# Definition:Harmonic Numbers/General Definition

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

It has been suggested that this page be renamed.In particular: Needs a "function" in its name after all. Ramanujan's function?To discuss this page in more detail, feel free to use the talk page. |

## Definition

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

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

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

### Complex Extension

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

For $z \in \C \setminus \Z_{< 0}$ the **harmonic numbers order $r$** can be extended to the complex plane as:

- $\ds \harm r z = \sum_{k \mathop = 1}^{\infty} \paren {\frac 1 {k^r} - \frac 1 {\paren {k + z}^r} }$

This article is complete as far as it goes, but it could do with expansion.In particular: I think it's worth adding a section here (usual transclusion rules apply of course) relating this to the harmonic numbers themselves, and perhaps to include a separate page for $\map {H^{\paren 1} } z$ for complex $z$ to be defined as a direct extension of $H_n$. Interesting exercise in elementary function theory.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 $\map \psi {n + 1} + \gamma$.

## Examples

### General Harmonic Number of Order $1$ at $\dfrac 1 2$

- $\harm 1 {\dfrac 1 2} = 2 - 2 \ln 2$

### General Harmonic Number of Order $1$ at $-\dfrac 1 2$

- $\harm 1 {-\dfrac 1 2} = -2 \ln 2$

### General Harmonic Number of Order $1$ at $\dfrac 1 3$

- $\harm 1 {\dfrac 1 3} = 3 - \dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$

### General Harmonic Number of Order $1$ at $-\dfrac 1 3$

- $\harm 1 {-\dfrac 1 3} = -\dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$

### General Harmonic Number of Order $1$ at $\dfrac 2 3$

- $\harm 1 {\dfrac 2 3} = \dfrac 3 2 - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$

### General Harmonic Number of Order $1$ at $-\dfrac 2 3$

- $\harm 1 {-\dfrac 2 3} = -\dfrac 3 2 \ln 3 - \dfrac \pi {2 \sqrt 3}$

### General Harmonic Number of Order $2$ at $\dfrac 1 2$

- $\harm 2 {\dfrac 1 2} = 4 - 2 \map \zeta 2$

### General Harmonic Number of Order $2$ at $-\dfrac 1 2$

- $\harm 2 {-\dfrac 1 2} = -2 \map \zeta 2$

## Also see

- Definition:Riemann Zeta Function: $\ds \harm r \infty = \lim_{n \mathop \to \infty} \harm r n$

- Results about
**the general harmonic numbers**can be found**here**.

## Technical Note

The $\LaTeX$ code for \(\harm {r} {z}\) is `\harm {r} {z}`

.

When either of the arguments is a single character, it is usual to omit the braces:

`\harm r z`

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(4)$