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	In [[mathematics]], the '''digamma function''' is defined as the [[logarithmic derivative]] of the [[gamma function]]:<ref name="AbramowitzStegun"/><ref name="DLMF5"/><ref name="Weissstein"/>		In [[mathematics]], the '''digamma function''' is defined as the [[logarithmic derivative]] of the [[gamma function]]:<ref name="AbramowitzStegun"/><ref name="DLMF5"/><ref name="Weissstein"/>

	:<math>\psi(z) = \frac~~{\mathrm~~{d}}{~~\mathrm{d}z~~}\ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}.</math>		:<math>\psi(z) = \frac{d}{dz}\ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}.</math>

	It is the first of the [[polygamma function]]s. This function is [[Monotonic function\|strictly increasing]] and [[Concave function\|strictly concave]] on <math>(0,\infty)</math>,<ref>{{Cite journal \|last1=Alzer \|first1=Horst \|last2=Jameson \|first2=Graham \|date=2017 \|title=A harmonic mean inequality for the digamma function and related results \|url=https://core.ac.uk/download/pdf/228202664.pdf \|journal=Rendiconti del Seminario Matematico della Università di Padova \|volume=137 \|pages=203–209\|doi=10.4171/RSMUP/137-10 }}</ref> and it [[Asymptotic analysis\|asymptotically behaves]] as<ref>{{cite web \|url=https://dlmf.nist.gov/5.11 \|title=NIST. Digital Library of Mathematical Functions (DLMF), 5.11.}}</ref>		It is the first of the [[polygamma function]]s. This function is [[Monotonic function\|strictly increasing]] and [[Concave function\|strictly concave]] on <math>(0,\infty)</math>,<ref>{{Cite journal \|last1=Alzer \|first1=Horst \|last2=Jameson \|first2=Graham \|date=2017 \|title=A harmonic mean inequality for the digamma function and related results \|url=https://core.ac.uk/download/pdf/228202664.pdf \|journal=Rendiconti del Seminario Matematico della Università di Padova \|volume=137 \|pages=203–209\|doi=10.4171/RSMUP/137-10 }}</ref> and it [[Asymptotic analysis\|asymptotically behaves]] as<ref>{{cite web \|url=https://dlmf.nist.gov/5.11 \|title=NIST. Digital Library of Mathematical Functions (DLMF), 5.11.}}</ref>
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	The digamma function is often denoted as <math>\psi_0(x), \psi^{(0)}(x) </math> or {{math\|Ϝ}}<ref>{{cite book \|last=Pairman \|first=Eleanor \|author-link=Eleanor Pairman \|date=1919 \|title=Tables of the Digamma and Trigamma Functions \|url=https://archive.org/details/cu31924001468416/page/n9/mode/2up \|publisher=Cambridge University Press \|page=5}}</ref> (the uppercase form of the archaic Greek [[consonant]] [[digamma]] meaning [[Gamma\|double-gamma]]).		The digamma function is often denoted as <math>\psi_0(x), \psi^{(0)}(x) </math> or {{math\|Ϝ}}<ref>{{cite book \|last=Pairman \|first=Eleanor \|author-link=Eleanor Pairman \|date=1919 \|title=Tables of the Digamma and Trigamma Functions \|url=https://archive.org/details/cu31924001468416/page/n9/mode/2up \|publisher=Cambridge University Press \|page=5}}</ref> (the uppercase form of the archaic Greek [[consonant]] [[digamma]] meaning [[Gamma\|double-gamma]]).
	~~Gamma.~~

	==Relation to harmonic numbers==		==Relation to harmonic numbers==
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	==Infinite product representation==		==Infinite product representation==
	The function <math>\psi(z)/\Gamma(z)</math> is an entire function,<ref name=MezoHoffman/> and it can be represented by the infinite product		The function <math>\psi(z)/\Gamma(z)</math> is an [[entire function]],<ref name=MezoHoffman/> and it can be represented by the infinite product

	:<math>		:<math>
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	:<math>\psi(1-x)-\psi(x)=\pi \cot \pi x</math>.		:<math>\psi(1-x)-\psi(x)=\pi \cot \pi x</math>.
	:<math>\psi'(-x)+\psi'(x) = \frac{\pi^2}{\sin^2(\pi x)}+\frac{1}{x^2}</math>.		:<math>\psi'(-x)+\psi'(x) = \frac{\pi^2}{\sin^2(\pi x)}+\frac{1}{x^2}</math>.
			:<math>\psi''(-x)-\psi''(x) = \frac{2\pi^3 \cot(\pi x)}{\sin^2(\pi x)}+\frac{2}{x^3}</math>.

