Lesson 1 - Laplace Transform Definition (Engineering Math)
Summary
TLDRIn this comprehensive tutorial series, Jason introduces students to the LL (Laplace) transform, a powerful mathematical tool used in science and engineering to solve differential equations more efficiently. Starting with the basics, the course builds up to using a table of LL transforms and properties for both direct and inverse transformations. The script emphasizes the practicality of the LL transform in various fields and guides learners through the process of transforming functions of time into the S domain, showcasing the transform's utility in simplifying complex problems.
Takeaways
- 📚 The course is designed to teach students how to master the use of the Laplace Transform, starting from scratch and building up to practical applications.
- 🔧 The Laplace Transform is a powerful tool used in science and engineering to solve complex problems, particularly differential equations, more efficiently.
- 📈 The course covers the Laplace Transform properties and how to apply both forward and inverse transforms to functions of time.
- 🎓 Students are expected to have basic knowledge of calculus and algebra, as these are fundamental to understanding and performing the Laplace Transform calculations.
- ⏳ The Laplace Transform takes a function of time and transforms it into a function in the 'S-domain', simplifying the process of solving differential equations.
- 📉 The integral definition of the Laplace Transform involves integrating a function of time multiplied by an exponential function over time from 0 to infinity.
- 📑 A table of common Laplace Transforms is an essential resource that students will learn to apply and derive, starting with simple functions and building up to more complex ones.
- 🌟 The Laplace Transform of an exponential function e^(λt) is particularly important and results in a simple function of 's', which is 1 / (s - λ), provided s > λ.
- 🔄 The Laplace Transform can be applied in a wide range of scientific and engineering disciplines, including electrical circuits, mechanical systems, and aerospace engineering.
- 📝 The process of solving problems with the Laplace Transform involves transforming the problem into the S-domain, solving it there with simpler algebraic methods, and then transforming the solution back to the time domain.
- 📚 The special case of the Laplace Transform when λ = 0, which is the transform of the constant function 1, results in 1 / s, valid for s > 0.
Q & A
What is the main objective of the LLas Transform Tutor course?
-The main objective of the LLas Transform Tutor course is to build students' skills so that they can master the use of the LLas transform, including how to use a table of LL transforms and the properties of the LLas transform to perform both direct and inverse transformations.
Why are LLas transforms important in science and engineering?
-LLas transforms are important in science and engineering because they provide a practical skill used to solve more complicated problems in various classes, including electrical engineering, mechanics, pure mathematics, and differential equations. They simplify the process of solving differential equations, which are ubiquitous in these fields.
What mathematical background is expected from students before starting the LLas Transform course?
-Students are expected to have a basic understanding of calculus, ideally at the level of 'calculus one', and proficiency in algebra. This background will enable them to perform the mathematical operations required for the LLas transform.
What is the basic concept behind the LLas transform?
-The basic concept behind the LLas transform is to take a function of time (or any function) and transform it into a new function in a different domain (the S domain), simplifying the process of solving differential equations and other complex problems.
How does the LLas transform help in solving differential equations?
-The LLas transform helps in solving differential equations by providing a unified method to convert a complex differential equation into a new form in the S domain, which is often easier to solve using algebraic methods. Once solved, the solution can be transformed back into the time domain.
What is the mathematical expression for the LLas transform of a function F(t)?
-The mathematical expression for the LLas transform of a function F(t) is given by the integral from 0 to infinity of e to the minus s t * F of t with respect to t, denoted as F(S) = L{F(t)} = ∫[0,∞] e^(-st) * F(t) dt.
What is an improper integral, and how is it related to the LLas transform?
-An improper integral is an integral where at least one of the limits is infinity, or the integrand has an unbounded point in the interval of integration. In the context of the LLas transform, the improper integral is used to transform a time-domain function into the S domain.
What is the significance of the LLas transform of an exponential function e^(Λt)?
-The LLas transform of an exponential function e^(Λt) is significant because it results in a simple function of S, which is 1 / (s - Λ). This result is a fundamental part of the LLas transform table and is used as a basis for transforming more complex functions.
