Joukowsky Transform - General Joukowsky Transform

General Joukowsky Transform

The Joukowsky transform of any complex number to is as follows

$\begin{align} z &= x + iy =\zeta+\frac{1}{\zeta} \\ &= \chi + i \eta + \frac{1}{\chi + i \eta} \\ &= \chi + i \eta + \frac{(\chi - i \eta)}{\chi^2 + \eta^2} \\ &= \frac{\chi (\chi^2 + \eta^2 + 1)}{\chi^2 + \eta^2} + i\frac{\eta (\chi^2 + \eta^2 - 1)}{\chi^2 + \eta^2}. \end{align}$

So the real (x) and imaginary (y) components are:

$\begin{align} x &= \frac{\chi (\chi^2 + \eta^2 + 1)}{\chi^2 + \eta^2} \qquad \text{and} \\ y &= \frac{\eta (\chi^2 + \eta^2 - 1)}{\chi^2 + \eta^2}. \end{align}$