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Can somebody give me an intuitive understanding of the Lemma of Jordan, which is:

$$\lim \limits_{R\rightarrow \infty} \int_{\gamma} \exp(i \omega z)\,\rm dz=0 $$

if:

$$\lim \limits_{z\rightarrow \infty} f(z) =0, \omega>0$$

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It seems you forgot $f(z)$ in the 1st integral, and you also don't specify what $\gamma$ is. Jordan lemma in the form you state it will be valid for $\gamma$ given by a semicircle of radius $R$ in the upper half-plane. That means you can parameterize $z=Re^{i\phi}$ with $\phi \in (0,\pi)$. Now if we write $$e^{i\omega z}=e^{i\omega R e^{i\phi}}=e^{i R \cos \phi}\times e^{-R\sin\phi}$$ The first factor is bounded, and the second very quickly goes to $0$ as $R\rightarrow\infty$ precisely because $\phi\in(0,\pi)$, as this guarantees $\sin\phi>0$. –  O.L. Oct 20 '13 at 15:56

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