**What is the Laplace transform?**

The **Laplace transform** In recent years, it has been of great importance in engineering, mathematics, and physics studies, among other scientific areas, since in addition to being of great theoretical interest, it provides a simple way to solve differential equations, transforming them into algebraic equations.

Originally the Laplace transform was presented by Pierre-Simon Laplace (1745-1827) in his study on the theory of probability, and at first it was treated as a mathematical object of purely theoretical interest.

Current applications arise when various mathematicians tried to give a formal justification to the «operational rules» used by Oliver Heaviside (1850-1925) in the study of equations of electromagnetic theory.

**Definition of the Laplace transform**

Let f be a function defined for t ≥ 0. The Laplace transform is defined as follows:

The Laplace transform is said to exist if the previous integral converges, otherwise the Laplace transform is said not to exist.

In general, lowercase letters are used to denote the function to be transformed and the uppercase letter corresponds to its transform. In this way we will have:

**examples**

Consider the constant function f

Whenever the integral converges, that is, whenever s > 0. Otherwise, s < 0, the integral diverges.

Let g

By integrating by parts and knowing that te-st tends to 0 when t tends to infinity and s > 0, together with the previous example we have that:

The transform may or may not exist, for example for the function f

Sufficient conditions to guarantee that the Laplace transform of a function f exists are that f is piecewise continuous for t ≥ 0 and is of exponential order.

A function is said to be piecewise continuous for t ≥ 0, when for any interval [a, b] with a > 0, there is a finite number of points tk, where f has discontinuities and is continuous on every subinterval [tk-1,tk].

On the other hand, it is said that a function is of exponential order c if there exist real constants M > 0, c and T > 0 such that:

As examples we have that f

Formally we have the following theorem:

**Theorem (Sufficient conditions for existence)**

If f is a piecewise continuous function for t > 0 and of exponential order c, then the Laplace transform exists for s > c.

It is important to highlight that this is a sufficiency condition, that is, it could be the case that there is a function that does not meet these conditions and even so its Laplace transform exists.

An example of this is the function f

**Laplace transform of some basic functions**

The following table shows the Laplace transforms of the most common functions.

**History of the Laplace transform**

The Laplace transform owes its name to Pierre-Simon Laplace, a French mathematician, astronomer and theorist who was born in 1749 and died in 1827. His fame was such that he was known as the Newton of France.

In 1744, Leonard Euler (1707-1783) devoted his studies to integrals with the form

as solutions of ordinary differential equations, but he quickly abandoned this investigation. Later, Joseph Louis Lagrange (1736-1813), who greatly admired Euler, also investigated this type of integral and related it to probability theory.

**1782, laplace**

In 1782 Laplace began to study these integrals as solutions to differential equations and, according to historians, in 1785 he decided to reformulate the problem, which later gave rise to the Laplace transforms as they are understood today.

Having been introduced into the field of probability theory, it was of little interest to scientists at the time, and was only seen as a mathematical object of theoretical interest only.

**Oliver Heaviside**

It was in the middle of the 19th century when the English engineer Oliver Heaviside discovered that differential operators can be treated as algebraic variables, thus giving them their modern application to Laplace transforms.

Oliver Heaviside was an English physicist, electrical engineer and mathematician who was born in London in 1850 and died in 1925. While trying to solve problems of differential equations applied to the theory of vibrations, and using Laplace’s studies, he began to shape the applications of the Laplace transforms.

The results presented by Heaviside quickly spread throughout the scientific community of the time, but since his work was not rigorous, he was quickly criticized by the more traditional mathematicians.

However, the usefulness of Heaviside’s work in solving equations in physics made his methods popular with physicists and engineers.

Despite these setbacks and after a few decades of failed attempts, by the early 20th century a rigorous justification could be given to the operational rules established by Heaviside.

These attempts paid off thanks to the efforts of various mathematicians, such as Bromwich, Carson, Van der Pol, among others.

**Properties of the Laplace transform**

Among the properties of the Laplace transform, the following stand out:

**linearity**

Let c1 and c2 be constants and f

Due to this property, the Laplace transform is said to be a linear operator.

**Example:**

**First translation theorem**

If it happens that:

And ‘a’ is any real number, so:

**Example:**

Since the Laplace transform of cos(2t) = s/(s^2 + 4) then:

**Second translation theorem**

Yeah

So

**Example:**

If f

is G(s)= 6e-2s/s^4

**scale change**

Yeah

And ‘a’ is a real other than zero, we have that

**Example:**

Since the transform of f

**Laplace transform of derivatives**

If f, f’, f»,…, f(n) are continuous for t ≥ 0 and are of exponential order and f(n)

**Laplace transform of integrals**

Yeah

So

**Multiplication by tn**

if we have to

So

**division by t**

if we have to

So

**periodic functions**

Let f be a periodic function with period T > 0, that is, f(t +T) = f

**Behavior of F(s) as s tends to infinity**

If f is piecewise continuous and of exponential order and

So

**Inverse transforms**

When we apply the Laplace transform to a function f

We know that the Laplace transforms of f

Some common inverse Laplace transforms are as follows

Furthermore, the inverse Laplace transform is linear, that is, it holds that

**Exercise**

find

To solve this exercise we must match the function F(s) with one of the previous table. In this case, if we take an + 1 = 5 and using the linearity property of the inverse transform, we multiply and divide by 4! Getting

For the second inverse transform we apply partial fractions to rewrite the function F(s) and then the linearity property, obtaining

As we can see from these examples, it is common that the function F(s) being evaluated does not match precisely one of the functions given in the table. For these cases, as can be seen, it is enough to rewrite the function until reaching the appropriate form.

**Applications of the Laplace transform**

**Differential equations**

The main application of Laplace transforms is to solve differential equations.

Using the property of the transform of a derivative it is clear that

And of the n-1 derivatives evaluated at t = 0.

This property makes the transform very useful for solving initial value problems involving differential equations with constant coefficients.

The following examples show how to use the Laplace transform to solve differential equations.

**Example 1**

Given the following initial value problem

Use the Laplace transform to find the solution.

We apply the Laplace transform to each member of the differential equation

By the property of the transform of a derivative we have

Expanding the entire expression and isolating Y(s) we are left with

Using partial fractions to rewrite the right hand side of the equation we get

Finally, our goal is to find a function y

**Example 2**

solve

As in the previous case, we apply the transform on both sides of the equation and separate term by term.

In this way we have as a result

Substituting with the given initial values and solving for Y(s)

Using simple fractions we can rewrite the equation as follows

And applying the inverse Laplace transform gives us as a result

From these examples one might mistakenly conclude that this method is not much better than traditional methods for solving differential equations.

The advantages of the Laplace transform is that it is not necessary to use variation of parameters or worry about the various cases of the method of undetermined coefficients.

Also, when solving initial value problems by this method, we use the initial conditions from the beginning, so no other calculations are necessary to find the particular solution.

**Systems of differential equations**

The Laplace transform can also be used to find solutions to simultaneous ordinary differential equations, as the following example shows.

**Example**

Solve

With the initial conditions x(0) = 8 and y(0) = 3.

if we have to

So

Solving gives us the result

And by applying the inverse Laplace transform we have

**Mechanics and electrical circuits**

The Laplace transform is of great importance in physics, mainly having applications for mechanics and electrical circuits.

A simple electrical circuit is composed of the following elements:

A switch, a battery or source, an inductor, a resistor and a capacitor. When the switch is closed an electric current is produced which is denoted by i

By Kirchhoff’s second law, the voltage produced by source E to the closed circuit must be equal to the sum of each of the voltage drops.

The electric current i