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The Joukowski mapping transforms a circle into

an ellipse in the complex plane.

To prove the asymptotic optimality, we begin with a lemma due to Zarantonello, which

deals with the particular case where the ellipse reduces to a circle. This particular case is

important in itself.

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P#

1 1

Let be a circle of center the origin and radius

£Y )

( X

and let be a point of not enclosed by . Then

¡Y )

¡

(

§

T

a±— „ ®

i

X H

¬g0 T 0¡ ¦ QH 46 U 2

5I6

C4 P3 (

U ¡

0 0

¦§¥

¤

¤ PG

X W P GX X

the minimum being achieved for the polynomial .

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H

£ 6 A

£

See reference [168] for a proof.

Note that by changing variables, shifting, and rescaling the polynomial, then for any

circle centered at and for any scalar such that , the following min-max result 0 0

¡ ¡

holds:

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X H

V0 0¡ ¦ P H 356 U 2

¬

F4I6

C

0

²

U ¡

0

§

¦¤

¤ P $"G

6

X W P GX T

9²

Now consider the case of an ellipse centered at the origin, with foci and semi-

Xy (y

c

major axis , which can be considered as mapped by from the circle , with the

£¡Y )

(

¦

¦

convention that . Denote by such an ellipse.

%

T

y

´ A Av ¡˜¤

¥¦ ¥ X

' P

1

¦

Consider the ellipse mapped from by the mapping and

£¡Y )

( ¦

%

let be any point in the complex plane not enclosed by it. Then

¡

T

a± „ ®

i

0X H“ w H

H T

FI¦ 6 U

C4 0¡ 9QH U 6 2

8 P 75 0 H “ ¥ w H 0¥

0¥ H0 ¤

¤ ¤

¤ ¤

X W P GX

¬X

in which is the dominant root of the equation .

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¥¦

¥

¤

£ 6 A

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We start by showing the second inequality. Any polynomial of degree satis-

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