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[Complex analysis] Contradiction in the definition of a branch
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[QUOTE="aheight, post: 6443587, member: 600309"] With regards to multi-valued functions, everything begins with ##\arg(z)## so good idea to have a good grasp of it. First, it's an infinite helical coil winding over the complex plane. But to get some practical handle on it, we can just look at a few windings keeping in mind they wind infinitely upward and downward. The plot shows three winding for ##z=re^{it}## with ##-3\pi\lt t\leq -\pi## red, ##-\pi\lt t\leq \pi## blue, and ##\pi\lt t\leq 3 \pi## green. And in the figure, there are three points on this 3-surface plot over every value of z. But in reality, since the coil are infinite, there are infinitely many but just for now, consider just the three in the plot. Now, this is the crucial question about branching: How can we "excise" a continuous "piece" of this infinite coil that is single-valued for all values of z except 0 and continuous in the interval ## \theta\lt t\leq 2\pi+\theta##?" That's easy: take any ## \theta\lt t\leq 2\pi+\theta## section of it. Right? By convention then we define the "principal" sheet as the section ##-\pi\lt t\leq \pi## and we name this section ##\text{Arg}(z)##. And keep in mind branches of multivalued functions are by their nature "discontinuous" on their branch-cuts. With regards to ##\text{Arg}(z)## the discontinuity is along the negative real axis. [ATTACH type="full" alt="argPlot.jpg"]276167[/ATTACH] [/QUOTE]
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[Complex analysis] Contradiction in the definition of a branch
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