## Fourier integrals vanishing on large circles

When evaluating contour integrals, it is often of interest to prove that Fourier-type integrals vanish on large enough semicircles (see the figure). This holds under the following condition: Theorem. Suppose that $$f(z)=O(|z|^{-a}), \quad a>0$$ for in the upper half-plane. Then for any $$\lambda > 0$$ we have $$\int_{\gamma_R} f(z)\mathrm{e}^{i\lambda z} \rightarrow 0, \quad R\to+\infty,$$ where …