Complex integration

Given a complex function $𝑓 : 𝑥 + 𝑖 𝑦 \mapsto 𝑢 + 𝑖 𝑣$ , its complex integral along a path $𝐶$ is

\int 𝐶 𝑓 (𝑧) 𝑑 𝑧 = \int 𝐶 (𝑢 + 𝑖 𝑣) (𝑑 𝑥 + 𝑖 𝑑 𝑦) = \int 𝐶 𝑢 𝑑 𝑥 - 𝑣 𝑑 𝑦 + 𝑖 \int 𝐶 𝑣 𝑑 𝑥 + 𝑢 𝑑 𝑦

However, if we require $𝑓$ be analytic, the Cauchy-Riemann equations imply that this integral should be path-independent.

Cauchy's Integral Theorem If $𝑓 (𝑧)$ is an analytic complex function for a closed region $𝑅 \subseteq ℂ$ , then
$\oint 𝜕 𝑅 𝑓 (𝑧) 𝑑 𝑧 = 0$
That is, $𝑓 (𝑧)$ is defined inside and on a closed path $𝜕 𝑅$ .

Note that if $𝑅$ contains a singularity and is hence non-analytic, then Cauchy's Integral Theorem does not apply and the path integral may be non-zero.¹ If $𝑓$ does have singularities but they lie outside $𝑅$ , the theorem still holds.

Core integration techniques

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2023. Advanced Mathematical Methods, p. 55 ↩

Complex integration

Core integration techniques

Footnotes