Initial value theorem for Laplace transforms
= Initial value theorem for Laplace transforms
{title2=$f(0+)=\lim_{p\to+\infty}p\mathcal Lf(p)$}
For a <function> continuous at zero and of exponential order, the displayed limit recovers its initial value from its <Laplace transform>. Substitute $u=pt$ to obtain $p\mathcal Lf(p)=\int_0^\infty e^{-u}f(u/p)\,du$ and apply the <dominated convergence theorem>; an exponential-order bound controls the tail. The limit along positive real $p$ suffices for this version.