Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2012 / 9 / ii / Solution

For $\mathrm{Re}p>0$, the complex exponential $e^{-pT}$ tends to zero, since its modulus is $e^{-T\mathrm{Re}p}$. Therefore direct integration givesThe condition on the real part also gives absolute convergence of this improper integral.