1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 573

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Fig. 8.6

Theorem 8.27 (Jensen-Walsh). Every nonreal zero of the derivative of
a polynomial P(z) with real coefficients lies in or on at least one of the
Jensen circles of P(z).

Proof Suppose that a= (J + h (-y i= 0) is a nonreal zero of P(z). Then 0:
is also a zero of the same polynomial, and since

1 z - 0:

z -a = lz -al^2

(x - (J) - i(y - -y)

= -'-----'---'---'"-'-

lz -a 12

where z = x + iy i= a, we have

Im -^1 - = -(y - 'Y)

z -a lz-al^2

Similarly,

Im -^1 - = -(y + 'Y)

z - 0: lz -0:12

so that

Im(-1-+ _1_) = -2y[(x-(J)2+y2--y2]

z-a z-a lz-al^2 lz-O:l^2

On the other hand, if a is a real zero, we obtain

and

1 z-a (x-a)-iy

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

z - a lz -al^2 lz -al^2

1 -y

Im--·=~~~

z -a lz-al^2

(8.10-2)

(8.10-3)

Since the equation of the Jensen circle corresponding to a = (J + i-y is

(x - (3)^2 + y^2 = -y2, it follows from (8.10-2) and (8.10-3) that whenever z