Question

Feeling queasy? Then compute the first three iterates of Z²+i for the following starting complex number z: 5+6i and guess whether the iterates of this number are periodic or tend to infinity if you continued the iteration process. please type

First iterate:

second iterate

third iterate  

Answer 1

Hello,

I have tried to solve the question and write the solution as detailed as I can.

The numerical answers are-

First iterate: (-11)+6i

Second iterate: 3600-(1341)i

Third iterate: 1,11,61,719-(96,55,199)i

When the iterations are repeated, the iterates are not periodic. The sequence tends to infinity. Please check the attached images for the detailed solutions.

Given function is f (2) = z2 ti.. Starting complex number z = 5+ 61 ) f(x) = f(5+61) = (5+61)2 +i . f2) = 5 + 2x5X61 + (61)2 + i = 25 + 60° + (-36) +i = (-11 + Gli i f (5+ 6i) = (-11) + di ») f(-11 + 611) = (-11+611) +i if (-11 + 61 i) = (-11)² + 2X (-11% 6lit (61i)²ti = 21 + (-13421) - 372) + i = (12) - 372)) + i (1 - 1342) = 3600 - 13411 .f(-11+61i) = 3600 – 13411 3) +(360o - 134 11) = (36 00 - 341) + = (3600)2 – 2x.3600 X 1341i + (13411) = 1296 X104 - (96,55,200)i +(-17,98,281) +i + = 12,960,000 - 96,55,200 i - 17,98,281 + i = 1,11,61,719 - 96,55, 1991- if(3600 - 13411) = 1,11961,719 - 96255, 1998 These are the first three iterations for the given functiona Now, observe that Re [f(x)] is always positive and is much larger than the previous iteration and I'm [4 (2)]

Pace No Date is negative and much lesser than the previous iterations This implies that if we continue the iterations, then Re [f (2)] will tend to co and Im [ f (2)] will tend to -oo. ? Hence, the iterates are not periodic