anchors | [{"section":"Pi function","snippet":"An alternative notation introduced by Gauss is the"},{"type":"math_alttext","value":"{\\displaystyle \\Pi (z)=\\Gamma (z+1)=z\\Gamma (z)=\\int _{0}^{\\infty }e^{-t}t^{z}\\,dt,}"}] | — |
modifiedReflection and multiplication via Pi85bea676e27e
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| anchors | [{"section":"Pi function","snippet":"Using the pi function, the reflection formula is"},{"type":"math_alttext","value":"{\\displaystyle \\Pi (z)\\Pi (-z)={\\frac {\\pi z}{\\sin(\\pi z)}}={\\frac {1}{\\operatorname {sinc} (z)}}}"},{"type":"math_alttext","value":"{\\displaystyle \\Pi \\left({\\frac {z}{m}}\\right)\\,\\Pi \\left({\\frac {z-1}{m}}\\right)\\cdots \\Pi \\left({\\frac {z-m+1}{m}}\\right)=(2\\pi )^{\\frac {m-1}{2}}m^{-z-{\\frac {1}{2}}}\\Pi (z)\\ .}"}] | — |
modifiedVolume of n-ellipsoid5aca1338283c
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| anchors | [{"section":"Pi function","snippet":"The volume of an"},{"type":"math_alttext","value":"{\\displaystyle V_{n}(r_{1},\\dotsc ,r_{n})={\\frac {\\pi ^{\\frac {n}{2}}}{\\Pi \\left({\\frac {n}{2}}\\right)}}\\prod _{k=1}^{n}r_{k}.}"}] | — |
modifiedUpper incomplete gamma function839911014f7a
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| anchors | [{"section":"Relation to other functions","snippet":"The upper incomplete gamma function is obtained by allowing the lower limit of integration to vary"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z,x)=\\int _{x}^{\\infty }t^{z-1}e^{-t}dt.}"}] | — |
modifiedBeta–Gamma relation03ce4f4a7bf5
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| anchors | [{"section":"Relation to other functions","snippet":"The gamma function is related to Euler's beta function by the formula"},{"type":"math_alttext","value":"{\\displaystyle \\mathrm {B} (z_{1},z_{2})=\\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\\,dt={\\frac {\\Gamma (z_{1})\\,\\Gamma (z_{2})}{\\Gamma (z_{1}+z_{2})}}.}"}] | — |
modifiedRelation to Riemann zeta6f1d5aa5126e
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| anchors | [{"section":"Relation to other functions","snippet":"The gamma function also shows up in an important relation with the Riemann zeta function"},{"type":"math_alttext","value":"{\\displaystyle \\pi ^{-{\\frac {z}{2}}}\\;\\Gamma \\left({\\frac {z}{2}}\\right)\\zeta (z)=\\pi ^{-{\\frac {1-z}{2}}}\\;\\Gamma \\left({\\frac {1-z}{2}}\\right)\\;\\zeta (1-z).}"},{"type":"math_alttext","value":"{\\displaystyle \\zeta (z)\\Gamma (z)=\\int _{0}^{\\infty }{\\frac {u^{z}}{e^{u}-1}}\\,{\\frac {du}{u}},}"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)=\\zeta _{H}'(0,z)-\\zeta '(0),}"}] | — |
modifiedLerch's formula for log Gamma4854401eba54
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| anchors | [{"section":"Relation to other functions","snippet":"The logarithm of the gamma function satisfies the following formula due to Lerch"},{"type":"math_alttext","value":"{\\displaystyle \\pi ^{-{\\frac {z}{2}}}\\;\\Gamma \\left({\\frac {z}{2}}\\right)\\zeta (z)=\\pi ^{-{\\frac {1-z}{2}}}\\;\\Gamma \\left({\\frac {1-z}{2}}\\right)\\;\\zeta (1-z).}"},{"type":"math_alttext","value":"{\\displaystyle \\zeta (z)\\Gamma (z)=\\int _{0}^{\\infty }{\\frac {u^{z}}{e^{u}-1}}\\,{\\frac {du}{u}},}"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)=\\zeta _{H}'(0,z)-\\zeta '(0),}"}] | — |
modifiedMoments of stretched exponentialf9c6a563bd08
