In the following equation $\{\beta_i\}_{i=1}^N$ and $\{\alpha_i\}_{i=1}^N$ are non-zero complex numbers: $\sum_{i=1}^N \beta_i e^{\alpha_i t} = 0$. I would like to know if the Lebesgue measure of the set $\{t\in \mathbb{R}\}$ such that the above equation holds is zero.
2026-04-05 19:38:49.1775417929
Lebesgue Measure of the set of roots of a complex exponential equation
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The set of zeros of a non-constant analytic function is discrete, therefore at most countable and of Lebesgue measure $0$.