Let $f(z)$ be an analytic function with no singularities (an entire function) and $\gamma$ be a closed curve in $\mathbb{C}$. Then $$\int_\gamma f(z)dz=0$$
But I'm failing to understand this visually. My understanding is that a line integral is essentially the area of the "curtain" formed between the curve and the plane, but that notion doesn't really work here since the complex function has both a real and imaginary output. Is there some intuitive, visual, easy to understand way to see why this integral is always zero? Of course I can see a proof with symbolic mathematics, but that doesn't help much.