Solution
= Solution
In the open strip $-c<\operatorname{Im}\lambda<c$, neither pole of $g$ is present. The two evaluations therefore give
$$
\boxed{H(\lambda)+J(\lambda)=\left(g(\lambda)-\frac{r_+}{\lambda-\alpha_+}\right)+\frac{r_+}{\lambda-\alpha_+}=g(\lambda)}.
$$
Both summands are <holomorphic> on this strip. This is a <half-plane decomposition by Cauchy integrals>; the pole subtraction in $H$ is what makes its extension <holomorphic> even at the upper pole of $g$.