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Can a single valued analytic branch of logz be defined on the following sets?

April 26, 2017analytic branch defined Mathematics Valued

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Can a single valued analytic branch of logz be defined on the following sets?

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To get a single branch the argument needs to be from the interval (0, 2π] or (-π, π] or (a, a+2π] for any real number a. a) No. 1< |z| < 2 is a ring and arguments of complex numbers are from the interval [0, 2π] so when Argz = 0 is one branch and Argz = 2π another.