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Cauchy's integral theorem can be derived from Green's theorem, as follows. Let $U$ be a simply connected open subset of $\mathbb{C}$, let $f : U \to \mathbb{C}$ be a holomorphic function with real and complex parts $f(z) = u(z) + i v(z)$, and let $C$ be a positively oriented contour in $U$. Then Cauchy's integral theorem states that

$\oint_C f(z) \, dz = \oint_C (u(z) + i v(z))(dx + i dy) = 0.$

Note that this can be expressed in terms of two real line integrals as

$\oint_C (u \, dx - v \, dy) + i \oint_C (v \, dx + u \, dy).$

Both of these integrals can be computed using Green's theorem, which gives that they are equal to

$\iint_D \left( - \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \, dx \, dy + i \iint_D \left( \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) \, dx \, dy$

where $D$ is the interior of the region bounded by $C$, and the integrands here both vanish by the Cauchy–Riemann equations. What this implies is that the "vector fields" (really 1-forms) $u \, dx - v \, dy$ and $v \, dx + u \, dy$ are the "gradients" (really differentials) of scalar functions, which turn out to be the real and imaginary parts of the Antiderivative of $f$.

Cauchy's integral formula is not hard to deduce from here.

References

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