Expansion of a complex function around branch point

[CAF123]

TL;DR

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.

 

[aheight]

You can generate power series for functions around branch points. The power series are (multi-valued) fractional Puiseux series. For example, consider $w=\sqrt{z(1−z)}$. A Puiseux series for this function with center=0 is:

$\begin{align*} w(z)&=0.5 z^{3/2}+0.125 z^{5/2}+0.0625 z^{7/2}+0.0390625 z^{9/2}+0.0273438 z^{11/2}\\ &+0.0205078 z^{13/2}+0.0161133 z^{15/2}+0.013092 z^{17/2}+0.01091 z^{19/2}\\ &+0.00927353 z^{21/2}+0.00800896 z^{23/2}+0.00700784 z^{25/2}+0.00619924 z^{27/2}\\ &+0.00553504 z^{29/2}+0.00498153 z^{31/2}-1. \sqrt{z}+\cdots \end{align*}$
having radius of convergence equal to distance to nearest singular point, that is 1. Keep in mind that will give only one of two solutions. Would need to conjugate it (replace z^{1/2} in each monomial by −z^{1/2}) to get the other solution. For example, can use the Mathematica code:

[CODE title="Mathematica"]Series[Sqrt[z (1 - z)], {z, 0, 5}][/CODE]

and this returns
$\sqrt{z}-\frac{z^{3/2}}{2}-\frac{z^{5/2}}{8}-\frac{z^{7/2}}{16}-\frac{5 z^{9/2}}{128}+O\left(z^{11/2}\right)$
That would be the conjugate series.For example, try z=1/3+1/4I

$\sqrt{z(1−z)}=\pm(−0.539−0.077I)$

and if you plug-in that value to the above power series you'll get ±(−0.539−0.077I).

 

[CAF123]

Thanks. I had never come across this term before. Maybe I can ask something more generally: Say I have a function defined in a subset of the complex plane, here z<0. Suppose I want to expand this function around z=0. As the function is defined only for z<0 can there be a valid expansion where the radius of convergence of such an expansion can be extended into the region z>0?

 

[aheight]

Thanks. I had never come across this term before. Maybe I can ask something more generally: Say I have a function defined in a subset of the complex plane, here z<0. Suppose I want to expand this function around z=0. As the function is defined only for z<0 can there be a valid expansion where the radius of convergence of such an expansion can be extended into the region z>0?

That sounds like analytic continuation. I'm no expert but if the function defined for z<0 has an analytic continuation into the right half-plane, I would suspect a power series for the analytic continuation centered at zero would have radius of convergence equal to the nearest singular point of the analytically-continued function and equal to the original function in the series domain of convergence.