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Help on Branch Points

  1. Jul 16, 2008 #1


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    [itex](1-z^3)^{1/2}[/itex] has three branch points at [itex]z=1, z=\omega[/itex]and [itex]z=\omega^2[/itex] where [itex]\omega = e^{2\pi i/3}[/itex]

    The branch points are the zeros of the function correct?

    So why are not [itex]z=\omega^4[/itex], [itex]z=\omega^6[/itex], [itex]z=\omega^8[/itex] etc also branch points as they are zeros of the function?
    [itex](e^{2\pi i/3^3})^6[/itex] =[itex]e^{2\pi i^6}[/itex] = [itex]e^{12\pi i}[/itex] = 1 so all even numbered powers of [itex]\omega[/itex] are zeros of the function

    also why does the latex code move around vertically like this?
  2. jcsd
  3. Jul 17, 2008 #2


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    Hi BWV! :smile:

    [tex](\omega^{n})^3 = 1[/tex] for any n:

    [tex](\omega^{n})^3 = (e^{\frac{2n\pi}{3}})^3 = e^{2n\pi} = (e^{2\pi})^n = 1[/tex] :smile:

    If you mean why doesn't it stay on the same line as ordinary text, the answer is:

    :biggrin: 'cos it doesn't! :biggrin:

    But you can make it stay on the line by using [noparse][itex] and [/itex] instead of [tex] and [/tex][/noparse] ("itex" stands for "inline tex") … but that has the disadvantage that it squeezes it vertically into the line, so fractions and powers get squashed. :mad:

    Alternatively, you can type the whole thing in latex, enclosing each bit of ordinary text inside the brackets of "\text{}":smile:
  4. Jul 17, 2008 #3


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    [itex]\omega= e^{2\pi /3}[/itex] is the "principal cube root" of 1. That is, [itex]\omega^3= 1[/itex]. We don't talk about [itex]\omega^4[/itex] because [itex]\omega^4= (\omega^3)(\omega)= (1)(\omega)= \omega[/itex]. Similarly, [itex]\omega^6= (\omega)^3(\omega^3)= (1)(1)= 1[/itex] and [itex]\omega^8= (\omega)^3(\omega^3)(\omega^2)= (1)(1)(\omega^2)= \omega^2[/itex].
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