nLab Cauchy integral theorem

Idea

Cauchy’s integral theorem states that contour integrals of holomorphic functions in the complex plane $\mathbb{C}$ are invariant under homotopy of paths. In particular, if a function is holomorphic on a simply connected subspace of $\mathbb{C}$, then its contour integral on a path depends only on the beginning and ending points of the path, and indeed can be given by subtracting the values there of an antiderivative (in accordance with the second Fundamental Theorem of Calculus).

Statement

Let $D$ be an open subset of the complex plane $\mathbb{C}$, let $a$ and $b$ be two points in $D$, let $\gamma_1$ and $\gamma_2$ be two curves in $D$ from $a$ to $b$, let the region between them also lie entirely within $D$, and let $f$ be a holomorphic function on $D$. Then we have

$$\int_{\gamma_1} f(z) \,\mathrm{d}z = \int_{\gamma_2} f(z) \,\mathrm{d}z .$$

In particular we have

$$\int_{\gamma_1} f(z) \,\mathrm{d}z = 0$$

if $a = b$ (because then $\gamma_2$ may be taken to be a constant); in other words, the contour integral of a holomorphic function is zero around any loop whose inside lies entirely within the function's domain.

Related concepts

• Cauchy's integral formula

• Cauchy's theorems

• Goursat theorem

• Stokes theorem

References

• Wikipedia, Cauchy integral theorem

category: analysis