Residue (mathematics): Difference between revisions
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In complex analysis, the '''residue''' of a function ''f'' [[holomorphic function|holomorphic]] in an open set <math>\Omega</math> with possible exception of a point <math>z_0\in\Omega</math> where the function may admit an [[isolated singularity]], is a particular number describing behaviour of ''f'' around <math>z_0</math>. | |||
More precisely, if a function ''f'' is holomorphic in a neighbourhood of <math>z_0</math> (but not necessarily at <math>z_0</math> itself), with either a [[removable singularity]] or a [[pole (complex analysis)|pole]] at <math>z_0</math>, then it can be represented as a [[Laurent series]] around this point, that is | |||
then it can be represented as | |||
:<math>f(z) = \sum_{n=-N}^\infty c_n (z-z_0)^n</math> | :<math>f(z) = \sum_{n=-N}^\infty c_n (z-z_0)^n</math> | ||
with some <math>N\in \mathbb{N}\cup\{\infty\}</math> and coefficients <math>c_n\in \mathbb{C}.</math> | with some <math>N\in \mathbb{N}\cup\{\infty\}</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)</math> or <math>\ | <math>\mathrm{Res}(f,z_0)</math> or <math>\underset{z=z_0}{\mathrm{Res}}f(z).</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 functions. | 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 applications in real analysis as well. | 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. | ||
Revision as of 16:56, 12 November 2008
In complex analysis, the residue of a function f holomorphic in an open set with possible exception of a point where the function may admit an isolated singularity, is a particular number describing behaviour of f around .
More precisely, if a function f is holomorphic in a neighbourhood of (but not necessarily at itself), with either a removable singularity or a pole at , then it can be represented as a 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.