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Thanks for all the help these last couple of days mfb

When theta gets to 2pi it repeats. So at 0 and 2pi the function is multivalued thus not continuous?

Of course! Thanks a bunch man!

Homework Statement If f is analytic on the closed disc, show that for r<1 we have f(re^{i\phi}) = \frac{1}{2\pi} \int_{0}^{2\pi}\frac{f(e^{i\theta})}{1-re^{i(\phi - \theta)}}d\theta Homework Equations The Attempt at a Solution I tried using cauchy integral formula and end up...

For part a it would be all z such that 0≤arg(z)≤pi. Therefore im(z) > 0. Is the distance formula just the standard Euklidian distance formula?

It seems the equation fails for values of z where when you square them the argument is outside of 0 to 2pi

Θ will go from 0 to 2pi I am trying to find the reason why v(r,Θ) not continuous on C\{0} implies u does not have a harmonic conjugate on C\{0}. Is it because if it is not continuous then it's partial derivative will not exist on C\{0}, thus implying that that Cauchy Riemann equations will...

ok can we consider 1/|f(z)|? which is analytic since we were given f is analytic and |f|≥ 1 on the whole complex plane? Then 1/|f(z)| ≤ 1 for all z. By liouville's theorem 1/f(z) is constant. Am i warmer?

If you don't mind, could you explain where the problem with v = Θ comes in? Also, since v = Θ is not continuously defined on C\{0} would that mean that it doesn't have a partial, and therefore cannot satisfy cauchy riemann, which in turn is the reason why u cannot have a harm conj on C\{0}?

Homework Statement Let f(z) be entire and let |f(z)| ≥ 1 on the whole complex plane. Prove f is constant.Homework Equations Theorem 1: Let f be analytic in the domain D. If |f(z)| = k, where k is a constant, then f is constant. Maximum Modulus Principle: Let f be analytic and non constant...

then wouldn't part a be true for all z then? It seems obvious that sqrt(z^2) = z for all z. Why would that not be the case?

Homework Statement Find harmonic conjugate of u(x,y)=ln(x^2+y^2) and specify the region it is defined then show u has no harm conj on C\{0} Homework Equations The Attempt at a Solution Ok so i found the harmonic conj by converting to polar and found it to be v(r,Θ) = Θ. I am...

For part a do we have to consider the principal nth root? Or particular branches of the square root function? In otherwords where this function is not multivalued?

Homework Statement Let f(z) = sqrt(z) be the branch of the square root function with sqrt(z) = (r^1/2) (e^iΘ/2), 0≤Θ<2\pi, r > 0 (a) for what values of z is sqrt(z^2) = z? (b) Which part of the complex plane stretches, and which part shrinks under this transformation? Homework...

Homework Statement Let X and Y be rv's with joint pdf f(x,y) = 6(1-y) for 0≤x≤y≤1 and 0 elsewhere find Pr(X≤3/4, Y≤1/2) Homework Equations The Attempt at a Solution Ok I am having trouble with finding the right limits of integration for dependent variables. If we let the...