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\item $\exp(x) = \sum_{n = 0}^\infty \frac{x^n}{n!}$.
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\item $\exp(x) \in G_0(A)$ with $\exp(x)^{-1} = \exp(-x)$.
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\item For any $y \in A$ commuting with $x$, $\exp(x + y) = \exp(x)\exp(y)$.
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\item Let $\ell: \complex \setminus (\infty, 0] \to \complex$ be the principal logarithm. If $\sigma_A(x) \subset \bracs{\lambda \in \complex| \text{Im}(\lambda) \in (-\pi, \pi)}$, then $\ell(\exp(x)) = x$.
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\item Let $\ell: \complex \setminus (-\infty, 0] \to \complex$ be the principal logarithm. If $\sigma_A(x) \subset \bracs{\lambda \in \complex| \text{Im}(\lambda) \in (-\pi, \pi)}$, then $\ell(\exp(x)) = x$.
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\end{enumerate}
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\end{proposition}
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\begin{proof}
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