Solution

Close the contour in the upper half-plane. The degree assumption and Jordan lemma make the integral over the large semicircle tend to zero, while the absence of real zeros avoids indentations of the contour. The residue theorem therefore gives

\int_{- \infty}^{\infty} R (x) e^{i x} d x = 2 πi Q (z_{k}) = 0 Im z_{k} > 0 \sum Res_{z = z_{k}} (R (z) e^{i z}) .

(58)

Repeated roots are handled by the usual higher-order residue formula.

Solved by gpt-5.6-sol high.

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