I'm working my way through a textbook on complex variables and I'm having trouble with the initial set-up for this problem. I have to compute the line integral
$$ \int_{\gamma}^{^{}}e^{z}\;dz$$ where $\gamma$ is the line segment from $0$ to $z_{0}$.
I know what to do once the problems been parameterized into x(t) and y(t) so that I can use z = x + iy. The part I keep getting stuck on is on getting the i to cancel out in z because most of the problems i've done so far have had i remain in z.
This is probably something really simple but i'm just not getting it so I'd appreciate any help.
Along $\gamma$, you have $z=tz_0$ ($0\leq t\leq 1$).
Then $e^z=e^{tz_0}$ and $dz=z_0\;dt$.
The integral becomes $$\int_{\gamma}e^z\;dz =\int_{t=0}^{t=1}e^{tz_0}z_0\;dt$$
It’s fine to integrate this directly—it doesn’t matter that there are complex numbers involved. You don’t have to take it all the way down to real and imaginary parts immediately. In fact it’s much easier to integrate it in this form (the same rules apply even with complex constants involved). Continuing:
$$=\left.\frac{e^{tz_0}}{z_0}z_0\right|_{t=0}^{t=1}$$ $$=\boxed{e^{z_0}-1}$$
Now at this point you may want to write this as $$e^{x_0}\cos y_0-1 +ie^{x_0}\sin y_0$$ but I would just leave it in the boxed form above.