I am working on debugging a program I made that would raise a complex number to a complex power Ex: (a+bi)^(c+di). I'm running into issues when trying to calculate the inverse tangent of (di/c) for the calculation of atan(d/c) in my final equation of Z = e^((a+bi)(ln(sqrt(c^2 + d^2) + i(atan(d/c))). Is this formula the correct way to go about finding a complex number to a complex power, and how would I calculate the atan of d/c by hand? Thank you!
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Well, ordinarily, when you want to calculate $z^w$, you need only a good hold on $\log(z)$. Are you sure you’re doing the right thing? – Lubin Mar 29 '21 at 18:25
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@Lubin I'm fairly sure I am, I found the formula I'm using in a seperate thread. This could be incorrect however. I can't seem to find a reliable source for ways to do this type of calculation anywhere https://math.stackexchange.com/a/3816175 – Davis Anderson Mar 29 '21 at 20:50
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For your second question:
Use the fact that $$\arctan z=-i\operatorname{artanh} iz$$ Letting $z=ix$, we find that $$\arctan ix=i\operatorname{artanh} x$$ where $\operatorname{artanh} x$ is the inverse hyperbolic tangent function.
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