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| anchors | [{"section":"Relation to other functions","snippet":"The gamma function is related to the stretched exponential function"},{"type":"math_alttext","value":"{\\displaystyle \\langle \\tau ^{n}\\rangle \\equiv \\int _{0}^{\\infty }t^{n-1}\\,e^{-\\left({\\frac {t}{\\tau }}\\right)^{\\beta }}\\,\\mathrm {d} t={\\frac {\\tau ^{n}}{\\beta }}\\Gamma \\left({n \\over \\beta }\\right).}"}] | — |
modifiedParticular values of the gamma functionb0f4772971f9
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| anchors | [{"section":"Particular values","snippet":"some particular values of the gamma function are"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{array}{rcccl}\\Gamma \\left(-{\\frac {3}{2}}\\right)&=&{\\frac {4{\\sqrt {\\pi }}}{3}}&\\approx &+2.36327\\,18012\\,07354\\,70306\\\\[6pt]\\Gamma \\left(-{\\frac {1}{2}}\\right)&=&-2{\\sqrt {\\pi }}&\\approx &-3.54490\\,77018\\,11032\\,05459\\\\[6pt]\\Gamma \\left({\\frac {1}{2}}\\right)&=&{\\sqrt {\\pi }}&\\approx &+1.77245\\,38509\\,05516\\,02729\\\\[6pt]\\Gamma (1)&=&0!&=&+1\\\\[6pt]\\Gamma \\left({\\frac {3}{2}}\\right)&=&{\\frac {\\sqrt {\\pi }}{2}}&\\approx &+0.88622\\,69254\\,52758\\,01364\\\\[6pt]\\Gamma (2)&=&1!&=&+1\\\\[6pt]\\Gamma \\left({\\frac {5}{2}}\\right)&=&{\\frac {3{\\sqrt {\\pi }}}{4}}&\\approx &+1.32934\\,03881\\,79137\\,02047\\\\[6pt]\\Gamma (3)&=&2!&=&+2\\\\[6pt]\\Gamma \\left({\\frac {7}{2}}\\right)&=&{\\tfrac {15{\\sqrt {\\pi }}}{8}}&\\approx &+3.32335\\,09704\\,47842\\,55118\\\\[6pt]\\Gamma (4)&=&3!&=&+6\\end{array}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {1}{\\Gamma (-3)}}={\\frac {1}{\\Gamma (-2)}}={\\frac {1}{\\Gamma (-1)}}={\\frac {1}{\\Gamma (0)}}=0.}"}] | — |
modifiedLog-gamma functionc5af602876f4
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| anchors | [{"section":"Log-gamma function","snippet":"many computing environments include a function that returns the natural logarithm of the gamma function"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)=-\\gamma z-\\log z+\\sum _{k=1}^{\\infty }\\left[{\\frac {z}{k}}-\\log \\left(1+{\\frac {z}{k}}\\right)\\right].}"}] | — |
modifiedFunctional equation for log-gamma318f2a19b71d
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| anchors | [{"section":"Log-gamma function","snippet":"the functional equation"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)=\\operatorname {log\\Gamma } (z+1)-\\log z}"}] | — |
modifiedRocktaeschel's approximation589cd52a5811
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| anchors | [{"section":"Log-gamma function","snippet":"Rocktaeschel (1922) proposed for"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z)\\approx (z-{\\tfrac {1}{2}})\\log z-z+{\\tfrac {1}{2}}\\log(2\\pi ).}"}] | — |
modifiedBöhmer's approximation833dbb078f05
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| anchors | [{"section":"Log-gamma function","snippet":"with a smaller"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (z-m)=\\operatorname {log\\Gamma } (z)-\\sum _{k=1}^{m}\\log(z-k).}"}] | — |
modifiedHermite's Stirling series96f033dccc3f
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| anchors | [{"section":"Log-gamma function","snippet":"The gamma function also has Stirling Series (derived by Charles Hermite in 1900)"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {log\\Gamma } (1+x)={\\frac {x(x-1)}{2!}}\\log(2)+{\\frac {x(x-1)(x-2)}{3!}}(\\log(3)-2\\log(2))+\\cdots ,\\quad \\Re (x)>0.}"}] | — |
modifiedIntegral of log-gamma via Barnes G9a9bd9ff6cd0
