= Solution
For real $z>0$, the exponential decays as $\operatorname{Re}t\to-\infty$, so two allowed contours run from $-\infty$ to $i$ and from $-\infty$ to $-i$. For real $z<0$, it decays as $\operatorname{Re}t\to+\infty$, so reverse the construction: run from $i$ or $-i$ to $+\infty$. These are the infinite contours in the diagram.
For $z>0$ their integrals are
$$
\int_{-\infty}^{\pm i}e^{zt}\,dt=\frac{e^{\pm iz}}{z};
$$
for $z<0$ the corresponding orientation changes only an overall sign. Taking the sum of the two exponentials produces a solution independent of $\sin z/z$:
$$
\boxed{\ w_2(z)=\frac{\cos z}{z}\ }.
$$
Its nonzero <Wronskian> with $w_1$ on $z\ne0$ proves <linear independence>.
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