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:

    At that,
    his small size,
    keen eyes,
    serviceable beak
    and general truculence
    assure his survival—
    William Carlos Williams (1883–1963)

    The source of our actions resides in an unconscious propensity to regard ourselves as the center, the cause, and the conclusion of time. Our reflexes and our pride transform into a planet the parcel of flesh and consciousness we are.
    E.M. Cioran (b. 1911)