Let $h$ be a real valued harmonic function on the twice punctured plane $Ω=\Bbb C \setminus \{0, 1\}$. Show that there exist unique real numbers $a_0, a_1$ such that $$u(z)=h(z)−a_0 \log |z|−a_1 \log |z−1|$$ is the real part of a holomorphic function on $Ω$.

All I can think of is use Cauchy-Riemann equations, but I cannot go anywhere with the integral. Can someone help me?

  • $\begingroup$ The point of the $a_0$ and $a_1$ is that you can add appropriate copies of $a_0\log |z|$ and $a_1 \log|z-1|$ and get something that extends to a harmonic function on the whole plane. Once you've got something that's harmonic on the whole plane, use the CR equations to show that you can make it the real part of a holomorphic function. $\endgroup$ – user98602 Apr 13 '15 at 21:10
  • $\begingroup$ @MikeMiller so I am going to guess $a_0$ and $a_1$ ? I thought I a, supposed to calculated. Moreover, assume I found $a_0$ and $a_1$, the CR equations are used to construct a harmonic conjugate, right? $\endgroup$ – Khoa Apr 13 '15 at 21:17
  • 3
    $\begingroup$ Duplicate math.stackexchange.com/questions/1231650/… $\endgroup$ – A.Γ. Jul 24 '15 at 20:00

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