1550251515-Classical_Complex_Analysis__Gonzalez_

268 Chapter 5

5.17 THEJOUKOWSKI FUNCTION w = 1/2(z + 1/z)

Letting z = rei^6 , w = u +iv, we have

u +iv= ~ (reiB + ~ e-iB)

which yields

u = ~ (r + ~) cos B, (5.17-1)

It follows that the circles lzl = r i= 1 (r constant) map into the ellipses

u2 v2

%(r + 1/r )2 + l/4(r - ~ )2 = 1 (5.17-2)

and. the circles Jzl = 1/r map into the same ellipses.

The semiaxes of the ellipses (5.17-2) are

and we find that c = .Ja^2 - b^2 = 1, so that the ellipses are confocal, the

foci being points (-1, 0) and ( +1, 0). As either r -t 0 or r -t oo, we have
a -t oo and b -t oo, and as r -t 1 we have a -t 1 and b -t 0. Hence both
the inside and the outside of the unit circle in the z-plane are mapped into
thew-plane with the segment (-1, 1] deleted (Fig. 5.21).
To the ray Arg z = () = const. corresponds the hyperbola
u2 v2
-----=1
cos^2 B sin^2 ()

(5.17-3)

y v

u

Fig. 5.21