The book I am studying claims that some arguments are valid because the following integral converges:
$$\lim_{R \rightarrow \infty} \int_\Gamma \frac{e^\lambda}{|\lambda|}d\lambda$$
where $\Gamma=Re^{i\theta}$, $\theta \in [-\phi,\phi]$ and $\frac{\pi}{2} < \phi < \pi$.
However, when I try integrating, I get:
$$\int_\Gamma \frac{e^\lambda}{|\lambda|}d\lambda = \int_{-\phi}^{\phi} \frac{e^{Re^{i\theta}}}{R}iRe^{i\theta}d\theta = \int_{-\phi}^{\phi} \frac{1}{R}\frac{d}{d\theta}\big(e^{Re^{i\theta}}\big)d\theta = \frac{1}{R}\big(e^{Re^{i\phi}}-e^{Re^{-i\phi}}\big)$$
If we take $z=Re^{i\phi}$, then:
$$\frac{1}{R}\big(e^{Re^{i\phi}}-e^{Re^{-i\phi}}\big) = \frac{1}{|z|}\big(e^{z}-e^{\bar z}\big) = \frac{e^z}{|z|}\big(1-e^{-|z|^2}\big),$$
which diverges when $|z|=R \rightarrow \infty$. Have I done something wrong?
In case someone is interested, I am studying Pazy's "Semigroups of Linear Operators and Applications to Partial Differential Equations", Theorem 1.7.7 (I know I probably should not be struggling with that at this level).
Everything is alright until you think the quantity diverges.
$$|e^z|=|e^{R\cos\phi}e^{iR\sin\phi}|=e^{R\cos\phi}$$
Note that for $\pi/2<\phi<\pi$, $\cos\phi<0$, thus $e^{R\cos\phi}\to0$ in the limit.
Therefore, we can conclude $e^z\to 0$ as $R\to\infty$.