De moivre theorem for irrational exponents

Question

De moivre theorem works well for integer exponents as well as rational exponents. But what if the exponent is irrational. Will it still work and if so how many values will it give. If it work for irrational exponents plz show me the way to prove de moivre theorem for irrational exponents.

Answer 1

You run into this problem concerning branches. Indeed, there become an infinite amount of solutions to problems like $e^\pi$ when we consider all possibilities. Thus, we isolate the solutions as follows:

$$1=\cos2k\pi+i\sin2k\pi\quad k\in\mathbb Z$$

So,

$$(1e)^\pi=e^\pi(\cos2k\pi^2+i\sin2k\pi^2)$$

And when $k=0$, we call it the primary branch. Of course, we are usually concerned with the primary branch.

This is why complex exponents get really tricky.

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For rational exponents, notice that

$$1^q=\cos2kq\pi+i\sin2kq\pi$$

And eventually, we will repeat due to periodic nature of trig functions.