In the complex field, is this function Harmonic?

oab729
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Homework Statement


Without directly verifying via the laplace equations, explain why Log|z| is harmonic in the punctured complex plane.

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The Attempt at a Solution


I thought it was because Log z is analytic on the complex plane except for the nonpositive real axis, so Log |z| would be analytic and hence harmonic since any |z| turns z into a positive real (for z=/= 0), hence Log |z| is like Log z for positive reals. But, if it's that, why would they ask to show its harmonic and not analytic.
 
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Do you have the theorem that if f=u+iv is analytic then u is harmonic?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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