In applied mathematics, the **Joukowsky transform**, named after Nikolai Zhukovsky is a conformal map historically used to understand some principles of airfoil design.

The transform is

where is a complex variable in the new space and is a complex variable in the original space. This transform is also called the **Joukowsky transformation**, the **Joukowski transform**, the **Zhukovsky transform** and other variations.

In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils. A **Joukowsky airfoil** is generated in the *z* plane by applying the Joukowsky transform to a circle in the plane. The coordinates of the centre of the circle are variables, and varying them modifies the shape of the resulting airfoil. The circle encloses the point = −1 (where the derivative is zero) and intersects the point = 1. This can be achieved for any allowable centre position by varying the radius of the circle.

Joukowsky airfoils have a cusp at their trailing edge. A closely related conformal mapping, the **Kármán–Trefftz transform**, generates the much broader class of Kármán–Trefftz airfoils by controlling the trailing edge angle. When a trailing edge angle of zero is specified, the Kármán–Trefftz transform reduces to the Joukowsky transform.

Read more about Joukowsky Transform: General Joukowsky Transform, Velocity Field and Circulation For The Joukowsky Airfoil, Kármán–Trefftz Transform

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