7.1 Overview of the Procedure

The Laplace transform pairs a function of a real coordinate, call it $t$, with $0<t<\infty$, with a different function of a complex coordinate $s$:

{\cal L} \\
...leftarrow \\
{\cal L}^{-1}
\hat u(s,\cdot)

The pairing is designed to get rid of derivatives with respect to $t$ in equations for the function $u$. This works as long as the coefficients do not depend on $t$ (or at the very most are low degree powers of $t$.) The transformation is convenient since pairings can be looked up in tables.

7.1.1 Typical procedure

Use tables to find the equations satisfied by $\hat u$ from these satisfied by $u$. Solve for $\hat u$ and look up the corresponding $u$ in the tables.

Table 7.1 lists important properties of the Laplace transform and table 7.2 gives example Laplace transform pairs. In the tables, $k>0$, $a$, $b$, and $c$ are constants, normally positive, $n$ is a natural number, and

\hbox{erfc}(x) \equiv \frac2{\sqrt{\pi}}\int_x^\infty e^{-\xi^2} { \rm d}\xi

Table 7.1 assumes that $a$ and $b$ are positive.

Table 7.1: Properties of the Laplace transform.
$\displaystyle u(t)_{\strut}^{\strut}$ $\displaystyle \hat u(s)$
0. $\displaystyle \frac{1}{2 \pi{\rm i}}
\int_{{\strut}c-{\rm i}\infty}^{{\strut}c+{\rm i}\infty}
e^{-st} \hat u(s){ \rm d}s$ $\displaystyle \int_0^\infty u(t)e^{-st}{ \rm d}t$
1. $\displaystyle C_1 u_1(t) + C_2 u_2(t) $ $\displaystyle C_1 \hat u_1(s) + C_2 \hat u_2(s)$
2. $\displaystyle u(at)$ $\displaystyle a^{-1} \hat u(s/a)$
3. $\displaystyle
\frac{\partial^{{\strut}n} u}{\partial_{\strut} t^n}(t)$ $\displaystyle s^n \hat u(s) - s^{n-1} u(0) - \ldots -
\frac{\partial^{n-1} u}{\partial t^{n-1}}(0)$
4. $\displaystyle t^n u(t)$ $\displaystyle (-1)^n
\frac{\partial^{{\strut}n} \hat u}{\partial_{\strut} s^n}$
5. $\displaystyle e^{ct} u(t)$ $\displaystyle \hat u(s-c)$
$\displaystyle \bar u(t-b) \equiv H(t-b) u(t-b)$
$\displaystyle =
\begin{array}{lr} u(t-b) & (t-b>0)  0 & (t-b<0) \end{array}$
$\displaystyle e^{-bs} \hat u(s)$
7. $\displaystyle
\int_{{\strut}0}^{{\strut}t} f(t-\tau)g(\tau){ \rm d}\tau$ $\displaystyle \hat f(s)\hat g(s)$

Table 7.2: Selected Laplace transform pairs.
$\displaystyle u(t)$ $\displaystyle \hat u(s)$
1. $\displaystyle 1 $ $\displaystyle \frac{1}{s}_{\strut}$
2. $\displaystyle t^n$ $\displaystyle \frac{n!}{s^{n+1}}_{\strut}$
3. $\displaystyle e^{bt}$ $\displaystyle \frac1{s-b}_{\strut}$
4. $\displaystyle \sin(at)$ $\displaystyle \frac{a}{s^2+a^2}_{\strut}$
5. $\displaystyle \cos(at)$ $\displaystyle \frac{s}{s^2+a^2}_{\strut}$
6. $\displaystyle \frac{1}{\sqrt{\pi t}}_{\strut}$ $\displaystyle \frac{1}{\sqrt{s}}$
7. $\displaystyle \frac{1}{\sqrt{\pi t}}_{\strut} e^{-k^2/(4t)}$ $\displaystyle \frac{1}{\sqrt{s}} e^{-k\sqrt{s}}$
8. $\displaystyle \frac{k}{\sqrt{4\pi t^3}}_{\strut} e^{-k^2/(4t)}$ $\displaystyle e^{-k\sqrt{s}}$
9. $\displaystyle \hbox{erfc}\left(\frac{k}{2\sqrt{t}}\right)_{\strut}$ $\displaystyle \frac1s e^{-k\sqrt{s}}$

7.1.2 About the coordinate to be transformed

In many cases, $t$ is physically time, since time is most likely to satisfy the constraints $0<t<\infty$ and coefficients independent of $t$. Also, the Laplace transform likes initial conditions at $t=0$, not boundary conditions at both $t=0$ and $t=\infty$.