What is meant by a "general line integral of form $\int_\gamma p\ dx + q\ dy$"? In his text on complex analysis, Ahlfors speaks of "general line integrals of form $\int_\gamma p\ dx + q\ dy$".  I'm curious exactly what is meant by this.  I take it that $p$ and $q$ are not strictly real-valued (as is frequently assumed with functions $u$ and $v$ when discussing $f = u + iv$)?
If $p$ and $q$ are complex-valued s.t. $f = p + q$, what is the point of splitting up $f$ in such a way?


what is the point of splitting up $f$ in such a way?

The notation  $\int_\gamma p\, dx + q\, dy$ gives a clue: $p$ is attached to $dx$ while $q$ is attached to $dy$. These functions get integrated in different ways. If you plug in a parametric equation $x=x(t)$, $y=y(t)$, the integral becomes 
$$
\int_\gamma (p\, x'  + q\,y')\,dt  = \int_\gamma \Big(p(x(t),y(t))\, x'(t)  + q(x(t),y(t))\,y'(t)\Big)\,dt
$$
(the second form is more precise but less pleasant to look at.) 
You could also ask yourself:  are the vectors $3\vec\imath + 7\vec\jmath$ and $4\vec\imath + 6\vec\jmath$ the same or different? Of course they are different, even though $3+7=4+6$. It's similar with $\int p\,dx + q\,dy$. 
