Extension of Harmonic Number to Non-Integer Argument

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Theorem

Let $\map H x$ be the real function defined as:

$\map H x = \gamma + \dfrac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }$

where:

$\map \Gamma x$ denotes the Gamma function
$\map {\Gamma'} x$ denotes the derivative of the Gamma function
$\gamma$ denotes the Euler-Mascheroni constant.


Then $H$ is an extension of the mapping $H: \N \to \Q$ defined as:

$\forall n \in \N: \map H n = H_n$

where $H_n$ denotes the $n$th harmonic number.


Proof

For $n \in \N$:

\(\ds \map H n\) \(=\) \(\ds \gamma + \frac {\map {\Gamma'} {n + 1} } {\map \Gamma {n + 1} }\)
\(\ds \) \(=\) \(\ds \gamma - \gamma + \sum_{m \mathop = 1}^\infty \paren {\frac 1 m - \frac 1 {n + m} }\) Reciprocal times Derivative of Gamma Function
\(\ds \) \(=\) \(\ds \lim_{k \mathop \to \infty} \sum_{m \mathop = 1}^k \paren {\frac 1 m - \frac 1 {n + m} }\)
\(\ds \) \(=\) \(\ds \lim_{k \mathop \to \infty} \paren {\sum_{m \mathop = 1}^k \paren {\frac 1 m} - \sum_{m \mathop = 1}^k \paren {\frac 1 {n + m} } }\) Linear Combination of Convergent Series
\(\ds \) \(=\) \(\ds \lim_{k \mathop \to \infty} \paren {\sum_{m \mathop = 1}^k \paren {\frac 1 m} - \sum_{m \mathop = n + 1}^{k + n} \paren {\frac 1 m} }\) reindexing sum
\(\ds \) \(=\) \(\ds \lim_{k \mathop \to \infty} \paren {\sum_{m \mathop = 1}^n \paren {\frac 1 m} - \sum_{m \mathop = k + 1}^{k + n} \paren {\frac 1 m} }\)
\(\ds \) \(=\) \(\ds \sum_{m \mathop = 1}^n \paren {\frac 1 m} - \lim_{k \mathop \to \infty} \sum_{m \mathop = k + 1}^{k + n} \paren {\frac 1 m}\) this limit will vanish since it is positive but less than $\dfrac n {k + 1}$
\(\ds \) \(=\) \(\ds H_n - 0\) Definition of Harmonic Number
\(\ds \) \(=\) \(\ds H_n\)

$\blacksquare$


Also see


Sources

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