Difference between revisions of "Derivative of Li 2(-1/x)"
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(Created page with "==Theorem== The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}x} \mathrm{Li}_2 \left( -\dfrac{1}{x} \right) = \dfrac{\log(1+\frac{1}{x})}{x} = \dfrac{\log(1+x)-\log(...") |
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\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},$$ |
where $\mathrm{Li}_2$ denotes the [[dilogarithm]] and $\log$ denotes the [[logarithm]]. | where $\mathrm{Li}_2$ denotes the [[dilogarithm]] and $\log$ denotes the [[logarithm]]. | ||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | {{BookReference|Polylogarithms and Associated Functions| | + | {{BookReference|Polylogarithms and Associated Functions|1981|ed=2nd|edpage=Second Edition|Leonard Lewin|prev=Relationship between dilogarithm and log(1-z)/z|next=Relationship between Li_2(-1/x),Li_2(-x),Li_2(-1), and log^2(x)}}: $(1.6)$ |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 04:21, 30 June 2016
Theorem
The following formula holds: $$\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
1981: Leonard Lewin: Polylogarithms and Associated Functions (2nd ed.) ... (previous) ... (next): $(1.6)$
