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The potential flow theory states that $$ F(z) = \phi + i\psi $$ where $z = x + iy$ In case of a source/sink, we have $$ F(z) = \frac{Q}{2\pi}\ln z\tag{1} $$ $$ \phi(x,y) = \frac{Q}{2\pi}\ln (\sqrt{x^2 + y^2}) \quad \text{or}\quad \phi(r,\theta) = \frac{Q}{2\pi}\ln(r)\tag{2} $$ $$ \psi(x,y) = \frac{Q}{2\pi}\arctan{\frac{y}{x}} \quad \text{or}\quad \phi(r,\theta) = \frac{Q}{2\pi}\theta\tag{3} $$ how do we go from (1) to (2) and from (1) to (3)?

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This is a matter of the definition of the complex logarithm. If $z = r e^{i \theta}$ for $\theta \in (-\pi, \pi],$ the principal value of the complex logarithm of $z$ is defined as \begin{equation*} \mathrm{Log}(z)= \mathrm{ln}(r) + i\theta. \end{equation*} So given $z = x + iy,$ \begin{equation} F(z) = \frac{Q}{2\pi}(\mathrm{ln}(r) + i \theta), \end{equation} where $r = |z| = \sqrt{x^2 + y^2}$ and $\theta = \mathrm{Arg}(z)$. Comparing to \begin{equation} F(z) = \phi + i \psi \end{equation} and equating the real and imaginary parts gives $(2)$ and $(3)$.

Note also that the formula $\mathrm{Arg}{(x + iy)} = \mathrm{arctan}{\frac{y}{x}}$ is only valid for $x > 0$. A more general formula makes use of the $\mathrm{atan2}$ function.

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    $\begingroup$ many thanks @Zac. For the sake of interest, what is the atan2 function. Could you provide a insighful link? $\endgroup$ – ecjb Jan 25 at 12:33
  • $\begingroup$ The Wikipedia page here on the $\mathrm{atan2}$ function gives a reasonable explanation. A problem with using the $\mathrm{arctan}$ formula is that the range of $\mathrm{arctan}$ is $(- \pi/2, \pi/2)$, so it will never give a correct argument for a complex number whose real component is less than zero (for example $-1 + i$.) The $\mathrm{atan2}$ function fixes this by adding or subtracting $\pi$ as appropriate. Often things will still work out if we stick with the $\mathrm{arctan}$ formula, but it can be good to know about its shortcomings! $\endgroup$ – Zac Jan 25 at 13:20

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