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Thread: Definition of a branch point

  1. #1
    Super Member Deadstar's Avatar
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    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?)
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  2. #2
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    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.
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  3. #3
    MHF Contributor chisigma's Avatar
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    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

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