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}

Read more about this topic:  Joukowsky Transform

Famous quotes containing the words general and/or transform:

    “There are two great rules in life, the one general and the other particular. The first is that every one can in the end get what he wants if he only tries. This is the general rule. The particular rule is that every individual is more or less of an exception to the general rule.”
    —Samuel Butler (1835–1902)

    “Americans, unhappily, have the most remarkable ability to alchemize all bitter truths into an innocuous but piquant confection and to transform their moral contradictions, or public discussion of such contradictions, into a proud decoration, such as are given for heroism on the battle field.”
    —James Baldwin (1924–1987)