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Jordan's lemma states that, for $b>0$, if $g(z)\to 0$ uniformly on large upper semicircles, then $\int g(z)e^{ibz}dz\to 0$ on those arcs. Indeed its absolute value is at most $R\max |g|{\int }_0^{\pi }{e}^{-bR\sin \theta }d\theta \le (\pi /b)\max |g|$. For negative $b$, use lower semicircles. The corresponding exponential-decay estimates can also be used on rectangular closures.

First take $a>0$ and $0<\epsilon <a$, and defineUse finite strip rectangles with vertical sides at $\pm R$, $R=N+1/2$, then let $N\to \infty$. Their orientation is positive. At a nonzero integer, the residue isThe origin is removable when $a\ne 0$ and contributes zero. The short vertical-side integrals tend to zero since $|\sin \pi (\pm R+iy)|=\cosh (\pi y)$ and $F=O(R^{-1})$ there. The residue theorem therefore gives the first evaluation,The sum is understood as the limit of symmetric partial sums. It converges for $|x|<\pi$ by the Dirichlet test: the oscillatory partial sums are bounded, and $n/(a^2+n^2)$ decreases eventually to zero.

For the second evaluation, close the top line outward into the upper half-plane and the bottom line outward into the lower half-plane. Because of the strip orientations, both outward closures are clockwise. The upper region contains only the pole $ia$, and the lower one only $-ia$. Their residues areTo justify discarding the added boundaries, use rectangles with far vertical sides at $\pm (N+1/2)$ and outer horizontal sides at imaginary height $\pm R$. The numerator grows at most like $e^{|x||\mathrm{Im}z|}$, while $1/\sin \pi z$ decays like $e^{-\pi |\mathrm{Im}z|}$. Thus the horizontal integral tends exponentially to zero, and the vertical integrals are bounded by a constant times $R^{-1}{\int }_{\epsilon }^R{e}^{-(\pi -|x|)y}dy\to 0$. The strict condition $|x|<\pi$ is essential. The clockwise residue calculation givesEquating the two evaluations proves the alternating sine series with a quadratic denominator:Both sides are even functions of $a$, so this proves the result for every nonzero real $a$. At $a=0$ the displayed zero-index summand and right-hand ratio are literally undefined; under their natural continuous extension, set that summand to zero and take the right-hand limit $-x$. This limit may be passed through the nonzero-index sum, since its difference from the $a=0$ sum is bounded by $2a^2{\sum }_{n\ge 1}{n}^{-3}$. Thus the intended limiting identity at zero is $2{\sum }_{n\ge 1}(-1{)}^n\sin (nx)/n=-x$. The separate contour evaluation above assumes distinct off-axis poles and should not be applied unchanged after they coalesce at the origin.

Set $M_R={\max }_{0\le \theta \le \pi }|f(R{e}^{i\theta })|$. The complex limit at infinity gives $M_R\to 0$. Parametrizing the semicircle and using $|e^{i\lambda R{e}^{i\theta }}|=e^{-\lambda R\sin \theta }$ yieldsBy symmetry and the supplied sine bound,The arc integral is therefore at most $(\pi /\lambda )M_R$, tending to zero. This proves the required vanishing, the relevant form of Jordan's lemma. No holomorphicity of $f$ is required just for this arc estimate.