Analytic continuation of a dilogarithm

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CAF123
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The correct analytical continuation of the dilog function is of the form $$\text{lim}_{\epsilon \rightarrow 0^+} \text{Li}_2(x \pm i\epsilon) = -\left(\text{Li}_2(x) \mp i\pi \ln x \right)$$
I read this in a review at some point which I can no longer find at the moment so just wondered if this is the correct expression for the continuation (up to signs I mean, the structure is definitely correct) but, I guess more importantly, how would one go about deriving this? Looks like resorting to integral representation of dilog is the best way to proceed but then the upper limit becomes complex for argument being ##x \pm i \epsilon##.
 

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  • #2
MathematicalPhysicist
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Try consulting a book on special functions.

I recommend Andrews' red book, though I am not 100% sure they cover the dilog function.
 

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