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a) Find the maximum value of the module | z n - 2 n | in the disk | z | <= 2 for an integer n >= 1. b) Find all points in the disk | z | <= 2 where...

  1. a) Find the maximum value of the module | zn - 2n | in the disk | z | <= 2 for an integer n >= 1.

b) Find all points in the disk | z | <= 2 where the maximum of | zn - 2n | is attained.

c) Indicate how many such points of maximum of | zn - 2n | exist in the disk | z | <= 2.


Subject is complex analysis, using maximum modulus principle.

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( 1 )
By triangular inequality, we know
|1 21 + 2 2 ) &lt; 1211 + 1 2 2 )
= 151 2 ~
( )
U
Now | 2 7 - 2 ~ | = 1 2 ~ + ( - 2 2) |
101 = 1
By
1
&lt; 12 ~ | + 1 - 2 7 /
(';' s is I've real No )
UZ
....

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