Residue (mathematics): Difference between revisions
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In complex analysis, the '''residue''' of a complex function ''f'' [[holomorphic]] in a neighbourhood $\Omega$ of a point <math>z_0\in\mathbb{C}<math> is a particular number characterising behaviour of ''f'' around this point. | In complex analysis, the '''residue''' of a complex function ''f'' [[holomorphic]] in a neighbourhood $\Omega$ of a point <math>z_0\in\mathbb{C}</math> is a particular number characterising behaviour of ''f'' around this point. | ||
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with some <math>N\in \mathbb{N}</math> and coefficients <math>c_n\in \mathbb{C}.</math> | with some <math>N\in \mathbb{N}</math> and coefficients <math>c_n\in \mathbb{C}.</math> | ||
The coefficient <math>c_{-1}<math> is the residue of ''f'' at <math>z_0<math>, denoted as | The coefficient <math>c_{-1}</math> is the '''residue''' of ''f'' at <math>z_0</math>, denoted as | ||
<math>\mathrm{Res}(f,z_0) or <math>\mathrm{Res}_{z_0}f.</math> | <math>\mathrm{Res}(f,z_0)</math> or <math>\displaystyle\mathop{\mathrm{Res}}_{z=z_0}f.</math> | ||
Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex | Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex functions. | ||
For example, the residue allows to evaluate [[path integral]]s of the function ''f'' via the [[residue theorem]]. This technique finds many | For example, the residue allows to evaluate [[path integral]]s of the function ''f'' via the [[residue theorem]]. This technique finds many applications in real analysis as well. | ||
[[Category:CZ Live]] | [[Category:CZ Live]] | ||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:Stub Articles]] | [[Category:Stub Articles]] | ||
Revision as of 18:12, 7 November 2007
In complex analysis, the residue of a complex function f holomorphic in a neighbourhood $\Omega$ of a point is a particular number characterising behaviour of f around this point.
More formally, if a function f is holomorphic in a neighbourhood of then it can be represented as the Laurent series around this point, that is
with some and coefficients
The coefficient is the residue of f at , denoted as or
Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex functions. For example, the residue allows to evaluate path integrals of the function f via the residue theorem. This technique finds many applications in real analysis as well.