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I'm doing a course in complex analysis - and in a question I come to a term that is $[\exp(2\pi i)]^f $ where $f$ is some fraction. I naively proceed with $[\exp(2\pi i)]^f=1^f=1$ which leads to an incorrect answer, where instead I should keep $[\exp(2\pi i)]^f = \exp(2 f \pi i)\ne 1$. This has confused me somewhat (and possibly embarrassingly) - is it incorrect to evaluate the exponential first?

For example, is it $[\exp(2\pi i)]^\frac{1}{4}=\exp(\frac{\pi}{2} i) = i$ or, is it $[\exp(2\pi i)]^\frac{1}{4}=1^\frac{1}{4} = 1$?

Trock
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2 Answers2

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Raising complex numbers to a power is problematic because there is an undeterminacy. Notice that

$$e^{i2\pi}=e^{i2k\pi}$$ for every integer $k$ so that

$$(e^{i2\pi})^f$$ could take infinitely many values

$$e^{i2k\pi f}.$$

When $f$ is irrational, the expression has no meaning and remains undefined.

When $f$ is rational, you can choose to consider all $q$ distinct values of

$$e^{i2k\pi p/q},$$ assuming the fraction irreducible.

You can also choose a particular "branch", by constraining the argument to certain range, such as $[0,2\pi)$. In this case

$$(e^{i2\pi})^f=(e^{i\,0})^f=e^{i\,0f}=1.$$

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Look what happens if you consider $$\left[e^{2 i\pi}\right]^f = \left[e^{i(2\pi + 2k \pi)}\right]^f, \ \ k\in\mathbb Z.$$

dfnu
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