DigammaFunctionZero

The $n$-th real zero of the digamma function $\psi$: $n=0$ names the zero on $(0,\infty)$ ($x_0 \approx 1.4616$), $n \ge 1$ the zero on $(-n, -n+1)$. Provided by @enumeratio/analytic.

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DigammaFunctionZero(n)the $n$-th real zero of $\psi$, $n \ge 0$.

Typed signatures

Type	Package	Applies
(integer) -> number	analytic	always

Domain: Special functions

Details

• $\psi$ is real, meromorphic, with simple poles at $0, -1, -2, \dots$ and strictly increasing between consecutive poles ($\psi' = $ trigamma $> 0$), so each interval carries exactly one zero — bisection on the native PolyGamma/Digamma finds it without a separate digamma implementation.
• $\psi(x_0) = 0$ at $x_0 \approx 1.4616321449683623$, sometimes called the digamma's positive real zero.

Plot

Examples

3 more kept as data, checked against the oracles

Basic #

the digamma function's positive real zero

live ↗

In$N(\mathrm{DigammaFunctionZero}(0))$Out$1.461\,632\,144\,968\,362\,2$

the zero in $(-1, 0)$

live ↗

In$N(\mathrm{DigammaFunctionZero}(1))$Out$-0.504\,083\,008\,264\,455\,4$

Properties #

$\psi$ vanishes exactly there, by construction

live ↗

In$\mathrm{chop}(N(\mathrm{Digamma}(\mathrm{DigammaFunctionZero}(2))))$Out$0.0$

1 / 4

$\psi(x_n) = 0$ exactly at digamma's own claimed zero, $n = 0$

live ↗

In$\mathrm{chop}(N(\mathrm{Digamma}(\mathrm{DigammaFunctionZero}(0))))$Out$0.0$

About

Examples	10
Crosswalk	FungrimFungrim entry ×2

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/digamma-zero.ts

mappedwolframpackages/oracle/src/mappings.ts

No direct equivalent head or built-in; N[x /. FindRoot[PolyGamma[0, x] == 0, {x, x0}]] would need a per-n bracket this row's fixed template can't supply. Wolfram Function Repository searched for a digamma-zero/root-finder resource specific to this; nothing found (only generic root-finding and PolyGamma-simplification resources turned up).

mappedexternalmpmathpackages/oracle/src/mappings.ts

findroot(digamma, ...) at the appropriate bracket — Wolfram has no direct equivalent head.

See also: PolyGamma