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8.7: Laurent Series - Mathematics LibreTexts

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The Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases …

Theorem \(\PageIndex{1}\) Laurent series Suppose that \(f(z)\) is analytic on the annulus \[A: r_1 < |z - z_0| < r_2. \nonumber \] Then \(f(z)\) can be expressed as a series \[f(z) = \sum_{n = 1}^{\infty} \dfrac{b_n}{(z - z_0)^n} + \sum_{n = 0}^{\infty} a_n (z - z_0)^n. \nonumber \] The coefficients have the formulus \[\begin{array} {l} {a_n = \dfrac{1}{2\pi i} \int_{\gamma} \dfrac{f(w)}{(w - z_0)^{n + 1}}\ dw} \\ {b_n = \dfrac{1}{2\pi i} \int_{\gamma} f(w) (w - z_0)^{n - 1}\ dw} \end{array} \nonumber \] where \(\gamma\) is any circle \(|w - z_0| = r\) inside the annulus, i.e. \(r_1 < r < r_2\

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