inverse Laplace transform of $\delta_1(\cdot)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:18:10Z http://mathoverflow.net/feeds/question/37658 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37658/inverse-laplace-transform-of-delta-1-cdot inverse Laplace transform of $\delta_1(\cdot)$ Fedor Petrov 2010-09-03T20:06:39Z 2010-09-03T20:06:39Z <p>Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify classes of functions, for which this should hold).</p> <p>In other words, $\tilde{\psi}(p)=\delta_1(p)$ in the sense of distributions.</p> <p>Or, Stiltjes transform $\int_0^{\infty} \frac{\psi(t)}{t+y} dt=e^{-y}, y>0$.</p> <p>(just represent $\frac1{t+y}=\int_0^{\infty} e^{-q(t+y)}dq$ and change order of integration).</p> <p>So the question(s) is(are):</p> <p>Does such function exist, if it exists, what are its properties, where is it written about all this stuff and so on.</p>