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CAF123

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- Expansion about a branch point

I’m coming at this question with a physics application in mind so apologies if my language is a bit sloppy in places but I think the answer to my question is grounded in math so I’ll post it here.

Say I have a function F(z) defined in the complex z plane which has branch points at z=0 and z = 1 and branch cuts from -infinity to zero and from 1 to infinity. That is, the function is purely real (i.e. analytic) in the interval of z from 0 to 1.

Express F(z) = f(z)/(z-1)/z, extracting the singular points.

Now consider the region z<0. Suppose I want to expand the imaginary part of f(z)/(z-1) around z=0. What does it mean to expand this around a branch point z=0? Expansions are usually defined within a radius of convergence so for infinitesimal z>0 I am in the region of the complex z plane where the function is now real so I don’t know why or if such an expansion makes sense.

Hope my question makes sense. Thanks in advance.

Say I have a function F(z) defined in the complex z plane which has branch points at z=0 and z = 1 and branch cuts from -infinity to zero and from 1 to infinity. That is, the function is purely real (i.e. analytic) in the interval of z from 0 to 1.

Express F(z) = f(z)/(z-1)/z, extracting the singular points.

Now consider the region z<0. Suppose I want to expand the imaginary part of f(z)/(z-1) around z=0. What does it mean to expand this around a branch point z=0? Expansions are usually defined within a radius of convergence so for infinitesimal z>0 I am in the region of the complex z plane where the function is now real so I don’t know why or if such an expansion makes sense.

Hope my question makes sense. Thanks in advance.