In complex analysis the fundamental theorem of calculus is only applicable to functions that have analytical antiderivatives. Since it may not be easy to know if that is the case, there are ways to calculate integrals without needing to know that it has an analytical antiderivative. This did make me wonder, is it possible to use the Cauchy-Riemann equations to investigate if the antiderivative is analytical for a certain $f(z)$ and calculate it using the Cauchy-Riemann equations?
I would think so, since Cauchy Riemann states that a function $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ has a derivative in a region equal to
$$ u_x-iu_y $$
if $u_x=v_y$ and $u_y=-v_x$ and all are continuous in that region. This can be used to show that a function $f(z)$ is analytical, since it is analytical in a region if it is differentiable in that region. To me this suggests that any function that can be written as $u_x-iu_y$ of some real $u$ has an analytical derivative that can be found by the Cauchy Riemann equations. Is this right or did I make a mistake/miss an assumption somewhere?