Lagmental Vicfred

The Residue Theorem Sums Local Laurent Coefficients by Vicfred

tags: mathematics, complex analysis, contour integration, residues

A contour integral of a meromorphic function equals 2 pi i times the enclosed residues. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Notation

A holomorphic function \(f:U\to\mathbf C\) has a complex derivative independent of direction. Cauchy's integral formula controls each value \(f^{(n)}(a)\) from boundary data.

$$ \operatorname{Res}(f,a)=[(z-a)^{-1}]f(z) $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \oint_\gamma f(z)\,dz=2\pi i\sum_a\operatorname{Ind}_\gamma(a)\operatorname{Res}(f,a) $$

Stress the formula

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ \oint_{|z|=2}\frac{e^z}{z(z-1)}\,dz=2\pi i\left(\operatorname{Res}_{z=0}+\operatorname{Res}_{z=1}\right)=2\pi i(e-1) $$

Interpretation

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad \operatorname{Res}(f,a)=[(z-a)^{-1}]f(z),\\[5pt] \mathsf{C}\;&:\quad \oint_\gamma f(z)\,dz=2\pi i\sum_a\operatorname{Ind}_\gamma(a)\operatorname{Res}(f,a). \end{aligned} $$

Limit of the argument

Contour formulas require orientation, winding number, and hypotheses about singularities. A pole on the contour cannot be ignored.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \oint_\gamma f(z)\,dz=2\pi i\sum_a\operatorname{Ind}_\gamma(a)\operatorname{Res}(f,a) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Sun 19 December 2010. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.