What does "V.P." stand for in "$V.P.\frac{1}{2\pi i}\int_{x-i\infty}^{x+i\infty}e^{st}\frac{f(s)}{s}\,ds$"?

Question

currently I am studying the Laplace transform and its inverse. Lately I came across the possible abbreviation "V.P." for example in $$ \int_0^tF(\tau)\,d\tau = V.P.\frac{1}{2\pi i}\int_{x-i\infty}^{x+i\infty}e^{st}\frac{f(s)}{s}\,ds. $$ However, I have no idea what this "V.P." should mean. And yes, I have tried to google it.

Best regards, Lukas

Answer 1

$V.P.$ is a French abbreviation for valeur principale – the Cauchy principal value. Here it signifies that the integral after $V.P.$ should be computed as $$\lim_{t\to\infty}\int_{x-it}^{x+it}e^{st}\frac{f(s)}s\,ds$$