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I am currently aware of the following two versions of the global Cauchy Theorem. Which one is stronger?

1.)If the region $U$ is simply connected, then for every closed curve contained therein, the integral of the holomorphic function $f$ defined on $U$ over the curve is zero.

2.) If the domain $D$ is an arbitrary open set, then for every closed curve contained therein with the property that the index is zero for points outside $D$, the integral of the holomorphic function $f$ defined on $D$ is zero. Thanks in advance.

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3 Answers 3

The second one is stronger, since any closed curve in a simply connected domain satisfies the condition imposed on the second version of Cauchy's theorem.

Somewhat more general versions of the theorem are called the homotopical version (corresponding to your first theorem) and the homological version (corresponding to your second one.) The homological version is stronger, since any null-homotopic curve is also null-homlogous.

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If $\Omega$ is simply connected then all closed curves in $\Omega$ have index $0$ with respect to points outside $\Omega$. Therefore the first statement follows from the second.

The second statement actually is the strongest form of Cauchy's theorem in the plane.

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The way I learned it is:

$$\int_{\Gamma}f(z)dz=0 $$

Iff

1.) $f$ is analytic in a simply connected domain D

2.) $\Gamma$ is any closed contour in D,

Where a domain D is defined by "an open connected set", D $\subseteq \mathbb{C}$.

A function $f=u(x,y) +iv(x,y)$, defined in some open set G, containing the point $z_0$ is differentiable at $z_0$ if the first partial derivatives of $u$ and $v$ exists in G, are continuous at $z_0$ and satisfiy the Cauchy-Riemann Equations.

Consequently, if the first partial derivatives are continuous and satisfy the Cauchy-Riemann Equations at all points of G, then $f$ is analytic in G

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