B Predicting outputs of f(x)=(1+i)^x

[AaronQ]

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?

 

[Fightfish]

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.

 

[AaronQ]

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?

 

[fresh_42]

$f(n) = (1+i)^n = (\sqrt{2} \, e^{i \frac{\pi}{4}})^n = 2^{\frac{n}{2}} e^{i \frac{n \pi}{4}}$
The real and imaginary part can be found by $r e^{i \varphi} = r \cos{\varphi} + i r \sin{\varphi}$

Perhaps you'll first have a read here:
https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/

or on the Wikipedia page on complex numbers.

 

[AaronQ]

$f(n) = (1+i)^n = (\sqrt{2} \, e^{i \frac{\pi}{4}})^n = 2^{\frac{n}{2}} e^{i \frac{n \pi}{4}}$
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.