Complex Analysis: Theory and Applications · Problem 10.8.2

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · joukowski-circles-ellipses-hyperbolas

Problem

Consider the Joukowski transformation w=z+1zw = z + \frac{1}{z}. (a) Show that the circles Cr={∣z∣=r:r>1}C_r = \{|z| = r : r > 1\} are mapped onto the ellipses u2(r+1r)2+v2(r−1r)2=1.\frac{u^2}{\left(r + \frac{1}{r}\right)^2} + \frac{v^2}{\left(r - \frac{1}{r}\right)^2} = 1. (b) Show that the ray r>0r > 0, θ=α\theta = \alpha is mapped onto a branch of the hyperbola u24cos⁡2α−v24sin⁡2α=1.\frac{u^2}{4\cos^2\alpha} - \frac{v^2}{4\sin^2\alpha} = 1.

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