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Isn't it because log(z) is undefined when z = 0 + 0i?Prove f((z) = log z cannot be analytic on any domain D that contains a piecewise smooth simple closed curve γ that surrounds the origin?
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The question is controversial and depends from the definition of complex logarithm. From De Moivre's writing $z= r\ e^{i\ \theta}$, where $r$ is the 'modulus' and $\theta$ is the 'argument', we derive $\ln z = \ln r + i\ \theta$. The point of controversial is the precise definition of $\theta$. If we accept the approach of the German mathematician Berhard Riemann, then the function $\ln z$ can be analitycally extended to the whole complex plane with the only exception of the point z=0. For details see...Prove f((z) = log z cannot be analytic on any domain D that contains a piecewise smooth simple closed curve γ that surrounds the origin?
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An illustrative example of Riemann's analytic extension of the function $\ln z$ is based on the following pitcure...The question is controversial and depends from the definition of complex logarithm. From De Moivre's writing $z= r\ e^{i\ \theta}$, where $r$ is the 'modulus' and $\theta$ is the 'argument', we derive $\ln z = \ln r + i\ \theta$. The point of controversial is the precise definition of $\theta$. If we accept the approach of the German mathematician Berhard Riemann, then the function $\ln z$ can be analitycally extended to the whole complex plane with the only exception of the point z=0. For details see...
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