LaplaceTransformOfIntegral 2008-09-23 12:51:15 2008-09-23 14:05:21 Derivation Laplace transform of derivative % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} On can show that if a real function \,$t \mapsto f(t)$\, is \PMlinkname{Laplace-transformable}{LaplaceTransform}, as well is $\displaystyle\int_0^tf(\tau)\,d\tau$.\, The latter is also continuous for\, $t > 0$\, and by the \PMlinkname{Newton--Leibniz formula}{FundamentalTheoremOfCalculusClassicalVersion}, has the derivative equal $f(t)$.\, Hence we may apply the formula for Laplace transform of derivative, obtaining $$F(s) \;=\; \mathcal{L}\{f(t)\} \;=\; s\,\mathcal{L} \left\{\int_0^t\!f(\tau)\,d\tau\right\}-\int_0^0\!f(t)\,dt \;=\; s\,\mathcal{L} \left\{\int_0^t\!f(\tau)\,d\tau\right\},$$ i.e. \begin{align} \mathcal{L} \left\{\int_0^t\!f(\tau)\,d\tau\right\} \;=\; \frac{F(s)}{s}. \end{align} \textbf{Application.}\, We start from the easily derivable rule $$\frac{1}{s} \;\curvearrowright\; 1,$$ where the curved \PMlinkescapetext{arrow points} from the Laplace-transformed function to the original function.\, The formula (1) thus yields successively $$\frac{1}{s^2} \;\curvearrowright\; \int_0^t\!1\,d\tau \;=\; t,$$ $$\frac{1}{s^3} \;\curvearrowright\; \int_0^t\!\tau\,d\tau \;=\; \frac{t^2}{2!},$$ $$\frac{1}{s^4} \;\curvearrowright\; \int_0^t\!\frac{\tau^2}{2!}\,d\tau \;=\; \frac{t^3}{3!},$$ etc.\, Generally, one has \begin{align} \frac{1}{s^n} \;\curvearrowright\; \frac{t^{n-1}}{(n\!-\!1)!} \quad \forall\, n \in \mathbb{Z}_+. \end{align}