PolyGamma

The polygamma function $\psi^{(m)}(z) = \dfrac{d^{m+1}}{dz^{m+1}} \ln\Gamma(z)$, the $m$-th derivative of the digamma.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPolygamma function↗MathWorldPolygammaFunction↗WikidataQ857956↗DLMF5.15↗FungrimPolyGamma✓↗Fungrim entryWolfram LanguagePolyGamma!↗

PolyGamma(m, z)the polygamma $\psi^{(m)}(z)$ at real $z$.

SageMathpsi(m, z)SymPypolygamma(m, z)↗mpmathpsi(m, z)↗Juliaenumeratio_ball(polygamma, m, z)

Typed signatures

Type	Package	Applies
(order: integer, complex | infinity) -> number	compute-engine	always
(number, number?) -> number	analyticreplaces compute-engine	always

Domain: Special functions

Details

• $\psi^{(0)}$ is the digamma — see Digamma — and $\psi^{(1)}$ the trigamma.
• For $m \geq 1$ it is a Hurwitz zeta in disguise: $\psi^{(m)}(z) = (-1)^{m+1}\, m!\, \zeta(m+1, z)$. See HurwitzZeta.
• So at $z = 1$ every order is a zeta value: $\psi^{(m)}(1) = (-1)^{m+1} m!\, \zeta(m+1)$, giving $\psi'(1) = \pi^2/6$ and $\psi''(1) = -2\zeta(3)$.
• Half-integer arguments pick up the alternating zeta: $\psi'(\tfrac12) = \pi^2/2$.
• Recurrence from $\Gamma$: $\psi^{(m)}(z+1) = \psi^{(m)}(z) + (-1)^m m!\, z^{-m-1}$, so $\psi'(1) - \psi'(2) = 1$.
• Poles at the nonpositive integers, of order $m+1$ — the poles of Gamma, differentiated.
• Real $z$ is compute-engine's own; @enumeratio/analytic adds complex $z$ through the Euler–Maclaurin Hurwitz kernel, and the WGSL lowering for GPU plotting.

Plot

Examples

47 more kept as data, checked against the oracles

Basic #

$\psi'(1) = \zeta(2) = \pi^2/6$

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In$\mathrm{PolyGamma}(1, 1)=ζ(2)$Out$\mathrm{True}$

$\psi''(1) = -2\zeta(3)$, Apéry's constant doubled and negated

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In$\mathrm{PolyGamma}(2, 1)=-2ζ(3)$Out$\mathrm{True}$

An inexact argument evaluates to a decimal: $\psi'(1/2) = 4.93480\ldots$

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In$\mathrm{PolyGamma}(1, 0.5)$Out$4.934\,802\,200\,544\,679\,309\,42$

A pole of Gamma, still a pole after differentiating

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In$\mathrm{PolyGamma}(1, 0)$Out$\tilde∞$

One argument is the digamma: $\psi(5) = H_4 - \gamma = \frac{25}{12} - \gamma$. See Digamma

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

$\psi^{(3)}(5) = \pi^4/15 - 22369/3456$

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In$\mathrm{PolyGamma}(3, 5)$Out$\frac{1}{17\,280}(1\,152π^4-111\,845)$

...numerically

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In$N(\mathrm{PolyGamma}(3, 5))$Out$0.021\,427\,828\,192\,755\,075\,021\,9$

$\psi''(1/2) = -14\zeta(3)$

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In$\mathrm{PolyGamma}(2, 0.5)$Out$-16.828\,796\,644\,234\,319\,995\,6$

Scope #

Threads over a list (each element a positive integer, so each takes its zeta closed form)

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In$\mathrm{PolyGamma}(1, \bigl\lbrack1, 2\bigr\rbrack)$Out$\bigl\lbrack\frac{π^2}{6}, \frac{π^2}{6}-1\bigr\rbrack$

One-argument digamma at a machine-precision point

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

One-argument digamma at a complex point

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

Complex argument, integer order

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In$\mathrm{PolyGamma}(1, 2.5+3i)$Out$0.155\,597\,888\,471\,945\,53-0.230\,379\,553\,082\,323\,5i$

Between poles: $\psi'(-1/2) = \pi^2/2 + 4$

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In$\mathrm{PolyGamma}(1, -0.5)$Out$8.934\,802\,200\,544\,679\,309\,42$

