If Γ is a closed Jordan curve that encloses a region with area A, show that $A=\frac{1}{2i} \int_{\Gamma} \overline{z}dz$.
96023
Prove: $\int_\gamma \bar z \, dz=2i\operatorname{Area}(G)$
64084
Existence of a simple closed curve which is not null-homotopic
67224
Why is the area under a curve the integral?
116507
Very quick question: function extends to 1/z on the boundary of unit disc
109784
Coloring a region bounded by curves
16582
Show that $n$ lines separate the plane into $\frac{n^2+n+2}{2}$ regions
66494
Curvature of a curve on a sphere
43177
How do I rigorously show $f(z)$ is analytic if and only if $\overline{f(\bar{z})}$ is?
128658
Analyticity of $\overline {f(\bar z)}$ given $f(z)$ is analytic
69203
Calculus proof for the area of a circle
58832
How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
120985
Prove that $\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$
117508
Prove that the closed unit ball of $L^2[a,b]$ is closed in $L^1[a,b]$
147429
Set of points in $\mathbb{C}$ s.t. $z^{n-1} = \overline z$
16820
Evaluate $\int_{\gamma} \frac{ |dz| }{z} $ where $\gamma $ is the unit circle
6100
Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
27347
Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.