# 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.

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## Glossary

# Introduction to Laplace transform

Let *f* be an arbitrary (complex-valued or real-valued) function defined on the semi-infinite interval
\( [0, \infty ) ; \) then the integral

*f*provided that the above integral converges for some value \( \lambda = s \) of a parameter λ. Therefore, the Laplace transform of a function (if it exists) depends on a parameter λ, which could be either a real number or a complex number. Saying that a function

*f(t)*has a Laplace transform

*f*

^{L}means that for some λ =

*s*, the limit

**If a function**

**Theorem**:*f*is absolutely integrable over any finite interval from \( [0, \infty ) ; \) and the Laplace integral \( \int_0^{\infty} f(t)\,e^{-\lambda t} \,{\text d} t \) converges for some complex number \( \lambda = s , \) then it converges in the half-plane \( \mbox{Re}\,\lambda > \mbox{Re}\, s , \) i.e., in \( \left\{ \lambda \in \mathbb{C} \, : \, \Re\, \lambda > \Re\, s \right\} . \)

**Theorem:**The Laplace transform is a linear operator.

**Definition**: A function

*f*is said to be

**piecewise continuous**or

**intermittent**on a finite interval [a,b] if the interval can be divided into finitely many subintervals so that

*f(t)*is continuous on each subinterval and approaches a finite limit at the end points of each subinterval from the interior.

**Definition:**The

**Heaviside function**

*H(t)*is the unit step function:

*u(t)*is assumed to be zero at

*t*= 0. ■

**Definition**: A function \( f(t), \ t \in [0, \infty ) \) is said to be a

**function-original**if it has a finite number of points of discontinuity (finite jumps) on every finite subinterval of \( [0, \infty ) \) and it grows not faster than an exponential, that is

*c*,

*M*, and

*T*. Moreover, we assume that at points of discontinuity the values of a function-original are equal to the corresponding mean values:

**image**.

**Definition**: We say that a function

*f*is of exponential order if for some constants

*c*,

*M*, and

*T*the inequality \( | f(t) | \le M\, e^{ct} \) holds. We abbreviate this as \( f = O\left( e^{ct} \right) \) or \( f \in O\left( e^{ct} \right) . \) A function

*f*is said to be of exponential order α, or

*eo*(α) for abbreviation, if \( f = O\left( e^{ct} \right) \) for any real number

*c*> α, but not when

*c*< α. ■

**Theorem:**The Laplace transform exists for a function of exponential order.

**Theorem:**The Laplace transform establishes a one-to-one correspondence between functions-originals and their images. ■

Although *Mathematica* has a built-in function HeavisideTheta (which is 1 for *t* > 0 and 0 for
*t* < 0), it is convenient to define the Heaviside function directly:

HVS[t_] := Piecewise[{{1, t>0}, {1/2, t==0}, {0, True}}]

*Mathematica*has

**UnitStep**function that can also be used to redefine the Heaviside function:

UnitStep[0] = 1/2;

Protect{UnitStep]

# Examples

**Example**. The Laplace transform of the unit function and the Heaviside function is the same and equals

**Example**. We find the Laplace transform of the power function:

**Example**. The Laplace transform of the exponential function is

**Example**. To find the Laplace transform of the trigonometric functions, we recall that they are real and imaginary parts of pure imaginary exponential functions:

**j**is the unit vector in positive vertical direction on the complex plane, so

**j**

^{2}= -1. Then using the previous example formula, we get

LaplaceTransform[HeavisideTheta[t - 1] t, t, lambda]

LaplaceTransform[DiracDelta[t], t, lambda]

LaplaceTransform[f''[t], t, lambda]

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