Basic Laplace Transforms

Reference: Simmons (Second Edition), Chapter 9

function transform  comments
f(x) F (p) = L[f(x)]  logically (Lf)(p)
defined for x > 0 defined for p > p_{0} (or p ≥ p_{0})
f(x) + g(x) F (p) + G(p)  L additive
cf(x) cF (p)  c constant
1 \frac{1}{p}
x \frac{1}{p^{2}}
x^{2} \frac{2}{p^{3}}
\frac{1}{\sqrt{x}} \begin{matrix}√ & \overline{π} \\ √ & \overline{p} \end{matrix}
x^{n} \frac{n!}{p^{n+1}} = \frac{Γ(n + 1)}{p^{n+1}}  constant n > −1
e^{ax} \frac{1}{p − a}  a constant
cos ax \frac{p}{p^{2} + a^{2}}  a constant
sin ax \frac{a}{p^{2} + a^{2}}  a constant
δ(x) 1  Dirac delta “function”
δ(x − a) e^{−ap}  shift by constant a
e^{ax}f(x) F (p − a)  a constant
f′(x) pF (p) − f(0+)
f′′(x) p^{2}F (p) − pf(0+) − f′(0+)
∫_{0}^{x} f(u)du \frac{F (p)}{p}
xf(x) −F′(p)
\frac{f(x)}{x} ∫_{p}^{∞} F (v)dv
(f ∗ g)(x) F (p)G(p)  ∫_{0}^{x} f(x − u)g(u)du

110.302 Differential Equations JMB File: laptran, Rev. A; 30 Aug 2001; Page 1