For $\lambda>0$, subtracting the two formulas from part (iv) yields
$$
f_+(\lambda)-f_-(-\lambda)
=\int_\lambda^\infty
\frac{e^{-ix}-e^{ix}}x\,dx
=-2i\int_\lambda^\infty\frac{\sin x}{x}\,dx.
$$
Using the stated <Dirichlet integral> and taking the <one-sided limit> gives
$$
\boxed{
\lim_{\lambda\downarrow0}
\bigl[f_+(\lambda)-f_-(-\lambda)\bigr]
=-2i\frac\pi2=-i\pi}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 327 / 1 / b / v / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-327/1/b/v/solution