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- TL;DR Summary
- I cannot understand the notation for analytically-continuing this function as given by Lewen and was wondering if someone could help me.

I am trying to understand the branching geometry of the Dilogarithm function as described in Branching geometry of Dilogarithm. In Theorem 8.6, the following (dilogarithm) definition is given by letting ##n=2##:

$$

\text{Li}_2(z)=\text{Li}_2^{(k_0,k_1)}(z)=\text{Li}_2^{(0)}(z)+\sum_{m=0}^1 k_m\binom{1}{m}(2\pi i)^{m+1}(\text{Log}(z))^{2-m-1}

$$

but it's not clear at all what ##k_m## is and was wondering if someone would know this?

For example, suppose I wanted to plot an analytically-continuous ##6\pi## route via ##z(t)=2e^{it}## over it's Riemann surface, how would I splice-together the pieces defined above over the branch-cuts ##(-\infty,0)## and ##(1,\infty)## beginning on the principal branch ##\text{Li}_2(2)## so that the path traverses an analytically-continuous over them?

I can differentially-continue the path above over the sheets of ##f(z)=\text{Li}_2(z)## via the IVP:

$$

\frac{df}{dt}=-\frac{1}{2}\log(1-z)\frac{dz}{dt};\quad f(0)=\text{Li}_2(2)

$$

and by using MapleSoft's definition:

stitch the MapleSoft sheets together by trial-and-error over this path to create the (real) plot below showing the analytically-continuous route through the function along this path in black. However that's not practical and was hoping to better understand the formula for ##\text{Li}_2(z)## above to do this systematically by supplying the correct indices ##k_m## I assume.

$$

\text{Li}_2(z)=\text{Li}_2^{(k_0,k_1)}(z)=\text{Li}_2^{(0)}(z)+\sum_{m=0}^1 k_m\binom{1}{m}(2\pi i)^{m+1}(\text{Log}(z))^{2-m-1}

$$

but it's not clear at all what ##k_m## is and was wondering if someone would know this?

For example, suppose I wanted to plot an analytically-continuous ##6\pi## route via ##z(t)=2e^{it}## over it's Riemann surface, how would I splice-together the pieces defined above over the branch-cuts ##(-\infty,0)## and ##(1,\infty)## beginning on the principal branch ##\text{Li}_2(2)## so that the path traverses an analytically-continuous over them?

I can differentially-continue the path above over the sheets of ##f(z)=\text{Li}_2(z)## via the IVP:

$$

\frac{df}{dt}=-\frac{1}{2}\log(1-z)\frac{dz}{dt};\quad f(0)=\text{Li}_2(2)

$$

and by using MapleSoft's definition:

Start with the principal sheet ##\text{Li}_n## (implemented in mathematica as ##\text{Polylog}[n,z]##. Now, each time the branch cut ##(1,\infty)## is crossed in the counter clockwise direction, subtract from the current expression ##\frac{2\pi i\text{Log}^{n-1}(z)}{\Gamma(n)}##. Each time the branch cut ##(-\infty,0)## is crossed in the counter clockwise direction, add ##2\pi i## to the ##\text{Log}## term.

stitch the MapleSoft sheets together by trial-and-error over this path to create the (real) plot below showing the analytically-continuous route through the function along this path in black. However that's not practical and was hoping to better understand the formula for ##\text{Li}_2(z)## above to do this systematically by supplying the correct indices ##k_m## I assume.

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