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We generalise power series to Laurent series by also allowing negative exponents, \(\displaystyle \sum _{k= 0}^\infty c_k (z-z_0)^k \ \longrightarrow \ \sum _{k= -\infty }^\infty c_k (z-z_0)^k \, . \) We do not do this arbitrarily, there is a close connection with the functions described by this: power series describe functions that are holomorphic on a circle around \(z_0\) , Laurent series describe functions that are holomorphic on an annulus around \(z_0\) .

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Laurent Series

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  • 发表日期：

    2026年

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  • Christian Karpfinger

摘要

We generalise power series to Laurent series by also allowing negative exponents, \(\displaystyle \sum _{k= 0}^\infty c_k (z-z_0)^k \ \longrightarrow \ \sum _{k= -\infty }^\infty c_k (z-z_0)^k \, . \) We do not do this arbitrarily, there is a close connection with the functions described by this: power series describe functions that are holomorphic on a circle around \(z_0\) , Laurent series describe functions that are holomorphic on an annulus around \(z_0\) .