General Harmonic Numbers/Examples
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Examples of General Harmonic Numbers
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 $-\dfrac 1 2$
- $\harm 2 {-\dfrac 1 2} = -2 \map \zeta 2$