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Thread: Recurrence relation question

  1. #1
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    Recurrence relation question

    Let $\displaystyle Z_{n+1}=Z_{n}^{2}+2i$ and $\displaystyle Z_0=0.5+02.i$. Find $\displaystyle Z^{10}$ value and hence deduce the $\displaystyle Z_n$ when $\displaystyle n \longrightarrow \alpha$

    here my calculations are
    $\displaystyle Z_1=(0.5+0.2i)^2+2i=(0.2+2.2i)$
    $\displaystyle Z_2=(0.2+2.2i)^2+2i=(-4.8+2.88i)$
    $\displaystyle Z_3=(-4.88+2.88i)^2+2i=(14.7456-25.648i)$
    .......

    Is that above answer is true?
    But I saw in other way like $\displaystyle Z_{n+1}=Z_{n}^{2} + \lambda $ and then $\displaystyle Z^2=r^2e^{i2\theta}$ bt I couldn't understand this..
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  2. #2
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    Quote Originally Posted by dhammikai View Post
    Let $\displaystyle Z_{n+1}=Z_{n}^{2}+2i$ and $\displaystyle Z_0=0.5+02.i$. Find $\displaystyle Z^{10}$ value and hence deduce the $\displaystyle Z_n$ when $\displaystyle n \longrightarrow \alpha$

    here my calculations are
    $\displaystyle Z_1=(0.5+0.2i)^2+2i=(0.2+2.2i)$
    (0.5+ 0.2i)^2= 0.5^2- 0.2^2+ (2(0.5)(0.2))i= 0.21+ .2i so (0.5+ 0.2i)+ 2i= 0.21+ 2.2i. You seem to have dropped the hundredths place and that might be important.

    $\displaystyle Z_2=(0.2+2.2i)^2+2i=(-4.8+2.88i)$
    $\displaystyle Z_3=(-4.88+2.88i)^2+2i=(14.7456-25.648i)$
    .......

    Is that above answer is true?
    But I saw in other way like $\displaystyle Z_{n+1}=Z_{n}^{2} + \lambda $ and then $\displaystyle Z^2=r^2e^{i2\theta}$ bt I couldn't understand this..
    If you think of a+ bi as represnting a point, (a, b) in the "complex plane", then it can also be written in polar coordinates with distance from the origin r and angle [itex]\theta[/itex]. Of course, $\displaystyle x= r cos(\theta)$ and $\displaystyle y= r sin(\theta)$ so $\displaystyle a+ bi= r cos(\theta)+ i r sin(\theta))$ But it is also true that [tex]e^{i\theta}= cos(\theta)+ i sin(\theta) so that can be written as $\displaystyle a+ bi= re^{i\theta}$. Then $\displaystyle (a+ bi)^2= r^2e^{i(2\theta)}f$, $\displaystyle (a+ bi)^3= r^3e^{i3\theta}$, etc.

    Unfortunately, that "polar form" doesn't work well with addition so you would have to change back each time to add the "2i".

    In fact, that recursion formula is the one used in calculating Julia sets and the Mandelbrot set so I don't see any method other than 'brute strength" calculation nor do I see that knowing $\displaystyle z_{10}$ will tell you very much about the limit as n goes to infinity.
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  3. #3
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    Thank you very much "HallsofIvy" now I have a clear answer. Thanks
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