Argument principle for conjugate of a function

Question

If I have a holomorphic function $\phi$ and it has N zeros insider a disk $D$. The Argument principle tell us that the integral $$\int _{\partial D} \frac{d\phi}{\phi}=2\pi i N$$ Can i get a similar result by taking conjugate to $\phi$? $$\int _{\partial D} \frac{d\bar{\phi}}{\bar{\phi}}=-2\pi i N$$

Answer 1

If $f=u+iv$ is analytic then $\bar f=u-iv$,and according to the CR equations relating the partial derivatives of $u$ and $v$, and $f'$ we have $$\eqalign{ df&=(u_x+iv_x)dx+(u_y+iv_y)dy=(u_x+iv_x)(dx+idy)=f'(z)\>dz\ ,\cr d\bar f&=(u_x-iv_x)dx+(u_y-iv_y)dy=(u_x-iv_x)(dx-idy)=\overline{f'(z)}\>\overline{dz}\ .\cr}$$ This shows that your second integral is just the complex conjugate of the first, so that its value is indeed $-2\pi i N$.