General Harmonic Numbers/Examples/Order 1/Half

Example of General Harmonic Number

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

where $\harm 1 {\dfrac 1 2}$ denotes the general harmonic number of order $1$ evaluated at $\dfrac 1 2$.

$\ds \harm 1 z$

$=$

$\ds \map \psi {z + 1} + \gamma$

$\ds \leadsto \ \ $

$\ds \harm 1 {\dfrac 1 2}$

$=$

$\ds \map \psi {\dfrac 1 2 + 1} + \gamma$

setting $z := \dfrac 1 2$

$\ds $

$=$

$\ds \map \psi {\dfrac 3 2} + \gamma$

$\ds $

$=$

$\ds \paren {-\gamma - 2 \ln 2 + 2} + \gamma$

$\ds $

$=$

$\ds 2 - 2 \ln 2$

$\blacksquare$