(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find where the function: [tex]f(z)=Log(z-2i+1)[/tex] is analytic and where it is differentiable.

2. Relevant equations

Cauchy-Riemann equations?

3. The attempt at a solution

Here's where I am so far:

[tex]Log(z-2i+1)=Log((x+1)+i(y-2))=ln(\sqrt{(x+1)^{2}+(y-2)^{2}})+iArg(z)[/tex]

since I'm only looking at the principal value of the logarithm, [tex]0<\theta\leq2\pi[/tex] (this is the text book's choice of principal arguement), then

[tex]ln(\sqrt{(x+1)^{2}+(y-2)^{2}})[/tex] will be discontinuous at [tex]x=-1[/tex] and [tex]y=2[/tex]... and the function is undefined everywhere on the positive real axis (because of the choice of argument). So, [tex]f(z)[/tex] is non-differentiable at [tex]z=-1+2i[/tex] and everywhere in the positive direction on the real axis extending from the point [tex]z=-1[/tex], because that is the "center" of my mapping.

Now, I'm not sure how to show that it is differentiable everywhere else... Am I supposed to apply the cauchy-riemann equations to [tex]u=ln(\sqrt{(x+1)^{2}+(y-2)^{2}})[/tex] and [tex]v=Arg(z)[/tex]? If so, how do I take a partial derivative of [tex]Arg(z)[/tex]?

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# Homework Help: Complex logaritm question

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