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Hi I formulated an equation that calculates powers of the imaginary unit.

enter image description here

Here it is actually written as a function of two complex variables.

enter image description here

Now because its a function, we can take the derivative to find the rate of change of such function.

Thus its possible to find the rate of change of powers of i^n.

enter image description here

The two complex variables are evaluated at the values shown in the original equation.

What this should give me is the rate at which powers of i change. If we look at the unit circle in the complex place then we see that the distance between any of the values of i^n is of the angle of pi/2 or 90 degrees.

Thus the rate of change should be 90 degree per n. 90/n or even the total angle divided by n.

But upon calculating the derivative above for any n value the result that I keep getting is pi/4 or 45 degrees instead of 90 degrees.

Can anyone shed some insight to this matter? Thanks.

Erock Brox
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    Imaginary square roots are not well-defined. This is not a function. – CyclotomicField Aug 12 '23 at 05:11
  • With the more familiar real numbers, we often talk of "the square root" but we really mean "the principal square root". We pick the positive one as preferred. This works quite nicely. Unfortunately, it is not so neat with the complex numbers, we can pick a principal one but not in a way that works very nicely. This has been discussed many times but I am struggling to find the best previous answer. Here is one: https://math.stackexchange.com/questions/664962/what-is-the-square-root-of-complex-number-i – badjohn Aug 12 '23 at 06:17
  • See this question as well: https://math.stackexchange.com/questions/1324582/why-is-sqrt-i-neq-i-sqrti/1324613#1324613 – badjohn Aug 12 '23 at 06:23
  • Erok, please do not put pictures containing formulas in your posts anymore. You should learn to use Mathjax in order to write formulas in an appropriate manner. – Angelo Aug 12 '23 at 06:52

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