Complex Methods | Part IB, 2002

(a) Using the inequality sinθ2θ/π\sin \theta \geq 2 \theta / \pi for 0θπ20 \leq \theta \leq \frac{\pi}{2}, show that, if ff is continuous for large z|z|, and if f(z)0f(z) \rightarrow 0 as zz \rightarrow \infty, then

limRΓRf(z)eiλzdz=0 for λ>0\lim _{R \rightarrow \infty} \int_{\Gamma_{R}} f(z) e^{i \lambda z} d z=0 \quad \text { for } \quad \lambda>0

where ΓR=Reiθ,0θπ\Gamma_{R}=R e^{i \theta}, 0 \leq \theta \leq \pi.

(b) By integrating an appropriate function f(z)f(z) along the contour formed by the semicircles ΓR\Gamma_{R} and Γr\Gamma_{r} in the upper half-plane with the segments of the real axis [R,r][-R,-r] and [r,R][r, R], show that

0sinxxdx=π2\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}

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