	==Recurrence formula and characterization==		==Recurrence formula and characterization==
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	:<math>F(x+1)=F(x)+\frac{1}{x}</math>		:<math>F(x+1)=F(x)+\frac{1}{x}</math>

	that is [[monotonic function\|monotonic]] on {{math\|'''[[Real number\|R]]'''<sup>+</sup>}} and satisfies {{math\|''F''(1) {{=}} −''γ''}}. This fact follows immediately from the uniqueness of the {{math\|Γ}} function given its recurrence equation and convexity restriction. This implies the useful difference equation:		that is [[monotonic function\|monotonic]] on {{math\|'''[[Real number\|R]]'''<sup>+</sup>}} and satisfies {{math\|''F''(1) {{=}} −''γ''}}. This fact follows immediately from the uniqueness of the {{math\|Γ}} function given its recurrence equation and convexity restriction{{Citation needed\|date=September 2025}}. This implies the useful difference equation:

	: <math> \psi(x+N)-\psi(x)=\sum_{k=0}^{N-1} \frac{1}{x+k}</math>		: <math> \psi(x+N)-\psi(x)=\sum_{k=0}^{N-1} \frac{1}{x+k}</math>
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	We also have <ref>{{cite book \|title=Classical topi s in complex function theorey \|pages=46}}</ref>		We also have <ref>{{cite book \|title=Classical topi s in complex function theorey \|pages=46}}</ref>
	:<math> 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{k-1}-\gamma=\frac{1}{k}\sum_{n=0}^{k-1}\psi\left(1+\frac{n}{k}\right), k=2,3, ...</math>		:<math> 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{k-1}-\gamma-\ln k=\frac{1}{k}\sum_{n=0}^{k-1}\psi\left(1+\frac{n}{k}\right), k=2,3, ...</math>

	==Gauss's digamma theorem==		==Gauss's digamma theorem==
	For positive integers {{mvar\|r}} and {{mvar\|m}} ({{math\|''r'' < ''m''}}), the digamma function may be expressed in terms of [[Euler's constant]] and a finite number of [[elementary function]]s<ref>{{cite journal\|first1=Junesang\|last1=Choi\|first2=Djurdje\|last2=Cvijovic\|title=Values of the polygamma functions at rational arguments\|journal=Journal of Physics A\|year=2007\|volume=40\|pages=15019\|doi=10.1088/1751-8113/40/50/007\|number=50\|bibcode=2007JPhA...4015019C \|s2cid=118527596 }}</ref>		For positive integers {{mvar\|r}} and {{mvar\|m}} ({{math\|''r'' < ''m''}}), the digamma function may be expressed in terms of [[Euler's constant]] and a finite number of [[elementary function]]s<ref>{{cite journal\|first1=Junesang\|last1=Choi\|first2=Djurdje\|last2=Cvijovic\|title=Values of the polygamma functions at rational arguments\|journal=Journal of Physics A\|year=2007\|volume=40\|pages=15019\|doi=10.1088/1751-8113/40/50/007\|number=50\|bibcode=2007JPhA...4015019C \|s2cid=118527596 }}</ref>
			<ref>{{cite journal\|journal=Ann. Math.\|year=1916\|first1= J. L. W. V.\|last1=Jensen\|first2=T. H.\|last2=Gronwall\|title=An elementary exposition of the theory of the Gamma Function\|volume=17\|number=3\|pages=124–166\|doi=10.2307/2007272 \|jstor=2007272 }}</ref>

	:<math>\psi\left(\frac{r}{m}\right) = -\gamma -\ln(2m) -\frac{\pi}{2}\cot\left(\frac{r\pi}{m}\right) +2\sum_{n=1}^{\left\lfloor \frac{m-1}{2} \right\rfloor} \cos\left(\frac{2\pi nr}{m} \right) \ln\sin\left(\frac{\pi n}{m}\right) </math>		:<math>\psi\left(\frac{r}{m}\right) = -\gamma -\ln(2m) -\frac{\pi}{2}\cot\left(\frac{r\pi}{m}\right) +2\sum_{n=1}^{\left\lfloor \frac{m-1}{2} \right\rfloor} \cos\left(\frac{2\pi nr}{m} \right) \ln\sin\left(\frac{\pi n}{m}\right) </math>
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	==Multiplication theorem==		==Multiplication theorem==
	The multiplication theorem of the <math>\Gamma</math>-function is equivalent to<ref>{{cite book\|title=Table of integrals, series and products\|first1=I. S. \|last1=Gradshteyn\|first2=I. M. \|last2=Ryzhik\|year=2015\|isbn=978-0-12-384933-5\|chapter=8.365.5\|publisher=Elsevier Science \|lccn=2014010276}}</ref>		The [[multiplication theorem]] of the <math>\Gamma</math>-function is equivalent to<ref>{{cite book\|title=Table of integrals, series and products\|first1=I. S. \|last1=Gradshteyn\|first2=I. M. \|last2=Ryzhik\|year=2015\|isbn=978-0-12-384933-5\|chapter=8.365.5\|publisher=Elsevier Science \|lccn=2014010276}}</ref>
	:<math>\psi(nz)=\frac{1}{n}\sum_{k=0}^{n-1} \psi\left(z+\frac{k}{n}\right) +\ln n .</math>		:<math>\psi(nz)=\frac{1}{n}\sum_{k=0}^{n-1} \psi\left(z+\frac{k}{n}\right) +\ln n .</math>

