A random question for mathematicians regarding i and the Fibonacci sequence.
#1
What is the Fibonacci root of i, or to put it another way, what is the equation of x when       
 
 \( \Large \quad \quad      {({1+\sqrt 5\over 2})^x-(-1)^x ( { 2\over 1+\sqrt 5})^x \over \sqrt 5}=n \)  

?

All I need is a formular and I am capable of just entering the eqution and i straight into Google calculator to get a result.

I'm interested as decimal numbers entered into the Fibonacci function manly get complex results which shows a link between the function and i, and suggests that the Fibonacci root of i might not be complex and might be of some importance.

I know this isn't really to do with Tetration, but I've asked this across the internet and have not have any replies and I'll be greatfull for any help.

Thanks.
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#2
(08/07/2011, 11:17 PM)robo37 Wrote: What is the Fibonacci root of i, or to put it another way, what is the equation of x when ((1+sqrt 5)/2)^x-((-1)^x/((1+sqrt 5)/2)^x)))/sqrt 5=n?
You did not rationalize your denominator.
The numerical solutions to Fibonacci(x)=i are
x ~ -.42709179162288762 - .22105741463869769 * I
and
x ~ 1.67226840550786450 + 3.2642638352534426 * I.
I do not know of any closed forms for those numbers.
For an inverse to the Fibonacci function, there are no result found in terms of standard mathematical functions.
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
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