I have this integral that goes from $ci$ to $0$, along the imaginary axis, and I'm finding that I seem to get different answers if I
(1) Parametrize and evaluate it.
(2) Multiply the whole thing by $-1$, flip the limits to going from $0$ to $ci$, and then parametrize and evaluate it.
So I'm wondering- does the trick of multiplying by -1 and flipping the end points work before parametrizing when integrating over the complex plane? Or does it have to be parametrized first?
Details: My function is $f=-\int_{z=ci}^{0} e^{-z^{2}+2icz} dz$
For (1), I parametrize via $z=(c-t)i$, and get $f=i\int_{t=0}^{c} e^{t^{2}-c^{2}}dt$
For (2), I first flip the limits, getting $f=\int_{z=0}^{ci} e^{-z^{2}+2icz} dz$, then parametrize via $z=it$, getting $f=i\int_{t=0}^{c} e^{t^{2}-2ct}dt$
Everything is as it should be. The two real integrals that you got are transformed into each other by the substitution $t\to c-t$; they yield the same result.