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Cauchy integral theorem
Theorem 1.
Let $U\subset\mathbb{C}$ be an open, simply connected domain, and let $f:U\rightarrow\mathbb{C}$ be a function whose complex derivative, that is
$\lim_{{w\rightarrow z}}\frac{f(w)f(z)}{wz},$ 
exists for all $z\in U$. Then, the integral around every closed contour $\gamma\subset U$ vanishes; in symbols
$\oint_{\gamma}f(z)\,dz=0.$ 
We also have the following, technically important generalization involving removable singularities.
Theorem 2.
Let $U\subset\mathbb{C}$ be an open, simply connected domain, and $S\subset U$ a finite subset. Let $f:U\backslash S\rightarrow\mathbb{C}$ be a function whose complex derivative exists for all $z\in U\backslash S$, and that is bounded near all $z\in S$. Let $\gamma\subset U\backslash S$ be a closed contour that avoids the exceptional points. Then, the integral of $f$ around $\gamma$ vanishes.
Cauchy’s theorem is an essential stepping stone in the theory of complex analysis. It is required for the proof of the Cauchy integral formula, which in turn is required for the proof that the existence of a complex derivative implies a power series representation.
The original version of the theorem, as stated by Cauchy in the early 1800s, requires that the derivative $f^{{\prime}}(z)$ exist and be continuous. The existence of $f^{{\prime}}(z)$ implies the CauchyRiemann equations, which in turn can be restated as the fact that the complexvalued differential $f(z)\,dz$ is closed. The original proof makes use of this fact, and calls on Green’s Theorem to conclude that the contour integral vanishes. The proof of Green’s theorem, however, involves an interchange of order in a double integral, and this can only be justified if the integrand, which involves the real and imaginary parts of $f^{{\prime}}(z)$, is assumed to be continuous. To this date, many authors prove the theorem this way, but erroneously fail to mention the continuity assumption.
In the latter part of the $19^{{\text{th}}}$ century E. Goursat found a proof of the integral theorem that merely required that $f^{{\prime}}(z)$ exist. Continuity of the derivative, as well as the existence of all higher derivatives, then follows as a consequence of the Cauchy integral formula. Not only is Goursat’s version a sharper result, but it is also more elementary and selfcontained, in that sense that it is does not require Green’s theorem. Goursat’s argument makes use of rectangular contour (many authors use triangles though), but the extension to an arbitrary simplyconnected domain is relatively straightforward.
Theorem 3 (Goursat).
Let $U$ be an open domain containing a rectangle
$R=\{x+iy\in\mathbb{C}:a\leq x\leq b\,,c\leq y\leq d\}.$ 
If the complex derivative of a function $f:U\rightarrow\mathbb{C}$ exists at all points of $U$, then the contour integral of $f$ around the boundary of $R$ vanishes; in symbols
$\oint_{{\partial R}}f(z)\,dz=0.$ 
Bibliography.

Ahlfors, L., Complex Analysis. McGrawHill, 1966.
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