Why is $-1=e^{\pi i }=e^{2\pi i\frac{1}{2}}\neq (e^{2\pi i})^{\frac{1}{2}}=1^{\frac{1}{2}}=1$ true?

Question

I thought there was this rule that $e^{xB}=(e^x)^B$?

Also what I don't understand is that $x^{\frac{1}{2}}$ is defined as the square root of $x$. And because the square root may also be in $\mathbb{C}$ it doesn't matter which number $x$ is because every complex number has a n-th root.

But for $1^{\frac{1}{2}}$ we have the roots $-1$ and $1$, so I could also write $-1$ on the right side then the Statement above would be true and not true at the same time.

But we said the Expression in the Question is correct and to write $-1$ is false. Can somebody explain the reason we choose one Version over another altough they are equivalent?

Answer 1

If one is working in real analysis, the expression $x^{1/2}$ ($x>0$) is by definition $\sqrt{x}$, which is defined to be the positive real number $y$ such that $y^2=x$. Hence one has $$ 1^{1/2}=1. $$

If one is working in complex analysis, $1^{1/2}$ can be viewed as the multivalued function $f(z)=z^{1/2}$ evaluated at $z=1$. In this context, since $f$ is multivalued, it is incorrect to write $1^{1/2}=1$.

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The identity $(e^x)^y = e^{xy}$ holds for real numbers $x$ and $y$, but assuming its truth for complex numbers leads to paradox like the one you have observed.

To quote Wikipedia:

Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions.

See Failure of power and logarithm identities for more examples.