So I'm learning conformal field theory and having a hard time to prove the conformal Ward identity. From the lectures notes from John Cardy, he express the integral
$$ \delta S = \frac{1}{2\pi} \int_C T_{\mu \nu} \alpha^{\mu}\eta^{\nu}dl$$
in complex coordinates as
$$ \delta S = \frac{1}{2\pi i} \int_C T(z)\alpha(z)dz + \text{ complex conjugate}$$
My problem is how to change the expression of the line element $dl$ in the $x,y$ coordinates to its form in the $z, \bar z$ coordinates, where $z = x +iy$ and $\bar z = x - iy$.
I don't know if someone is gonna to pass by the same problem, but after a few tips from my professor I finally made it. The pdf with the answer is here. Would take to long for me to pass to math latex.