Difference between revisions of "Derivative of Li 2(-1/x)"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{Li}_2 \left( -\dfrac{1}{x} \right) = \dfrac{\log(1+\frac{1}{x})}{x} = \dfrac{\log(1+x)-\log(x)}{x},$$ | + | $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{Li}_2 \left( -\dfrac{1}{x} \right) = \dfrac{\log \left(1+\frac{1}{x} \right)}{x} = \dfrac{\log(1+x)-\log(x)}{x},$$ |
where $\mathrm{Li}_2$ denotes the [[dilogarithm]] and $\log$ denotes the [[logarithm]]. | where $\mathrm{Li}_2$ denotes the [[dilogarithm]] and $\log$ denotes the [[logarithm]]. | ||
Revision as of 23:56, 3 June 2016
Theorem
The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{Li}_2 \left( -\dfrac{1}{x} \right) = \dfrac{\log \left(1+\frac{1}{x} \right)}{x} = \dfrac{\log(1+x)-\log(x)}{x},$$ where $\mathrm{Li}_2$ denotes the dilogarithm and $\log$ denotes the logarithm.
Proof
References
1926: Leonard Lewin: Polylogarithms and Associated Functions (2nd ed.) ... (previous) ... (next): (1.6)
