My Coding > Mathematics > Complex numbers > Find (a + bi)^(c + id)

Find (a + bi)^(c + id)

How to calculate the complex power of a complex number? In this tutorial, I will show you the general equation for this.

\[ (a + bi)^{c + id} = ? \\ r = \sqrt{a^2 + b^2}; \Theta = \theta + 2\pi n = Arg(a, b) \\ (a + bi)^{c + id} = (r\times e^{i\Theta})^{c + id} = r^{c + id}\times e^{i\Theta(c + id)} = \\ = r^c\times r^{id}\times e^{ic\Theta}\times e^{i^2d\Theta} \]

Now we need to apply the following facts and tricks:

\[i^2 = -1; r^{id} = e^{\ln(r^{id})} = e^{id\ln(r)}\]

and we will have:

\[ = r^c\times e^{id\ln(r)}\times e^{ic\Theta}\times e^{-d\Theta} = \\ = r^c\times e^{i(c\Theta + d\ln(r))}\times e^{-d\Theta} = \\ = r^c\times e^{-d\Theta}\times [\cos(c\Theta + d\ln(r)) + i\sin(c\Theta + d\ln(r))]\]

That it! The final answer is:

\[(a + bi)^{c + id} = \\=r^c\times e^{-d(\theta + 2\pi n)}\times [\cos(c(\theta + 2\pi n) + d\ln(r)) + i\sin(c(\theta + 2\pi n) + d\ln(r))]\]

For more detailed explanation, please go to watch our video:


Published: 2023-05-10 00:23:48
Updated: 2023-05-10 00:52:37

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