Vtyjkyuk

I'm reading a paper and the author defines $Delta=arg(f(z))$ to be the increment of $arg(f(z))$ when $arg(z)$ for $z=e^iphi$ changes from $0$ to $2pi$. I understand that we're looking to find the difference of the arguments of $f(z)$ after $z$ makes one full rotation around the unit circle, but I'm also confused: $e^i0=e^i2pi$ so $f(e^i0)=f(e^i2pi)$, so would $Delta$ be anything besides 0?!

Thanks for the help, I'm new to complex analysis.

Yes, but the increment is the difference between the arguments of $f(z)$, not between the values of $f$. You have a function $f:G to mathbbC$ when $G subset mathbbC$ an open domain which contains the unit circle. Now let's define a curve $gamma:[0,2pi] to mathbbC$ by the rule $gamma(t)=f(e^it)$. Then by your definition the increment is $arg(gamma(2pi))-arg(gamma(0))$. Yes, the values of $gamma(2pi)$ and $gamma(0)$ are the same, but their arguments can still be different. For example, if $f$ is the identity function ($f(z)=z$) then you will get $gamma(0)=e^i0$ but $gamma(2pi)=e^2ipi$. The location of these points on the complex plane is the same but their arguments are different-$0$ and $2pi$ respectively. So the increment here will be $2pi$. Intuitively the increment is $2pi$ multiplied by the number of rotations that the curve $gamma$ does around the origin. A rotation counterclockwise counts as 1 rotation, a rotation clockwise counts as -1 rotation.