Is logarithm the only analytic function such that Re[f(z)]=Re[f(|z|)]? Complex logarithm has an amazing property: its real part is rotationally invariant in the complex plane:
$$\Re\ln z=\Re\ln|z|.$$
But even squaring $\ln$ leads to a non-symmetric real part. So I wonder: is logarithm the only nonconstant analytic function with such property?
 A: If we're considering complex analytic functions, then yes. Consider such an $f(z)$. Then $g(z) = \exp(f(z))$ is an analytic function with the radial property $|g(z)| = g(|z|)$. It remains to show that $g(z) = cz^k$ for some constants $c,k$. Therefore $f(z) = \log(c) + k \log{z}$.
To rigorously prove that $g(z) = cz^k$, we employ the polar form of the Cauchy-Riemann equations: 
$$u_r = r^{-1} v_{\theta}, v_r = -r^{-1} u_{\theta},$$
where $g(r,\theta) = u(r,\theta) + iv(r,\theta)$ with $u,v$ purely real functions on $\mathbb{C}$. Then a radially symmetric $g(r,\theta)$ is written as $a(r) \exp(i b(\theta))$, where $a$ and $b$ are real functions of a single real variable. Plugging this into the Cauchy-Riemann equations yields $a_r = r^{-1} a b_{\theta}$. Since both $a,b$ are functions of a single variable, then the above expression must not depend on $\theta$; it follows that $b_{\theta} = k$ for some constant $k$, and hence $a(r) = c r^k$. Therefore $g(z) = c r^k e^{ik\theta} = cz^k$, as desired.
