Definition

There are two types of Laplace transformations.

Unilateral Laplace transform takes in an equation, f(t) defined for $t\ge0$, and gives a function as output with a different domain. The transform is given by,

$\ell \{f(t)\}=\int _{ 0 }^{ \infty }{ { e }^{ -st }f(t)dt }$

where $s$ is referred to as frequency.

Bilateral Laplace transform is just an extension of the former, which takes even the other side of the real axes into consideration. Hence it is given by,

$\ell \{f(t)\}=\int _{ -\infty }^{ \infty }{ { e }^{ -st }f(t)dt }$

The Laplace transformation of a function, $f(t)$ is denoted as $F(s)$ i.e.

$\ell \{f(t)\}=F(s)$

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Motivation

We already saw a method by which we can get the constituent frequencies of some sound. We also have a method to get the frequency distribution of different constituent pure tones of a given sound. (Refer Fourier Transform)

While drawing out the frequency distribution from a time-domain function, we know how we can use the complex exponentials. Is there anything to generalize in that particular equation? (Fourier Transform equation)

Let us look at another perspective.

Consider some complex differential equations, say some differential equations dealing with the flow of heat through an insulated conductor (w.r.t time). Since it is always difficult solving such equations, we can try to convert them as algebraic functions that are easy to handle (easy to simply and solve). So is there any tool to solve such differential equations by making them analogous to algebraic equations?

In these sets of sections, we check how “Laplace Transformations” can be such a useful tool. We also check out the wide range of applications it has over many fields.

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Bird's eye view

Laplace transformations are a tool used in many fields. They take in an equation and convert it to its alternative form.

To be precise, they change functions with time-domain into functions with frequency domain. Hence,

$\ell f(t)=F(s)$

where $\ell$ denotes the Laplace transformation.

Hence we also define Inverse Laplace transformation where we pass in a function with frequency domain and get a function with time-domain.

$\ell^{-1}(F(s))=f(t)$

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Video 1 : Basic LaplaceTransform intuition

Context of the Definition

By simply using the term Laplace Transform, in most of the cases, we mean Unilateral Laplace transform. (unless stated explicitly)

Since the transformation consists of integrals, therefore we say the Laplace transform exists for a function, $f(t)$ given

$lim _{ t\longrightarrow \infty }{ \int _{ 0 }^{ t }{ { e }^{ -st }f(t)dt }}$

exists. If the integral doesn’t exist, we say the function fails to have a Laplace transform!.

In all cases, $s$ represents the frequency which consists of magnitude as well as a phase. (simply $s$ is a complex number).

NOTE: The region in $s$ for which the integral exists is referred to as Region of Convergence.

Apart from changing the domain, these transformations also help in simplifying the complex differential equations!

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Video 2 : Basic intuition of laplaceTransform in differential equations

Therefore we generally tableize the standard Laplace transformations (their inverses as well) which can be used to convert the differential equations as just algebraic problems.

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NOTE: Here remember that $\ell \{ c_{ 1 }f(t)+c_{ 2 }g(t)\} =c_{ 1 }\ell \{ f(t)\} +c_{ 2 }\ell \{ g(t)\}$ $and$ $\ell \{ f'(t)\} =s\ell \{ f(t)\} -f(0)$

Here there is also a way to compute the inverse Laplace transformation (as for normal one), which is referred to as Convolution, though we aren’t going to dive into it in this lecture note.

While dealing with our friend, we might come across two functions, “Unit Step function” and “Dirac function” which have a very wide range of usage.

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Unit Step Function

These functions have a unit jump in their graphs. These are given by

$\mu _{ c }(t)=0\quad t<c$

$\mu _{ c }(t)=1\quad t\ge c$

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Video 3 : Intuition of a Unit Step function

Dirac Delta Function

Consider an impulse applied over an object. We do get some differential equations. But what would be the Laplace transformation of the applied impulse? (as its value tend to $\infty$ at the time impulse is applied and remains $0$ for the rest).

Hence these functions are used to depict the sudden jump in the value at a particular instance, while will be $0$ for the rest. These are given by

$\delta (x)= \infty \quad x=0$
$\delta (x)=0\quad x\neq 0$

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Video 4 : Standard Dirac Delta function

Here we can get the area bounded under this curve, which is equal to $1$!

$\int _{ -\infty }^{ \infty }{ \delta (x)dx } = 1$

Now is it actually possible to construct such an equation? Lets consider the following:

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Video 5 : Formation of Dirac Delta Function

Here we are able to get a standard result from their Laplace transformations, which makes them unique and are often used widely in physics.

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Applications

• They are extensively used in solving complex differential equations as simple polynomial equations. Hence they have a major role in simplifying calculations in system modeling, solving digital signal processing problems.
• In pure physics, Laplace transformations are used to change a function from the time domain to the frequency domain.
• Laplace transforms are used to transform the signals (which can be made analogous to equations!) while sending signals over any communication medium such as FM, cellular phones.
• They are also used in many fields of physics and digital electronics.
• They make it possible to study the analytic part of Nuclear physics and are also used in solving the radioactive decay equations.
• They are also used to solve the equations obtained in control systems which regulates the behavior of other systems such as a home heating controller.

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History

Though these transformations got a formal method of solving from the work of Laplace, “Théorie analytique des probabilités (1812)”, they were even used extensively by Léonard Euler in the 17th century itself. He mainly worked on the integral transforms by considering them as inverse Laplace transform in solving linear ordinary differential equations and the same has also been given credit, by Laplace in his work.

Later in the late 18th century, it was Spitzer who attached the name, Laplace to the equation

$y=\int _{ a }^{ b }{ { e }^{ sx } } f(s) ds$

employed by Euler.

In the 19th century, the idea was extended to its all possible complex forms and was eventually extended to two variables. In no time, they were also used to transform equations of radioactive decay, hence simplifying much of complex equations.

In 1920, Bernstein used the expression (in his work on theta functions)

$f(s)=\int _{ 0 }^{ \infty }{ { e }^{ -su }\phi (u) du }$

and called it a Laplace transformation.

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References

• The Laplace Transform: Theory and Applications by Joel L. Schiff

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• Schaum's Outline of Laplace Transforms
• Brian Davies. Integral Transforms and their Applications
• L. Debnath, D. Bhatta, Integral Transforms and Their Applications

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The following notes and their corrosponding animations were created by the above-mentioned contributor and are freely avilable under CC (by SA) licence. The source code for the said animations is avilable on GitHub and is licenced under the MIT licence.

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