B Predicting outputs of f(x)=(1+i)^x

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The discussion focuses on the function f(x)=(1+i)^x and the exploration of its integer inputs, revealing patterns in both natural and negative numbers. Participants suggest converting the function into polar form to simplify the extraction of real and imaginary parts, leading to a parametric equation resembling a logarithmic spiral. A layman's explanation is requested, emphasizing the use of complex number principles to derive these parts. Resources like physics forums and Wikipedia on complex numbers are recommended for further understanding. The conversation highlights the mathematical intricacies of predicting outputs from complex functions.
AaronQ
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I got bored a while back and deiced to create a table of the integer inputs of f(x)=(1+i)^x and I noticed quiet a few patterns which I am trying to catalog here, although most of my work so far deal with Natural inputs, all patterns continue into the negative, see here, I was wondering if anyone on the forum had any ideas on and possible ways or equations that I could use to predict future real parts or imaginary numbers?
 
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The graph that you have is probably the logarithmic spiral. In order to get the real and imaginary parts of your equation, first convert it into polar form so that you can exponentiate it easily. You should end up with a parametric equation of a spiral.
 
Fightfish said:
The graph that you have is probably the logarithmic spiral. In order to get the real and imaginary parts of your equation, first convert it into polar form so that you can exponentiate it easily. You should end up with a parametric equation of a spiral.
Sorry a lot of that went over my head, could you give me it in more of laypeople terms?
 
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fresh_42 said:
##f(n) = (1+i)^n = (\sqrt{2} \, e^{i \frac{\pi}{4}})^n = 2^{\frac{n}{2}} e^{i \frac{n \pi}{4}}##
The real and imaginary part can be found by ##r e^{i \varphi} = r \cos{\varphi} + i r \sin{\varphi}##
Thank you that should work.
 
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