# Preface

This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in normal font. This means that you can copy and paste all commands into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

# Library of Laplace transforms

It is useful to have a “library” of Laplace transforms to hand; some common ones are listed below.
f(t)   fL(λ)
H(t)  1/λ
$$e^{at}$$   1/(λ -a)
sin ωt $$\frac{\omega}{\lambda^2 + \omega^2}$$
sinh ωt   $$\frac{\omega}{\lambda^2 - \omega^2}$$
$$e^{kt}\,\sin \omega t$$   $$\frac{\omega}{(\lambda -k )^2 + \omega^2}$$
tp  $$\frac{\Gamma (p+1)}{\lambda^{p+1}}$$
f(t)   fL(λ)
δ(t)
$$t^p\, e^{kt}$$   $$\frac{\Gamma (p+1)}{(\lambda -k)^{p+1}}$$
cos ωt  $$\frac{\lambda}{\lambda^{2} + \omega^2}$$
cosh ωt  $$\frac{\lambda}{\lambda^{2} - \omega^2}$$
$$e^{kt}\,\cos \omega t$$   $$\frac{\lambda}{(\lambda -k)^{2} + \omega^2}$$

# Elementary Properties of the Laplace Transforms

1. Linearity: $${\cal L} \left[ \alpha\,f(t) + \beta\, g(t) \right] = \alpha\, {\cal L} \left[ f \right] + \beta\,{\cal L} \left[ g \right] = \alpha\, f^L + \beta \, g^L .$$
2. The derivative rule: $${\cal L} \left[ f^{(n)} (t) \right] = \lambda^n - \sum_{k=1}^n \lambda^{n-k} f^{(k-1)} (+0) .$$
3. Convolution rule: $${\cal L} \left[ f \ast g \right] = f^L \,g^L .$$
Recall that the convolution of two functions f and g is
$\left( f \ast g \right) (t) = \int_0^t f(t-\tau )\,g(\tau ) \,{\text d} \tau = \int_0^t g(t-\tau )\,f(\tau ) \,{\text d} \tau = \left( g \ast f\right) (t) .$
4. Shift rule: $${\cal L} \left[ f(t-a)\,H(t-a) \right] = e^{-a\lambda} \,f^L (\lambda ) .$$
5. Similarity rule: $${\cal L} \left[ f(kt) \right] = \frac{1}{k}\, f^L \left( \frac{\lambda}{k} \right) .$$
6. Attenuation rule: $${\cal L} \left[ e^{-at} \, f(t) \right] = f^L \left( \lambda +a \right) .$$
7. Differentiation rule: $$\frac{{\text d}}{{\text d} \lambda} \, f^L (\lambda ) = - {\cal L} \left[ t\, f(t) \right] .$$
8. Integration rule: $${\cal L} \left[ t^n \ast f(t) \right] = \frac{n!}{\lambda^{n+1}} \, f^L (\lambda ) .$$
9. The Laplace transform of periodic functions.
If $$f(t) = f(t+ \omega ) ,$$ then
$f^L (\lambda ) = \int_0^{\infty} f(t )\,e^{-\lambda\,t} \,{\text d} t = \frac{1}{1- e^{-\omega\lambda}} \, \int_0^{\omega} \,f(t ) \,e^{-\lambda \, t} \,{\text d} t .$