Difference between revisions of "Derivative of Li 2(-1/x)"

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==References==
 
==References==
 
{{BookReference|Polylogarithms and Associated Functions|1926|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)
 
{{BookReference|Polylogarithms and Associated Functions|1926|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)
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[[Category:Theorem]]
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[[Category:Unproven]]

Revision as of 20:24, 27 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)