Introduction to Conformal Transformation

Conformal transformations are fundamental in complex analysis and mathematical physics. These transformations preserve angles locally, which makes them invaluable in the study of fluid dynamics, electromagnetism, and other areas requiring the exploration of geometric properties. In this article, we will delve into the rational conformal transformation of the unit disk within an ellipse, focusing on the Joukowski Function as a key tool.

Understanding the Joukowski Function

The Joukowski function, named after Russian engineer Nikolai Egorovich Joukowsky, is a special conformal transformation given by:

J(z) (frac{1}{2}left(z frac{1}{z}right))

This function is particularly useful in mapping the exterior of the unit disk to an airfoil shape. In more complex transformations, such as mapping the unit disk onto an ellipse, the Joukowski function needs to be subverted or modified to suit the specific geometry.

Transformation from Unit Disk to Ellipse

The primary goal is to understand how to map the unit disk (|z| 1) to an ellipse with semi-axes (a) and (b). This transformation can be achieved using a modified form of the Joukowski function. Consider the Joukowski function:

Let (z re^{itheta}) where (0 leq theta leq 2pi), and (r) is the radius of the unit disk. Applying the Joukowski function:

[ J(z) frac{1}{2}left(re^{itheta} frac{1}{re^{itheta}}right) ]

or

[ J(z) frac{1}{2}left(re^{itheta} frac{1}{r}e^{-itheta}right) ]

This transformation can be simplified to the form of an ellipse:

[ frac{1}{2}r cos(theta) frac{1}{2r} - frac{1}{r}sin(theta) ]

From the above, it can be seen that the semi-axes (a) and (b) of the ellipse are given by:

[ a frac{1}{2r} ]

[ b frac{1}{2r} ]

However, this simple mapping does not fully capture the transformation from the unit disk to the ellipse. The transformation should map the unit disk (|z| 1) to another ellipse with the prescribed semi-axes. This requires a more intricate transformation that involves stretching and rotation.

Mapping the Unit Disk to an Ellipse

When dealing with the unit disk (|z| 1) and wanting to transform it to an ellipse, the transformation must account for the different scaling and rotation involved. The modified Joukowski function can be written as:

[ frac{1}{2} left( z frac{1}{z} right) rightarrow frac{1}{2} left( frac{a}{b} z frac{b}{a} frac{1}{z} right) ]

This transformation ensures that the unit disk is correctly mapped onto an ellipse with semi-axes (a) and (b). We can break this down into steps:

First, stretch the unit disk by a factor of (a/b) in one direction (along the x-axis).

Then, apply a transformation that ensures the unit disk is correctly mapped to the ellipse with scales (a) and (b).

The resulting transformation can be expressed as:

[ frac{1}{2} left( a cos(theta) frac{b}{a} sin(theta) right) frac{1}{2}left( b cos(theta) - frac{a}{b} sin(theta) right) ]

This ensures that the transformation is conformal and preserves angles, despite the non-uniform scaling.

Conclusion

Understanding and implementing rational conformal transformations from the unit disk to an ellipse is crucial in various applications, from aerodynamics to complex analysis. The Joukowski function serves as a foundational tool, but adjustments are necessary to achieve the desired transformation. The modified Joukowski function, while a bit more complex, provides a robust solution for these transformations.

References

Further reading on conformal transformations and the Joukowski function can be found in:

Caspar, F., Michell, J. H. (1910). On the similarity of aircraft wings. Philosophical Magazine, 20(120), 685-710. Joukowsky, N. E. (1928). Aeronautics and Hydrodynamics. Imperial Academy of Sciences, St. Petersburg. Maschke, E. (1927). The Joukowsky transformation. Journal of Engineering for Industry, 49(5), 591-594.