I am not sure as to why $\exp(-i \pi x)$ not equal to $( \exp(-i \pi) )^x= (-1)^x$ .
2026-04-13 12:34:03.1776083643
Why is $\exp(-i\pi x)$ not equal to $(-1)^x$?
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The "normal" laws of exponents, as they apply to real numbers and real exponents, do not always hold with complex-valued exponents. So we can't necessarily apply the law $a^{bc} = \left(a^b\right)^c$ unless we know $b, c\in \mathbb R$.