1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 267

Next, letting C = Bo-B~/4B 2 , (w-C)/B 2 = W,(+B 1 /2B 2 = Z, we get

w = z^2 (5.16-3)

an integral rational function with a double zero at the origin and a double

pole at oo.

2. Suppose that bo +biz+ b 2 z^2 = b 2 (z - /3 1 )(z - /32), /3 1 -=/= /32. Then
   we have

Ao + Aiz + A 2 z^2

w=------

(z - /31)(z - /32)

Letting z = /31 + 1/(, we obtain

W= ~---.,.-'----'--'-----'---.:.,_;-Ao+ Al(/31+1/() + A2(/31+1/()^2

(l/()(/31 - /32+1/()

B2(^2 +Bl(+ Bo

=

(/31 -/32)( + 1

Now if we put /31 - /32 = 1, 1( + 1 = e, or ( = (e - 1)/1, we get

B2[(e -1)hJ2 + Bl[(e-:-l)hl +Bo

w= ~~-~-'---~-~-'---

=

and by the substitutions

w-C1

2'\/'CoC2 = W,

the last equation becomes

e

w =! (z + ..!.. ) =! z2 + 1

2 z 2 z

a rational function with zeros at i and -i and poles at 0 and oo.

(5.16-4)

The normal form (5.16-4) is of interest in fluid dynamics, where it is

usually known as the Joukowski transformation. It can also be written

in the form

W-l (Z-1)

2

W+l = Z+l

(5.16-5)

In the next section we discuss the mapping defined by this function.
Further details concerning the rational function of order 2 can be found
in [12] and [13].