Past exam of the short options course of the University of Oxford / 2023 / 5 / b / ii / Solution

The integral $I$ is the convolution of the positive-half-line kernels $x^{a-1}$ and $x^{b-1}$. The convolution theorem for Laplace transforms therefore gives, for $p>0$,On the other hand, the substitution $t=xu$ gives $I(x)=x^{a+b-1}B(a,b)$, where $B(a,b)={\int }_0^1{u}^{a-1}(1-u{)}^{b-1}du$ is the beta function. Applying part (i) again yields $\mathcal{L}I(p)=B(a,b)\Gamma (a+b)p^{-(a+b)}$. Comparing the two expressions provesAll integrands used in the real-$p$ convolution argument are nonnegative, so Tonelli's theorem justifies the interchange; the endpoint singularities are integrable because $a,b>0$.