Computing ∫₀∞ sin(x)/x dx

Complex Analysis · Axiom Academy

Evaluate the famous Dirichlet integral with an indented contour and Jordan's lemma. Evaluate the Dirichlet integral . There is no elementary antiderivative, but contour integration of gives the value exactly. The pole of sits at z=0 , right on the path. The tiny semicircle detours around it through the upper half-plane, so the closed contour encloses no poles. We send and . Nice work. You tamed an integral with no elementary antiderivative by routing it through the complex plane. Complexify wisely: integrate , not — only e^ iz decays in the upper half-plane, which is what makes the big arc disappear. Indent a pole on the path: a small semicircle around a simple pole contributes — exactly half the full , with the sign set by orientation. Match real and imaginary parts: ; the cosine part is an odd P.V. that cancels, leaving the sine integral as the imaginary part. Result: — a classic result that appears all over physics and signal processing. Indent around the pole, kill the big arc, read off the half-residue — the same recipe evaluates many improper integrals with a singularity on the real axis.

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