Let D be an arbitrary region and $R=\{x+iy:a\le x \le b, c\le y\le d\}$ be a rectangle .show that if $R\subseteq D $ and f is differentiable in D, then $$\int _{\partial R} f=0$$
where $\partial R=[z_1,z_2]\cup[z_2,z_3]\cup[z_3,z_4]\cup[z_4,z_1]$
i don't know how to solve this please ..thank you
It depends what your region $D$ looks like (its properties!). If it is a nice enough region, you can use Cauchys theorem, or Greens theorem. You can write $f(x+iy)=u(x,y)+iv(x,y)$ where $u,v$ are real functions. Now, use Greens theorem to evaluate $\int\limits_{\partial{R}} [u(x,y)+iv(x,y)] (dx+idy)=\int\limits_{\partial{R}}[u(x,y)dx-v(x,y)dy]+i \int\limits_{\partial{R}}[u(x,y)dy+v(x,y)dx]$. You might also want to remember the Cauchy Riemann equations!