Analytic continuation of a dilogarithm

In summary, the correct analytical continuation of the dilog function is of the form $$\text{lim}_{\epsilon \rightarrow 0^+} \text{Li}_2(x \pm i\epsilon) = -\left(\text{Li}_2(x) \mp i\pi \ln x \right)$$. This expression was mentioned in a review, but its derivation is not clear. To find the derivation, it is recommended to consult a book on special functions, such as Andrews' red book. However, it is uncertain whether the dilog function is covered in this book. The best approach may be to use the integral representation of the dilog function, but this may result in a complex upper limit for an
  • #1
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
Try consulting a book on special functions.

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

FAQ: Analytic continuation of a dilogarithm

What is analytic continuation?

Analytic continuation is a mathematical technique used to extend the domain of definition of a function beyond its original domain. This is done by using the properties of complex numbers and complex analysis.

What is a dilogarithm?

A dilogarithm is a special function that is defined as the integral of the logarithm of a complex number. It is also known as the Spence function and is denoted by Li2(z).

How is analytic continuation applied to a dilogarithm?

Analytic continuation is used to extend the domain of definition of the dilogarithm function beyond its original domain. This is done by using the Cauchy-Riemann equations and the Cauchy integral theorem.

Why is analytic continuation important in the study of dilogarithms?

Analytic continuation allows us to define the dilogarithm function in a larger domain, which in turn allows us to use it in a wider range of mathematical applications. It also helps in understanding the behavior of the function in different regions of the complex plane.

Are there any real-life applications of analytic continuation of a dilogarithm?

Yes, analytic continuation of a dilogarithm has various applications in physics, particularly in quantum field theory and statistical mechanics. It is also used in number theory, combinatorics, and complex analysis.

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