Given some curve $\gamma:\mathbb{R\to R^2}$ positively oriented, one can write its arclength as $$L(\gamma)=\int_\gamma \vert \gamma^\prime(t)\vert dt.$$
I saw somewhere that if $\gamma$ was a complex curve, that is $\gamma:\mathbb{R\to C}$, one could write the length as $$L(\gamma)=\int_\gamma\frac{dz}{iN(z)}$$ where $N(z)$ is the outer unit normal.
I understand that $\gamma^\prime(t)dt=dz$ and that $iN(z)$ is the unit tangent to the curve but still I don't understand why the second formula is valid. Can one explain the transition from the first formula to the second please?
Hint: If $\gamma $ is oriented positively, with $\gamma'(t)$ never $0,$ then the outward unit normal is $-i\gamma'(t)/|\gamma'(t)|.$