Complex General Harmonic Numbers extend Integer version

Theorem

The following definitions of the concept of General Harmonic Numbers are equivalent:

Valid over Integers

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:

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

Proof

For $x \in \C$ the harmonic numbers order $r$ are defined as follows:

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

\(\ds \harm r x\)

\(=\)

\(\ds \sum_{k \mathop = 1}^\infty \paren {\dfrac 1 {k^r} - \dfrac 1 {\paren {k + x}^r} }\)

Let $x \in \Z_{>0} $

\(\ds \)

\(=\)

\(\ds \paren {\dfrac 1 {1^r} - \dfrac 1 {\paren {1 + x}^r} } + \paren {\dfrac 1 {2^r} - \dfrac 1 {\paren {2 + x}^r} } + \cdots\)

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^x \frac 1 {k^r}\)

\(\ds \)

\(=\)

\(\ds \harm r x\)

$\blacksquare$