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Amplification factor of a two-sided one-step stencil

For

\sum_{k = r}^{s} a_{k} u_{m + k}^{n + 1} = \sum_{k = r}^{s} b_{k} u_{m + k}^{n},

the Fourier convention

u (θ) = \sum_{m} e^{- im θ} u_{m}

gives

H (θ) = \frac{\sum _{k = r}^{s} b _{k} e ^{ik θ}}{\sum _{k = r}^{s} a _{k} e ^{ik θ}},

provided the denominator does not vanish.

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