Compute the following integral: $$\int_{\delta D_1(0)} \frac{e^{z}}z dz $$
So I rewrote the formula in terms of $x$ and $y$ since $z = x + iy$ I got $$f(z) = \frac{e^xysin(y) + e^xxcos(y)}{x^2+y^2}+i\frac{e^xxsin(y) + e^xycos(y)}{x^2+y^2}$$
From here I wasn't 100% sure what to do, should I convert to polar and then go from there?
And just to double check does the notation $\int_{\delta D_1(0)}$ mean $\int_{-1}^1$? This particular notation confuses me a bit. Any help is appreciated, thanks.
Assuming that $\delta D_1(0)$ is the boundary of the unit disk, counter-clockwise oriented, the residue theorem gives:
$$\int_{\delta D_1(0)}\frac{e^z}{z}\,dz = 2\pi i\cdot\text{Res}\left(\frac{e^z}{z},z=0\right)=2\pi i.$$