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Cauchy's Integral Formula Reconstructs Interior Values by Vicfred

Last updated: Wed 26 February 2025

tags: mathematics, complex analysis, contour integration, residues

A holomorphic function inside a positively oriented contour is recovered from one boundary integral. A small computation will anchor the general statement before the abstraction takes over.

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.

$$ a\ \text{inside }\gamma,\qquad f\ \text{holomorphic on and inside }\gamma $$

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 \(\Longleftrightarrow\).

$$ f(a)=\frac1{2\pi i}\oint_\gamma\frac{f(z)}{z-a}\,dz $$

Stress the formula

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

$$ f^{(n)}(a)=\frac{n!}{2\pi i}\oint_\gamma\frac{f(z)}{(z-a)^{n+1}}\,dz $$

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad a\ \text{inside }\gamma,\qquad f\ \text{holomorphic on and inside }\gamma,\\[5pt] \mathsf{C}\;&:\quad f(a)=\frac1{2\pi i}\oint_\gamma\frac{f(z)}{z-a}\,dz. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] f(a)=\frac1{2\pi i}\oint_\gamma\frac{f(z)}{z-a}\,dz \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Tue 07 December 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.