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The book uses the notation of a complex flow $F(z)=u-iv$, with a function $W=\phi + i\psi$ such that $F=\frac{dW}{dz}$.

The book gives the following example:

Choose $$W(z) =\frac{\Gamma}{2\pi i}\log(z-z_0)$$ where $\Gamma$ and $z_0$ are some constants. Then [...] $\psi$ is constant on any circle centered at $z_0$.

Shouldn't it be $\phi$ that's constant? The real part of $W$ is $\phi=\frac{\Gamma}{2\pi i}\log(|z-z_0|)$, which is constant on circles centred at $z_0$. The imaginary part, $\psi$ is a function of $\arg(z)$, which is certainly not constant on any circle.

The book also has the following diagram:

enter image description here

What am I missing here?

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    $\begingroup$ I think you forgot the factor of $-i$ in front of the log, assuming $\Gamma$ is real. $\endgroup$ – Ted Shifrin May 1 '17 at 2:08
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Let $z = x+iy$ be a complex number. Define $$ \ln z = \frac{1}{2} \ln \color{blue}{\left( x^{2} + y^{2} \right)} + i \text{Arg } (x + y) $$

When $\color{blue}{\left( x^{2} + y^{2} \right)}$ is constant, the $\color{blue}{real}$ part of the logarithm is constant.

On the circular domains, the real portion of the logarithm is constant.

A simple case is $$\ln e = 1, \qquad \ln (i e) = 1 + i \frac{\pi}{2}$$.

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