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| anchors | [{"section":"Integration over log-gamma","snippet":"can be expressed in terms of the Barnes"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{z}\\operatorname {log\\Gamma } (x)\\,dx}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{z}\\operatorname {log\\Gamma } (x)\\,dx={\\frac {z}{2}}\\log(2\\pi )+{\\frac {z(1-z)}{2}}+z\\operatorname {log\\Gamma } (z)-\\log G(z+1)}"}] | — |
modifiedIntegral of log-gamma via Hurwitz zetab17666e15ef4
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| anchors | [{"section":"Integration over log-gamma","snippet":"It can also be written in terms of the Hurwitz zeta function"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{z}\\operatorname {log\\Gamma } (x)\\,dx={\\frac {z}{2}}\\log(2\\pi )+{\\frac {z(1-z)}{2}}-\\zeta '(-1)+\\zeta '(-1,z).}"}] | — |
modifiedEspinosa–Moll integral of squared log-gammab3cbde9cdbf6
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| anchors | [{"section":"Integration over log-gamma","snippet":"Espinosa and Moll derived a similar formula for the integral of the square of"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{1}\\operatorname {log\\Gamma } (x)\\,dx={\\frac {1}{2}}\\log(2\\pi ),}"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{1}\\log ^{2}\\Gamma (x)dx={\\frac {\\gamma ^{2}}{12}}+{\\frac {\\pi ^{2}}{48}}+{\\frac {1}{3}}\\gamma L_{1}+{\\frac {4}{3}}L_{1}^{2}-\\left(\\gamma +2L_{1}\\right){\\frac {\\zeta ^{\\prime }(2)}{\\pi ^{2}}}+{\\frac {\\zeta ^{\\prime \\prime }(2)}{2\\pi ^{2}}},}"}] | — |
modifiedBailey et al. evaluation40f12818402b
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| anchors | [{"section":"Integration over log-gamma","snippet":"and his co-authors"},{"type":"math_alttext","value":"{\\displaystyle L_{n}:=\\int _{0}^{1}\\log ^{n}\\Gamma (x)\\,dx}"}] | — |
modifiedLanczos approximatione02ea8e85a9d
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| anchors | [{"section":"Approximations","snippet":"Complex values of the gamma function can be approximated using Stirling's approximation or the Lanczos approximation"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)\\sim {\\sqrt {2\\pi }}z^{z-1/2}e^{-z}\\quad {\\hbox{as }}z\\to \\infty {\\hbox{ in }}\\left|\\arg(z)\\right|<\\pi .}"}] | — |
modifiedFixed precision computation via integration by parts71751b56c6cf
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| anchors | [{"section":"Approximations","snippet":"The gamma function can be computed to fixed precision"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\Gamma (z)&=\\int _{0}^{x}e^{-t}t^{z}\\,{\\frac {dt}{t}}+\\int _{x}^{\\infty }e^{-t}t^{z}\\,{\\frac {dt}{t}}\\\\&=x^{z}e^{-x}\\sum _{n=0}^{\\infty }{\\frac {x^{n}}{z(z+1)\\cdots (z+n)}}+\\int _{x}^{\\infty }e^{-t}t^{z}\\,{\\frac {dt}{t}}.\\end{aligned}}}"}] | — |
modifiedPower-times-linear integral evaluation95de53de812c
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| anchors | [{"section":"Integration problems","snippet":"a simple change of variables"},{"type":"math_alttext","value":"{\\displaystyle \\int _{0}^{\\infty }t^{b}\\,e^{-at}\\,dt={\\frac {1}{a^{b}}}\\int _{0}^{\\infty }u^{b}\\,e^{-u}\\,d\\left({\\frac {u}{a}}\\right)={\\frac {\\Gamma (b+1)}{a^{b+1}}}.}"}] | — |
modifiedBinomial coefficient via gamma2f00bfa64fc4
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| anchors | [{"section":"Calculating products","snippet":"we can write"},{"type":"math_alttext","value":"{\\displaystyle (1+z)^{n}=\\sum _{k=0}^{\\infty }{\\frac {\\Gamma (n+1)}{k!\\Gamma (n-k+1)}}z^{k},}"},{"type":"math_alttext","value":"{\\displaystyle (1+z)^{n}=\\sum _{k=0}^{n}{\\frac {n!}{k!(n-k)!}}z^{k}=\\sum _{k=0}^{n}{\\binom {n}{k}}z^{k}.}"}] | — |
modifiedProducts of rational functions via gammab1c55aeb416a
