# Tagged Questions

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### Derivatives, Cauchy-Riemann Equations [on hold]

Given the function, $w=z^4$ and I want to find the following solutions for this equation, Find real functions u and v such that w=u+iv Show that Cauchy-Riemann equation holds at all points in the ...
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### Need help about harmonic functions!

I have trouble on solving the following problem: Show that there doesn't exist a non-constant function $u$ such that $u$ is harmonic on C and for $z=x+iy$ in C that $u(z)>4x^2+9y^2+1$.
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### Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?
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### Harmonic Conjugate in Star Domain

I have been given that $u(x,y)$ is a harmonic function on a star shaped domain $D$. I have to show that it has harmonic conjugate $v(x,y)$ on same domain given up to additive constant by ...
### $u(x,y)$ harmonic and bounded in punctured disc; show $0$ is a removable singularity [duplicate]
I'm working on a problem from p. 166 of Lars Ahlfors' Complex Analysis: If $u$ is harmonic and bounded in $0 < |z| < \rho$, show that the origin is a removable singularity in the sense that ...