$\psi'(\infty) = 0$

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In$\mathrm{PolyGamma}(1, ∞)$Out$0$

Interval arithmetic: the trigamma is decreasing

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In$\mathrm{PolyGamma}(1, \mathrm{Interval}(3.45, 3.46))$Out$\mathrm{Interval}(0.334\,742\,897\,540\,441\,2, 0.335\,857\,280\,611\,811\,7)$

To 30 significant digits, every one correctly rounded

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In$N(\mathrm{PolyGamma}(1, \frac{1}{3}), 30)$Out$10.095\,597\,125\,427\,094\,081\,792\,004\,099\,9$

To 25 significant digits, every one of them right

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In$N(\mathrm{PolyGamma}(3, 5), 25)$Out$0.021\,427\,828\,192\,755\,075\,021\,948\,11$

$\psi'(1)=\zeta(2)=\pi^2/6$

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In$\mathrm{PolyGamma}(1, 1)$Out$\frac{π^2}{6}$

$\psi'(2)=\pi^2/6-1$, by the shift $\psi^{(n)}(m)=(-1)^{n+1}n!\,(\zeta(n+1)-\sum_{k<m}k^{-n-1})$

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In$\mathrm{PolyGamma}(1, 2)$Out$\frac{π^2}{6}-1$

One argument is the digamma: $\psi(0.7)$ evaluates directly

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In$\mathrm{PolyGamma}(0.7)$Out$-1.220\,023\,553\,697\,934\,614\,75$

Applications #

$\psi'(1+i) = 0.46300\ldots - 0.79423\ldots i$ — complex argument, from @enumeratio/analytic

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In$N(\mathrm{PolyGamma}(1, 1+i))$Out$0.463\,000\,096\,622\,763\,76-0.794\,233\,542\,759\,319i$

Properties #

$\psi'(1/2) = \pi^2/2$

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In$\mathrm{PolyGamma}(1, \frac{1}{2})=\frac{π^2}{2}$Out$\mathrm{True}$

Recurrence at $m = 1$, $z = 1$: $\psi'(z) - \psi'(z+1) = z^{-2}$

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In$\mathrm{PolyGamma}(1, 1)-\mathrm{PolyGamma}(1, 2)=1$Out$\mathrm{True}$

$\psi^{(0)}(1) = -\gamma$ -- PolyGamma(0, z) routes through Digamma(z) (#113) whenever Digamma has an exact value

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In$\mathrm{PolyGamma}(0, 1)$Out$-γ$

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

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In$\mathrm{PolyGamma}(0, \frac{1}{2})$Out$-2\ln(2)-γ$

$\psi'(1/2) = \pi^2/2$ in closed form

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In$\mathrm{PolyGamma}(1, \frac{1}{2})$Out$\frac{π^2}{2}$

$\psi'(1/4) = \pi^2 + 8G$

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In$\mathrm{PolyGamma}(1, \frac{1}{4})$Out$8\operatorname{G}+π^2$

$\frac{d}{dx}\psi^{(n)}(x) = \psi^{(n+1)}(x)$

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In$\frac{\mathrm{d}\mathrm{PolyGamma}(n, x)}{\mathrm{d}x}$Out$\mathrm{PolyGamma}(n+1, x)$

Possible issues #

An exact non-integer argument is left unevaluated under plain evaluation -- pair with N() or use an inexact argument

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In$\mathrm{PolyGamma}(1, \frac{5}{3})$Out$\mathrm{PolyGamma}(1, \frac{5}{3})$

About

Examples	76
Checked against	Wolfram Language 74 agree2 differSageMath 66 agree2 differ3 errorSymPy 62 agree8 differ1 unevaluated1 errormpmath 56 agree10 errorJulia 60 agree7 differ1 unevaluated2 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×28Wolfram LanguageSageMathSymPympmathJulia

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mpmath and Sage call it psi; the argument order matches.

mappedsympypackages/oracle/src/mappings.ts

mpmath and Sage call it psi; the argument order matches.

mappedmpmathpackages/oracle/src/mappings.ts

mpmath and Sage call it psi; the argument order matches.

mappedsagepackages/oracle/src/mappings.ts

mpmath and Sage call it psi; the argument order matches.

mappedjuliapackages/oracle/src/mappings.ts

See also: Digamma, Gamma, HurwitzZeta, Zeta