= Regular perturbation of a simple algebraic root
{title2=$\lambda(\varepsilon)=\lambda_0+\lambda_1\varepsilon+\cdots$}
= Regular perturbation root
{synonym}
If $P(\lambda_0,0)=0$ and $P_\lambda(\lambda_0,0)\ne0$, the <implicit function theorem> gives a local root branch. For analytic coefficients it has an ordinary power series in the perturbation parameter. Its corrections follow by substituting that series into the polynomial and comparing coefficients. A vanishing leading root may require its first nonzero correction to capture a slow exponential mode.
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