HarmonicNumber

The harmonic number $H_n = \sum_{k=1}^n \tfrac1k$ , exact at a non-negative integer $n$ ; with a second argument, the generalized $H_n^{(r)} = \sum_{k=1}^n k^{-r}$ . Continued off the integers by $H_z = \psi(z+1) + \gamma$ . Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimHarmonicNumber✓Fungrim entry00c02a✓f50c74Wolfram LanguageHarmonicNumber!

HarmonicNumber(n)the harmonic number

H_n = \sum_{k=1}^n 1/k

SageMathharmonic_number(n)SymPyharmonic(n)mpmathharmonic(n)Juliaenumeratio_ball(x -> digamma(x + 1) + const_euler(RR), n)Rustharmonic(n)

HarmonicNumber(n, r)the generalized harmonic number

H_n^{(r)} = \sum_{k=1}^n k^{-r}

SageMathharmonic_number(n, r)SymPyharmonic(n, r)Juliaenumeratio_ball((n, r) -> r == 1 ? digamma(n + 1) + const_euler(RR) : zeta(r) - zeta(r, n + 1), n, r)Rustgeneralized_harmonic(n, r)

Typed signatures

Type	Package	Applies
`(number, number?) -> number`	`analytic`	always

Domain: Special functions

Details

At a non-negative integer $n$ (and an integer $r$ of either sign, in the two-argument form) the sum is exact: $H_0 = 0$ , $H_1 = 1$ , $H_2 = \tfrac32$ , $H_{10} = \tfrac{7381}{2520}$ .
Continued off the lattice by the standard digamma identity $H_z = \psi(z+1) + \gamma$ (PolyGamma, EulerGamma) and its generalization $H_z^{(r)} = \zeta(r) - \zeta(r, z+1)$ (Zeta, HurwitzZeta) — both reduce to the same exact values at the integers, so there is one formula, not a case split.
Order $r = 1$ is the one place that generalization can't be used directly: $\zeta(1)$ is itself a pole, at every $z$ , even though $H_z^{(1)} = H_z$ is perfectly finite. HarmonicNumber(z, 1) is routed straight to the one-argument $\psi(z+1) + \gamma$ instead.
Negative integer $n$ has no sum and is a pole: $H_{-1} = H_{-2} = \cdots = \mathrm{ComplexInfinity}$ , in both the one- and two-argument forms — matching Wolfram, which does not extend the sum by the continuation there.
Complex $z$ and complex/non-integer $r$ are supported numerically, via PolyGamma's digamma ( $r$ absent) and HurwitzZeta ( $r$ present).
A non-integer $r$ (or non-integer $z$ with $r$ present) stays symbolic under plain evaluation even at an otherwise-exact $n$ — the sum $\sum k^{-r}$ has no rational value there — and only reduces under N() or a floating-point argument, the same gate every head in this package uses.
Differs from Wolfram: $HarmonicNumber(1/4)$ reduces to its closed form on evaluation; Wolfram keeps HarmonicNumber[1/4] until FunctionExpand. Both agree as identities.

Plot

Examples

25 more kept as data, checked against the oracles

Basic #

$H_{10} = \tfrac{7381}{2520}$

Examples	48
Checked against	Wolfram Language 46 agree2 differSageMath 41 agree3 differ2 unevaluatedSymPy 44 agree2 differ1 unevaluatedmpmath 39 agree1 unevaluated4 errorJulia 38 agree1 unevaluated5 errorRust 12 agree3 differ1 unevaluated20 error
Crosswalk	Fungrim Fungrim entry ×2 Wolfram LanguageSageMath ×2SymPy ×2mpmathJulia ×2Rust ×2

HarmonicNumber ​

Plot

Examples

Implementation

HarmonicNumber