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Title: Polygamma Function -- from Wolfram MathWorld

Open Graph Title: Polygamma Function -- from Wolfram MathWorld

X Title: Polygamma Function -- from Wolfram MathWorld

Description: A special function mostly commonly denoted psi_n(z), psi^((n))(z), or F_n(z-1) which is given by the (n+1)st derivative of the logarithm of the gamma function Gamma(z) (or, depending on the definition, of the factorial z!). This is equivalent to the nth normal derivative of the logarithmic derivative of Gamma(z) (or z!) and, in the former case, to the nth normal derivative of the digamma function psi_0(z). Because of this ambiguity in definition, two different notations are sometimes (but...

Open Graph Description: A special function mostly commonly denoted psi_n(z), psi^((n))(z), or F_n(z-1) which is given by the (n+1)st derivative of the logarithm of the gamma function Gamma(z) (or, depending on the definition, of the factorial z!). This is equivalent to the nth normal derivative of the logarithmic derivative of Gamma(z) (or z!) and, in the former case, to the nth normal derivative of the digamma function psi_0(z). Because of this ambiguity in definition, two different notations are sometimes (but...

X Description: A special function mostly commonly denoted psi_n(z), psi^((n))(z), or F_n(z-1) which is given by the (n+1)st derivative of the logarithm of the gamma function Gamma(z) (or, depending on the definition, of the factorial z!). This is equivalent to the nth normal derivative of the logarithmic derivative of Gamma(z) (or z!) and, in the former case, to the nth normal derivative of the digamma function psi_0(z). Because of this ambiguity in definition, two different notations are sometimes (but...

Opengraph URL: https://mathworld.wolfram.com/PolygammaFunction.html

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Domain: mathworld.wolfram.com

DC.TitlePolygamma Function
DC.CreatorWeisstein, Eric W.
DC.DescriptionA special function mostly commonly denoted psi_n(z), psi^((n))(z), or F_n(z-1) which is given by the (n+1)st derivative of the logarithm of the gamma function Gamma(z) (or, depending on the definition, of the factorial z!). This is equivalent to the nth normal derivative of the logarithmic derivative of Gamma(z) (or z!) and, in the former case, to the nth normal derivative of the digamma function psi_0(z). Because of this ambiguity in definition, two different notations are sometimes (but...
DC.Date.Modified2007-12-11
DC.Subject33B15
DC.RightsCopyright 1999-2026 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement.
DC.Formattext/html
DC.Identifierhttps://mathworld.wolfram.com/PolygammaFunction.html
DC.Languageen
DC.PublisherWolfram Research, Inc.
DC.Relation.IsPartOfhttps://mathworld.wolfram.com/
DC.TypeText
Last-Modified2007-12-11
og:imagehttps://mathworld.wolfram.com/images/socialmedia/share/ogimage_PolygammaFunction.png
og:typewebsite
twitter:cardsummary_large_image
twitter:image:srchttps://mathworld.wolfram.com/images/socialmedia/share/ogimage_PolygammaFunction.png
Noneie=edge

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derivativehttps://mathworld.wolfram.com/Derivative.html
logarithmhttps://mathworld.wolfram.com/Logarithm.html
gamma functionhttps://mathworld.wolfram.com/GammaFunction.html
factorialhttps://mathworld.wolfram.com/Factorial.html
logarithmic derivativehttps://mathworld.wolfram.com/LogarithmicDerivative.html
digamma functionhttps://mathworld.wolfram.com/DigammaFunction.html
Hurwitz zeta functionhttps://mathworld.wolfram.com/HurwitzZetaFunction.html
digamma functionhttps://mathworld.wolfram.com/DigammaFunction.html
trigamma functionhttps://mathworld.wolfram.com/TrigammaFunction.html
Wolfram Languagehttp://www.wolfram.com/language/
PolyGammahttp://reference.wolfram.com/language/ref/PolyGamma.html
PolyGammahttp://reference.wolfram.com/language/ref/PolyGamma.html
recurrence relationhttps://mathworld.wolfram.com/RecurrenceRelation.html
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harmonic numbershttps://mathworld.wolfram.com/HarmonicNumber.html
Hurwitz zeta functionhttps://mathworld.wolfram.com/HurwitzZetaFunction.html
Euler-Mascheroni constanthttps://mathworld.wolfram.com/Euler-MascheroniConstant.html
digamma functionhttps://mathworld.wolfram.com/DigammaFunction.html
digamma functionhttps://mathworld.wolfram.com/DigammaFunction.html
trigamma functionhttps://mathworld.wolfram.com/TrigammaFunction.html
Clausen functionshttps://mathworld.wolfram.com/ClausenFunction.html
rationalhttps://mathworld.wolfram.com/RationalNumber.html
Catalan's constanthttps://mathworld.wolfram.com/CatalansConstant.html
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Euler-Mascheroni Integralshttps://mathworld.wolfram.com/Euler-MascheroniIntegrals.html
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Gauss's Digamma Theoremhttps://mathworld.wolfram.com/GausssDigammaTheorem.html
Harmonic Numberhttps://mathworld.wolfram.com/HarmonicNumber.html
Periodic Zeta Functionhttps://mathworld.wolfram.com/PeriodicZetaFunction.html
q-Polygamma Functionhttps://mathworld.wolfram.com/q-PolygammaFunction.html
Riemann Zeta Functionhttps://mathworld.wolfram.com/RiemannZetaFunction.html
Stirling's Serieshttps://mathworld.wolfram.com/StirlingsSeries.html
Trigamma Functionhttps://mathworld.wolfram.com/TrigammaFunction.html
https://functions.wolfram.com/GammaBetaErf/PolyGamma2/https://functions.wolfram.com/GammaBetaErf/PolyGamma2/
area between y=x^3-10x^2+16x and y=-x^3+10x^2-16xhttps://www.wolframalpha.com/input/?i=area+between+y%3Dx%5E3-10x%5E2%2B16x+and+y%3D-x%5E3%2B10x%5E2-16x
domain and range of z = x^2 + y^2https://www.wolframalpha.com/input/?i=domain+and+range+of+z+%3D+x%5E2+%2B+y%5E2
grad sin(x^2 y)http://www.wolframalpha.com/input/?i=grad+sin%28x%5E2+y%29
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.http://www.amazon.com/exec/obidos/ASIN/0486612724/ref=nosim/ericstreasuretro
Mathematical Methods for Physicists, 3rd ed.http://www.amazon.com/exec/obidos/ASIN/0120598760/ref=nosim/ericstreasuretro
Ramanujan's Notebooks: Part I.http://www.amazon.com/exec/obidos/ASIN/0387961100/ref=nosim/ericstreasuretro
A Table of Series and Products.http://www.amazon.com/exec/obidos/ASIN/0138819386/ref=nosim/ericstreasuretro
Methods of Theoretical Physics, Part I.http://www.amazon.com/exec/obidos/ASIN/007043316X/ref=nosim/ericstreasuretro
Polygamma Functionhttps://www.wolframalpha.com/input/?i=polygamma+function
Weisstein, Eric W.https://mathworld.wolfram.com/about/author.html
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