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Math Help - Region of analyticity

  1. #1
    Junior Member platinumpimp68plus1's Avatar
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    Region of analyticity

    The region of analyticity of log(sqrt(z)). (using principle branch)

    I know the domain of log(z) is the complex plane minus the origin and the negative x-axis (C\{x+iy | x =< 0, y=0}). I'm not sure what domain of sqrt(z) produces this as an image, though. I know z --> sqrt(z) cuts angles in half... sooo would the domain be everything but the positive x-axis? I'm a little confused
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  2. #2
    Member mabruka's Avatar
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    Because you are using the principal branch of log, you take all the arguments of the numbers sqrt(z) from (-\pi.\pi] , therefore there is no z whose square root sqrt(z) lies on the negative real axis.


    So log(sqrt(z)) analicity region is \mathbb C-\{0\}
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  3. #3
    Junior Member platinumpimp68plus1's Avatar
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    Quote Originally Posted by mabruka View Post
    Because you are using the principal branch of log, you take all the arguments of the numbers sqrt(z) from (-\pi.\pi] , therefore there is no z whose square root sqrt(z) lies on the negative real axis.


    So log(sqrt(z)) analicity region is \mathbb C-\{0\}
    Ooh okay, I see. I kind of had a feeling it was that but was afraid I was missing something. Thank you!!
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  4. #4
    Member mabruka's Avatar
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    So you get more idea of whats happening see this applet:

    Math 132 Applet 4

    choose sqrt as a function and select principal branch.


    You will see how there isnt a single point z whose image sqrt lies on the negative real line.
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