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)

Comments: ( Back to comment list | You must be logged in to post comments.)

re: computer assisted solution

Comment 2 of 2 |

(In reply to computer assisted solution by Charlie)

The three cycle can be verified by evaluating the compositions G(G(z)) and G(G(G(z))).

G(G(z)) = (G(z)+i)/(G(z)-i) = (z+1)/(z-1)

Then G(G(G(z))) = (G(G(z))+i)/(G(G(z))-i) = z

Since G(G(G(z))) = z this proves the values repeat with a period of three.

Posted by Brian Smith on 2016-04-13 23:15:18

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