Raising to a complex power

Raising a number to a complex exponent can be tricky, especially if you want to understand what is really going on.

In fact, raising to any non-integer power requires one to let go of the idea that exponentiation is just repeated multiplication. There is more that is going on.

First of all, I would like to introduce this identity:

Actually, the general form is this:

This identity is very important for complex exponents. If you've fully understood the concepts of real exponentiation and logarithms, then this makes sense. If not, here's the proof.

First, assume that the identity is true. This means that whatever manipulations we apply to both sides should yield us something that is still coherent. Both sides have to be equivalent. If at any point both sides are obviously not equal, then our assumption is wrong, and the statement at the beginning is incorrect.

We then begin the manipulation:

We see that the result of the manipulation is clearly valid, and hence our assumption was correct.

You could also swap out the base of the logarithm for any other real number, since this applies for all real numbers:

This makes sense, too, because logarithms are the inverse of exponentiation. Applying them to the same number cancels out.

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Now we can start looking at how the complex powers can get manipulated.

At this point, we should also know about this formula, better known as Euler's formula:

We can then proceed to do this:

Now we can express the number in Cartesian form,

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Let's try some examples.

#math #complex #exponentiation #imaginary #power #complexnumbers