# Zhukovskii function

The rational function

$$w=\lambda(z)=\frac12\left(z+\frac1z\right)$$

of the complex variable $z$. It is important for its applications in fluid mechanics, which were discovered by N.E. Zhukovskii (see [1], [2]), particularly in constructing and studying the Zhukovskii profile (Zhukovskii wing). Suppose that a circle $K$ is given in the $z$-plane passing through the points $z=\pm1$ (Fig. a), together with a circle $K'$ touching $K$ on the outside at $z=1$, with centre $\alpha$ and radius $\rho$. Under the mapping $w=\lambda(z)$, the image of $K'$ is a closed curve $L'$ with a cusp at the point $w=1$, touching an arc of the circle $L$ (the image of $K$) at that point; this image is represented in Fig. band is the Zhukovskii profile.

Figure: z099280a

Figure: z099280b

The function $w=\lambda(\rho t+\alpha)$ maps the exterior of the unit circle in the $t$-plane to the exterior of $L'$. To obtain a Zhukovskii profile of a more general shape and disposition, the generalized Zhukovskii function is applied (see [3], [4], [5]):

$$w=\frac12(a-b)z+\frac12(a+b)\frac1z,\quad a>b>0.$$

#### References

 [1] N.E. Zhukovskii, "Collected works" , 2. Hydrodynamics , Moscow-Leningrad (1949) (In Russian) [2] N.E. Zhukovskii, "Collected works" , 6. The theoretical foundations of flying , Moscow-Leningrad (1950) (In Russian) [3] A.I. Markushevich, "Theory of functions of a complex variable" , 1–2 , Chelsea (1977) (Translated from Russian) [4] L.J. [L.I. Sedov] Sedov, "Two-dimensional problems in hydrodynamics and aerodynamics" , Acad. Press (1965) (Translated from Russian) [5] N.E. Kochin, I.A. Kibel', N.V. Roze, "Theoretical hydrodynamics" , 1 , Interscience (1964) (Translated from Russian)