I'm doing a problem related with Milne-Thompson theorem which tells that: "A cylinder of radius $a$ is immersed in a counter-clockwise whirlpool, which we model here as a potential vortex of intensity $\Gamma$ whose center is at a distance $L$ from the axis of the cylinder". First it is necessary to find the potential associated to the vortex which I got that:
$f(z) =-\dfrac{i\Gamma}{2\pi}\ln(z+L)$
The Milne-Thomson theory says that:
$f_{c}(z)=f(z)+\bar{f} \left ( \dfrac{a^2}{z} \right)$
So it is necessary to get the complex conjugate of the function. My problem is that I get two different results $\bar{f}(z)=\dfrac{i\Gamma}{2\pi}\ln(z+L)$ (applying properties of conjugates that I saw and through Wolfram) and $\bar{f}(z) =-\dfrac{i\Gamma}{2\pi}\ln(z+L)$ (using $z=re^{i\theta}$, solving and then applying the conjugate definition). So I'm doing something wrong in one of the methods and I would like to know what is the correct solution. Thank you for your help!
The logarithmic function is multi-valued in the complex plane, which makes any answer correct only up to an integer multiple of $2\pi i$, unless one somehow specifies the branch to use.
Neither of two answers can be correct, because they present a holomorphic function, while $\bar f$ is anti-holomorphic.
Suppose we use the principal branch of logarithm, which is real on the positive real axis. The reflection principle says that $\overline{f(z)} = f(\bar z)$. Hence,
$$\overline{f(z)} =\dfrac{i\Gamma}{2\pi}\ln(\bar z+L)$$
For other branches, there may be $+2\pi i k$ with some integer $k$.