# Definition of a branch point

• Feb 23rd 2010, 10:02 AM
Definition of a branch point
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?) $\displaystyle z$ is a point(s?) of an analytic function $\displaystyle f$ such that $\displaystyle f$ is holomorphic everywhere except at $\displaystyle z$? (or perhaps near z?)
• Feb 23rd 2010, 10:56 AM
shawsend
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 $\displaystyle f(z)=\sqrt{z}$ starting at the point $\displaystyle z=1$ and analytically continuing the function around the unit circle using the differential equation $\displaystyle \frac{df}{dt}=1/2 i f(t)$ (just differentiate the function and let $\displaystyle z=e^{it}$). Upon integrating from zero to $\displaystyle 2\pi$, $\displaystyle f(0)\ne f(2\pi)$. Therefore, there is a branch point in the unit circle.
• Feb 23rd 2010, 08:52 PM
chisigma
A multivalued function is a complex function that for a given $\displaystyle z$ 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 $\displaystyle f(z)= \sqrt{z}$. The point $\displaystyle z=0$ is a branch point for it because, setting $\displaystyle z= \rho\cdot e^{i\cdot (\theta + 2k\pi)}$, is...

$\displaystyle \sqrt{z}= \sqrt{\rho}\cdot e^{i\cdot (\frac {\theta}{2} + k\pi)}= \pm \sqrt{\rho}\cdot e^{i\cdot \frac {\theta}{2}}$ (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 $\displaystyle z=0$, that for this reason is called 'branch point'...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$