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| anchors | [{"section":"Calculating products","snippet":"Generally, this works for any product wherein each factor is a rational function of the index variable"},{"type":"math_alttext","value":"{\\displaystyle \\prod _{i=a}^{b}{\\frac {P(i)}{Q(i)}}=\\left(\\prod _{j=1}^{m}{\\frac {\\Gamma (b-p_{j}+1)}{\\Gamma (a-p_{j})}}\\right)\\left(\\prod _{k=1}^{n}{\\frac {\\Gamma (a-q_{k})}{\\Gamma (b-q_{k}+1)}}\\right).}"}] | — |
modifiedRiemann zeta functional equationa352b15100b6
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|---|
| anchors | [{"section":"Analytic number theory","snippet":"A fundamental property of the Riemann zeta function is its functional equation"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma \\left({\\frac {s}{2}}\\right)\\,\\zeta (s)\\,\\pi ^{-{\\frac {s}{2}}}=\\Gamma \\left({\\frac {1-s}{2}}\\right)\\,\\zeta (1-s)\\,\\pi ^{-{\\frac {1-s}{2}}}.}"}] | — |
modifiedBernoulli's product representation3dc0c5b64d4e
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| anchors | [{"section":"18th century: Euler and Stirling","snippet":"Bernoulli introduced the product representation"},{"type":"math_alttext","value":"{\\displaystyle x!=\\lim _{n\\to \\infty }\\left(n+1+{\\frac {x}{2}}\\right)^{x-1}\\prod _{k=1}^{n}{\\frac {k+1}{k+x}},}"}] | — |
modifiedEuler's infinite product (historical)2fa399e7147c
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| anchors | [{"section":"18th century: Euler and Stirling","snippet":"the first was not his integral but an infinite product that is well defined for all complex numbers"},{"type":"math_alttext","value":"{\\displaystyle n!=\\prod _{k=1}^{\\infty }{\\frac {\\left(1+{\\frac {1}{k}}\\right)^{n}}{1+{\\frac {n}{k}}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle n!=\\int _{0}^{1}(-\\log s)^{n}\\,ds,}"}] | — |
modifiedEuler's integral representation (historical)be89dd38aa69
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| anchors | [{"section":"18th century: Euler and Stirling","snippet":"to announce his discovery of the integral representation"},{"type":"math_alttext","value":"{\\displaystyle n!=\\prod _{k=1}^{\\infty }{\\frac {\\left(1+{\\frac {1}{k}}\\right)^{n}}{1+{\\frac {n}{k}}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle n!=\\int _{0}^{1}(-\\log s)^{n}\\,ds,}"}] | — |
modifiedGauss's rewriting of Euler's productf76d563d48c0
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| anchors | [{"section":"19th century: Gauss, Weierstrass, and Legendre","snippet":"Carl Friedrich Gauss rewrote Euler's product as"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)=\\lim _{m\\to \\infty }{\\frac {m^{z}m!}{z(z+1)(z+2)\\cdots (z+m)}}}"}] | — |
modifiedWeierstrass's product representation23c94273d347
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|---|
| anchors | [{"section":"19th century: Gauss, Weierstrass, and Legendre","snippet":"starting from yet another product representation"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)={\\frac {e^{-\\gamma z}}{z}}\\prod _{k=1}^{\\infty }\\left(1+{\\frac {z}{k}}\\right)^{-1}e^{\\frac {z}{k}},}"}] | — |
modifiedWeierstrass factorization theoremcfbbe8fdbf05
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|---|
| anchors | [{"section":"19th century: Gauss, Weierstrass, and Legendre","snippet":"he proved what is known as the Weierstrass factorization theorem"},{"type":"math_alttext","value":"{\\displaystyle \\Gamma (z)={\\frac {e^{-\\gamma z}}{z}}\\prod _{k=1}^{\\infty }\\left(1+{\\frac {z}{k}}\\right)^{-1}e^{\\frac {z}{k}},}"}] | — |