Line integrals are quite useful in the study of complex functions $f:Bbb Cto Bbb C$. Here the complex plane $Bbb C$ is identified with $Bbb R^2$ by setting $z=x+iy$, $zinBbb C$, $(x,y)inBbb R^2$. It is convenient to introduce the complex differential form $dz = dx+idy$ and to write
$$ f(z) =u(x,y)+iv(x,y) = u+iv.$$
Then the complex form
$$f(z)dz = (u+iv)(dx+idy) = (udx-vdy)+i(udy+vdx)$$
has $udx-vdy$ as its real part and $udy+vdx$ as its imaginary part. Define
$$int_c f(z) dz = int_c(udx-vdy)+iint_c(udy+vdx)$$
Assume that $u,vin C^1$. Recall that $f$ is holomorphic if and only if $u_x = v_y, u_y = -v_x$
Question. If $f$ is holomorphic, the function $f'(z)$ (the derivative of $f$ in $z$) given by the equation $df:=du+idv = f'(z)dz$ is well defined and $f'(z) = u_x-iu_y$.
In the question I what should I check for well definedness?