Complex Analysis: Confirmation for a question in my textbook? I'm being told that $$\frac{\exp{(1+i3\pi})}{\exp{(-1+i\pi /2})}= \exp(2)i$$
I keep getting $-\exp(2)i$. I have no idea how they didn't get that to be negative.

$$\frac{e^{1+3i\pi}}{e^{-1+i\pi/2}} = \frac{e \cdot e^{3i\pi}}{\frac{1}{e}e^{i\pi/2}}$$ Now by Euler's identity we have $e^{i\pi/2} = i$ and $e^{3i\pi} = -1$. Hence $$\frac{e \cdot e^{3i\pi}}{\frac{1}{e}e^{i\pi/2}} = \frac{-e}{\frac{i}{e}}$$ Now multiply the expression by $\frac{ei}{ei}$ to clean it up, and we'll see $$\frac{-e}{\frac{i}{e}} \cdot \frac{ei}{ei} = \frac{-e^2i}{-1} = ie^2$$
