perplexus.info :: Sequences : Complex Ratio and Number Resolution

Complex Ratio and Number Resolution (Posted on 2016-04-13)

Consider the function:
G(Z) = (Z+i)/(Z-i) for all complex numbers Z ≠ i, and:
The sequence {Z_n} is defined as:
G(Z_n-1) = Z_n, whenever n is a positive integer, and:
Z₀ = 1/137 + i

Find each of A, B, C and D, given that:
Z₂₀₁₄ = A+B*i, and:
Z₂₀₁₅ = C+D*i

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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computer assisted solution

| Comment 1 of 2

10 Real=1//137:Imag=1

20 for I=1 to 2015

30 a=real:c=real

40 b=imag+1:d=imag-1

50 real=(a*c+b*d)//(c*c+d*d)

60 imag=(b*c-a*d)//(c*c+d*d)

70 if i=2014 or i=2015 then

80 :print i,real;imag

90 next

Uses symbols a,b,c and d as in Wolfram Mathworld's article on complex division.

It finds

2014: 1 + 274*i

2015: 37538/37265 + 1/37265 * i

In fact the 2014th value is preceded by

2013: 1/137 + i

so actually it has been a 3-cycle since:

0: 1/137 + i

1: 1 + 274*i

2: 37538/37265 + 1/37265 * i

3: 1/137 + i

Posted by Charlie on 2016-04-13 14:37:15

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