Digamma

The digamma function $\psi(z) = \dfrac{d}{dz}\ln\Gamma(z)$, the logarithmic derivative of Gamma.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDigamma function↗MathWorldDigammaFunction↗WikidataQ905326↗Encyclopedia of MathematicsPsi-function↗DLMF5.2.E2↗FungrimDigammaFunction✓↗Fungrim entryWolfram LanguagePolyGamma!↗

Digamma(z)the digamma function $\psi(z) = \Gamma'(z)/\Gamma(z)$.

SageMathpsi(z)SymPydigamma(z)mpmathdigamma(z)Juliaenumeratio_ball(digamma, z)Rustdigamma(z)

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity) -> number	compute-engine	always

Domain: Special functions

Details

• Digamma is the order-zero polygamma: $\psi(z) = \psi^{(0)}(z)$. The higher derivatives are PolyGamma.
• $\psi(1) = -\gamma$, the negative Euler-Mascheroni constant.
• Recurrence inherited from Gamma's functional equation: $\psi(z+1) = \psi(z) + 1/z$.
• $\psi(1/2) = -\gamma - 2\ln 2$.
• Poles at the nonpositive integers, the same poles as Gamma.
• compute-engine's plain evaluation would otherwise leave Digamma at an exact integer argument symbolic except at the poles; @enumeratio/analytic overrides it at every positive integer via $\psi(n) = H_{n-1} - \gamma$, reusing HarmonicNumber. A rational, non-integer argument, or N(), still goes through the numeric path.
• Differs from Wolfram: $Digamma(1/4)$ reduces to its closed form on evaluation; Wolfram keeps Digamma[1/4] until FunctionExpand. Both agree as identities.

Plot

ComplexPlot

real slice

Examples

1 more kept as data, checked against the oracles

Basic #

A pole of Gamma, shared by its log-derivative

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In$\mathrm{Digamma}(0)$Out$\tilde∞$

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In$\mathrm{Digamma}(-1)$Out$\tilde∞$

A floating-point argument evaluates directly to a high-precision decimal

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In$\mathrm{Digamma}(0.5)$Out$-1.963\,510\,026\,021\,423\,479\,44$

A machine-precision argument

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In$\mathrm{Digamma}(100.5)$Out$4.605\,174\,352\,581\,845\,211\,87$

$\psi(5) = H_4 - \gamma = \frac{25}{12} - \gamma$

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In$\mathrm{Digamma}(5)$Out$\frac{25}{12}-γ$

Scope #

Threads over a list, reducing each positive-integer element to its closed form in $\gamma$

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In$\mathrm{Digamma}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack-γ, 1-γ, \frac{3}{2}-γ\bigr\rbrack$

Inexact complex arguments evaluate, as $\psi^{(m)}$ does for $m \geq 1$ (see PolyGamma)

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In$\mathrm{Digamma}(2.5+3i)$Out$1.281\,273\,919\,066\,231\,4+0.979\,805\,315\,344\,559\,6i$

$\psi(\infty) = \infty$, growing like $\ln z$

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In$\mathrm{Digamma}(∞)$Out$∞$

Gauss's digamma theorem: $\psi(1/4) = -\gamma - \pi/2 - 3\ln 2$

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In$\mathrm{Digamma}(\frac{1}{4})$Out$-3\ln(2)-\frac{π}{2}-γ$

$\psi(1/3) = -\gamma - \frac{\pi}{2\sqrt3} - \frac32\ln 3$

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In$\mathrm{Digamma}(\frac{1}{3})$Out$-\frac{3\ln(3)}{2}-\frac{\sqrt{3}π}{6}-γ$

Interval arithmetic: $\psi$ is increasing on $(0, \infty)$

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In$\mathrm{Digamma}(\mathrm{Interval}(1.23, 1.24))$Out$\mathrm{Interval}(-0.251\,669\,430\,696\,100\,1, -0.239\,493\,679\,125\,936\,8)$

To 30 significant digits, every one correctly rounded

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In$N(\mathrm{Digamma}(\frac{1}{3}), 30)$Out$-3.132\,033\,780\,020\,806\,322\,996\,419\,074\,29$

Applications #

Harmonic numbers via Digamma: $\psi(n+1) + \gamma = H_n$, here $H_5$ at $n=5$

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In$\mathrm{Digamma}(6)+γ=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}$Out$\mathrm{True}$

Properties #

$\psi(1) = -\gamma$

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In$\mathrm{Digamma}(1)+γ=0$Out$\mathrm{True}$

Recurrence: $\psi(z+1) = \psi(z) + 1/z$, here at $z = 1$

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In$\mathrm{Digamma}(2)=\mathrm{Digamma}(1)+1$Out$\mathrm{True}$

$\psi(1/2) = -\gamma - 2\ln 2$

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In$\mathrm{Digamma}(\frac{1}{2})+γ+2\ln(2)=0$Out$\mathrm{True}$

$\psi'(x)$ is the trigamma, $\psi^{(1)}(x)$

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In$\frac{\mathrm{d}\mathrm{Digamma}(x)}{\mathrm{d}x}$Out$\mathrm{Trigamma}(x)$

Reflection: $\psi(1-x) - \psi(x) = \pi\cot(\pi x)$, here $\pi\cot(0.3\pi) = 2.2825\ldots$

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In$\mathrm{Digamma}(0.7)-\mathrm{Digamma}(0.3)$Out$2.282\,500\,668\,502\,198\,374\,21$

1 / 3

$\psi(1) = -\gamma$.

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In$\mathrm{Digamma}(1)$Out$-γ$

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In$\mathrm{Digamma}(\frac{1}{3})-\mathrm{Digamma}(\frac{2}{3})$Out$-\frac{\sqrt{3}}{3}π$

Known value · Wolfram MathematicalFunctionData, PolyGamma:Digamma, Reflection formula

Neat examples #

Between the poles at 0 and -1, $\psi$ is still finite: $\psi(-1/2) \approx 0.036$

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In$\mathrm{Digamma}(-0.5)$Out$0.036\,489\,973\,978\,576\,520\,559$

About

Examples	24
Checked against	Wolfram Language 22 agree1 differSageMath 18 agree2 differ2 errorSymPy 21 agree3 differmpmath 19 agree3 errorJulia 19 agree3 errorRust 14 agree2 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsDLMFFungrimFungrim entry ×45Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Gamma, Zeta, GammaLn