MathLiveResidue theorem
It states that the contour integral of a function around a simple closed curve, traversed counterclockwise (positively oriented), equals 2πi times the sum of the residues of the function at the poles enclosed by the curve. It is used to evaluate complex contour integrals, often real integrals, by computing residues instead.
\frac{1}{2 \pi \mathrm{i}} \oint_{C} f(z) \, \mathrm{d}z = \sum_{a \in A} \operatorname{Res}(f, a)What the symbols mean
| a function meromorphic on and inside the contour | |
| a simple closed contour in the complex plane | |
| complex variable | |
| set of poles of f lying inside C | |
| a pole of f inside C | |
| residue of f at the pole a | |
| imaginary unit |
About this formula
- FieldMathematics
- TopicComplex analysis
- LevelUniversity
- Also calledCauchy residue theorem, Cauchy's residue theorem, contour integral of residues, sum of residues contour integral, residue theorem complex analysis
- SourceWikidata Q830513
Write it in Obsidian
MathLive is a free visual math editor for Obsidian. Run Find a formula, type “residue theorem”, and it goes into your note as editable LaTeX. You can also scan formulas from screenshots, PDFs and handwriting.