Definition of a branch point

**Deadstar** · February 23rd 2010, 10:02 AM

I can't seem to find a good definition for a branch point. I have several thoughts and have found various definitions but would like to hear MHFs opinions on this one!

I'm thinking something along the lines of...

A branch point is a point of a (analytic?) function that is undefined/discontinuous?

Or perhaps...

A branch point(s?) $z$ is a point(s?) of an analytic function $f$ such that $f$ is holomorphic everywhere except at $z$ ? (or perhaps near z?)

**shawsend** · February 23rd 2010, 10:56 AM

A point is called a branch-point if analytic continuation over a closed curve around it can produce a different value upon reaching the starting point. Take for example $f(z)=\sqrt{z}$ starting at the point $z=1$ and analytically continuing the function around the unit circle using the differential equation $\frac{df}{dt}=1/2 i f(t)$ (just differentiate the function and let $z=e^{it}$ ). Upon integrating from zero to $2\pi$ , $f(0)\ne f(2\pi)$ . Therefore, there is a branch point in the unit circle.

**chisigma** · February 23rd 2010, 08:52 PM

A multivalued function is a complex function that for a given $z$ can assume several different values. Typical examples are 'nth root', logarithm, inverse circular functions and so on. A branch point of a multivalued function is a point in the complex plane from which depart two or more branches of a multivalued function. Let consider for example the multivalued function $f(z)= \sqrt{z}$ . The point $z=0$ is a branch point for it because, setting $z= \rho\cdot e^{i\cdot (\theta + 2k\pi)}$ , is...

$\sqrt{z}= \sqrt{\rho}\cdot e^{i\cdot (\frac {\theta}{2} + k\pi)}= \pm \sqrt{\rho}\cdot e^{i\cdot \frac {\theta}{2}}$ (1)

In (1) the sign '+' is for k even and the sign '-' for k odd and at different signs they correspond two different branches that have in common the point $z=0$ , that for this reason is called 'branch point'...

Kind regards

$\chi$ $\sigma$

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