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	Moreover, equality holds if and only if {{math\|''s'' {{=}} 1}}.<ref>{{cite journal \|doi=10.1016/j.jmaa.2013.05.045 \|doi-access=free\|title=Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities \|year=2013 \|last1=Laforgia \|first1=Andrea \|last2=Natalini \|first2=Pierpaolo \|journal=Journal of Mathematical Analysis and Applications \|volume=407 \|issue=2 \|pages=495–504 }}</ref>		Moreover, equality holds if and only if {{math\|''s'' {{=}} 1}}.<ref>{{cite journal \|doi=10.1016/j.jmaa.2013.05.045 \|doi-access=free\|title=Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities \|year=2013 \|last1=Laforgia \|first1=Andrea \|last2=Natalini \|first2=Pierpaolo \|journal=Journal of Mathematical Analysis and Applications \|volume=407 \|issue=2 \|pages=495–504 }}</ref>

	Inspired by the harmonic mean value inequality for the classical gamma function, Horzt Alzer and Graham Jameson proved, among other things, a harmonic mean-value inequality for the digamma function:		Inspired by the [[harmonic mean]] value inequality for the classical gamma function, Horzt Alzer and Graham Jameson proved, among other things, a harmonic mean-value inequality for the digamma function:

	<math> -\gamma \leq \frac{2 \psi(x) \psi(\frac{1}{x})}{\psi(x)+\psi(\frac{1}{x})} </math> for <math>x>0</math>		<math> -\gamma \leq \frac{2 \psi(x) \psi(\frac{1}{x})}{\psi(x)+\psi(\frac{1}{x})} </math> for <math>x>0</math>
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	:<math>\operatorname{Im} \psi(\tfrac{1}{2}+bi) = \frac{\pi}{2}\tanh (\pi b).</math>		:<math>\operatorname{Im} \psi(\tfrac{1}{2}+bi) = \frac{\pi}{2}\tanh (\pi b).</math>

	Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the [[imaginary unit]] the numerical approximation		Apart from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the [[imaginary unit]] the numerical approximation {{OEIS2C\|A248177}}
	:<math>\operatorname{Re} \psi(i) = -\gamma-\sum_{n=0}^\infty\frac{n-1}{n^3+n^2+n+1} \approx 0.09465.</math>		:<math>\operatorname{Re} \psi(i) = -\gamma-\sum_{n=0}^\infty\frac{n-1}{n^3+n^2+n+1} \approx 0.09465.</math>

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	* {{OEIS el\|1=A020759\|2=Decimal expansion of (-1)*Gamma'(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function}}—psi(1/2)		* {{OEIS el\|1=A020759\|2=Decimal expansion of (-1)*Gamma'(1/2)/Gamma(1/2) where Gamma(x) denotes the Gamma function}}—psi(1/2)
	:{{OEIS2C\|A047787}} psi(1/3), {{OEIS2C\|A200064}} psi(2/3), {{OEIS2C\|A020777}} psi(1/4), {{OEIS2C\|A200134}} psi(3/4), {{OEIS2C\|A200135}} to {{OEIS2C\|A200138}} psi(1/5) to psi(4/5).		:{{OEIS2C\|A047787}} psi(1/3), {{OEIS2C\|A200064}} psi(2/3), {{OEIS2C\|A020777}} psi(1/4), {{OEIS2C\|A200134}} psi(3/4), {{OEIS2C\|A200135}} to {{OEIS2C\|A200138}} psi(1/5) to psi(4/5).
			* [https://www.gnu.org/software/gsl/doc/html/specfunc.html#psi-digamma-function Implementation in the GNU Scientific library]
			* [https://github.com/splicebox/PsiCLASS/blob/master/gamma.cpp C function with the Chebyshev series]
			* [https://arxiv.org/src/0908.3030v4/anc/org/nevec/rjm/BigDecimalMath.java Java implementation with Taylor series]

	[[Category:Gamma and related functions]]		[[Category:Gamma and related functions]]

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