I've been trying to learn about the Argument Principle. To practice, I've been trying to compute the integral from the definition for the simple function $$ f(z) = z + 1 $$ According to wikipedia,
if $f(z)$ is a meromorphic function inside and on some closed contour $C$, and $f$ has no zeros or poles on $C$, then $$ \frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}dz = Z - P $$ where $Z$ and $P$ denote respectively the number of zeros and poles of $f(z)$ inside the contour $C$, with each zero and pole counted as many times as its multiplicity and order, respectively, indicate.I wanted to compute this using a circle contour, defined by $$z = re^{i\theta}$$ where $\theta \in [0, 2\pi]$. Since I picked a simple function that I already know the roots of, I know that if I let $|r| \gt 1$, the integral should come out to $1$, because there is one root at $z = -1$ inside that circle, and if I let $|r| \lt 1$, the integral should come out to $0$, because there is no root inside that circle.
Unless I made a mistake, the integral definition for this choice of function and contour comes out to this: $$ \frac{1}{2\pi i} \int_{0}^{2\pi} \frac{ire^{i\theta}}{1 + re^{i\theta}}d\theta $$ This evaluates to $$ \frac{1}{2\pi i} \left( \log(1+re^{2\pi i}) - \log(1+re^{0}) \right) $$ This seems to be $0$ for any choice of $r \ne -1$.
What did I do wrong? Am I evaluating the logarithm wrong? Does it matter what contour I choose? How should I go about computing integrals for the Argument Principle?