Close the contour in the upper half-plane. The enclosed simple poles are eⁱπ/⁴ and e³ⁱπ/⁴, each with residue 1/(4z³). Their sum is −i/(2√2). The large semicircle contributes zero because the integrand decays as 1/z⁴. Hence the real integral is 2πi×[−i/(2√2)]=π/√2, A.
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