Definition:Harmonic Numbers

This page is about Harmonic Numbers. For other uses, see Harmonic.

Definition

The harmonic numbers are denoted $H_n$ and are defined for positive integers $n$:

$\ds \forall n \in \Z, n \ge 0: H_n = \sum_{k \mathop = 1}^n \frac 1 k$

From the definition of vacuous summation it is clear that $H_0 = 0$.

General Harmonic Numbers

Let $r \in \R_{>0}$.

For $n \in \N_{> 0}$ the harmonic numbers order $r$ are defined as follows:

$\ds \map {H^{\paren r} } n = \sum_{k \mathop = 1}^n \frac 1 {k^r}$

Notation

There appears to be no standard notation for the harmonic numbers.

The notation given here, and used on $\mathsf{Pr} \infty \mathsf{fWiki}$ throughout, is an adaptation for $\mathsf{Pr} \infty \mathsf{fWiki}$ of an idea by Donald E. Knuth, where he used $H_n^{\paren r}$.

Knuth's notation proves unwieldy when extended to the complex numbers, and so we have adopted the more conventional mapping notation $\harm r n$ and hence $\harm r z$.

Ramanujan used $\ds \map {\phi_r} n$.

Other notations that can also be found in the literature include $h_n$, $S_n$ and $\map \psi {n + 1} + \gamma$.