1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

6.6 • THE THEOREMS OF MORERA AND LJOUVILLE, AND EXTENSIONS 247

t Corollary 6.4 Let P be a polynomial of degree n 2: 1. Then P can be expressed
as the product of linear factors. That is,

P (z) = A (z -z1) (z -z2) · · · (z - z,.),

where z 1 , z2,... , Zn are the zeros of P, counted according to multiplicity, and
A is a constant.

-------.. EXERCISES FOR SECTION 6.6

1. Factor ea.ch polynomial as a product of linear factors.
   (a) P(z) = z^4 +4.
   (b) P(z) =z^2 +(l +i)z+5i.
   (c) P(z) = z^4 - 4z^3 +6z^2 - 4z+5.
   (d) P(z) = z^3 - (3+3i)z^2 +(-1+6i)z+ 3 - i. Hint: ShowthatP(i) =0.


2. Let f (z) = az" + b, where the region is the disk R = {z: lzl ~ 1}. Show that
   max I/ (z)I = lal + lbl.
   1•151


3. Show that cos z is not a bounded function.


4. Let f (z) = z^2 • Evaluate the following, where R represents the rectangular region
   defined by the set R = {z = x +iy: 2 ~ x ~ 3 and 1~y~3}.
   (a) maxi/ (z)I.
   
   
   • ER
     (b) min I/ (z)I.
     •ER
     (c) maxRe[/(z)].
   
   
   • ER
     (d) min Im[! (z)].
     •ER