What is the LLas transform of the constant function '1'?
-The LLas transform of the constant function '1' is 1 / s, which is derived from the special case of the exponential function e^(Λt) when Λ equals zero.
Why is it necessary to consider the condition 'S greater than Λ' when discussing the LLas transform of e^(Λt)?
-The condition 'S greater than Λ' ensures that the denominator in the resulting function of S from the LLas transform does not become zero, thus avoiding undefined or infinite values in the transform.
Can you provide an example of how the LLas transform simplifies the solution of a real-world problem?
-While the script does not provide a specific real-world example, the LLas transform simplifies real-world problems by allowing engineers and scientists to convert complex time-domain differential equations into simpler algebraic equations in the S domain. Solving these equations is typically more straightforward, and the solutions can then be transformed back to the time domain to provide practical solutions to real-world problems.
Outlines
📚 Introduction to LLas Transforms
Jason introduces the LLas Transform Tutor series, aiming to build students' skills in mastering the LLas transform, a tool used in science and engineering. He emphasizes that no prior knowledge is expected but by the end, students will understand and be able to apply the transform, its table, and its properties to solve complex problems. The LLas transform is a calculus-based technique for converting functions of time into a different domain, simplifying the solution of differential equations, which are ubiquitous in various fields of study.
🔍 Understanding the LLas Transform Concept
The script delves into the definition of the LLas transform, explaining it as an integral from 0 to infinity of a function of time multiplied by an exponential term. This process transforms a time-domain function into a new function in the S-domain. The explanation includes the mathematical notation and the significance of each component in the transform formula. It also touches on the concept of improper integrals and how limits of integration are applied in the context of the LLas transform.
📈 Calculating the LLas Transform of Exponentials
The tutorial focuses on calculating the LLas transform of exponential functions, a fundamental skill in mastering the transform. The process involves substituting the exponential function into the LLas transform formula and performing an integral with respect to time. The explanation includes a step-by-step guide on how to simplify the integral, apply a substitution to make it more manageable, and evaluate the limits of integration to arrive at a function of S, which is the result of the transform.
📘 Special Case: LLas Transform of One
The script highlights a special case of the LLas transform where the exponential function becomes a constant one, leading to the conclusion that the LLas transform of one is 1/s, with the condition that S is greater than zero to avoid undefined expressions. This result is emphasized as an important takeaway from the tutorial, marking the simplicity of the transform for constant functions.
🔧 LLas Transform as a Problem-Solving Tool
The tutorial concludes by reinforcing the practical applications of the LLas transform, particularly in solving differential equations across various scientific and engineering disciplines. It positions the LLas transform as a valuable skill that simplifies complex problem-solving by transforming time-domain functions into a more algebraically manageable domain. The script encourages students to follow through with the tutorial to gain proficiency in applying the LLas transform to real-world problems.
Mindmap
Keywords
💡LLas Transform
💡Differential Equations
💡Time Domain
💡S Domain
💡Improper Integral
💡Exponential Function
💡Algebraic Simplification
💡Inverse LLas Transform
💡Table of LLas Transforms
💡Engineering Applications
Highlights
Introduction to the LLas Transform Tutor series, aimed at building practical skills for mastering the LLas transform.
Expectation of no prior knowledge of the LLas transform, with the course aiming to teach both its concept and application.
The LLas transform is a practical skill used in science and engineering to solve complex problems.
The LLas transform is a unified method for solving differential equations more easily than traditional methods.
The LLas transform involves calculus and algebra, requiring proficiency in these areas.
The LLas transform takes a function of time and transforms it into a new function in the S domain.
The usefulness of the LLas transform is in its ability to simplify the process of solving differential equations.
The LLas transform is integral to various branches of science and engineering, including electrical, mechanical, and aerospace.
The LLas transform formula is introduced, explaining the process of transforming a time function into the S domain.
Explanation of the improper integral involved in the LLas transform and its mathematical rigor.
The process of creating a table of LLas transforms for common functions to simplify problem-solving.
Detailed walkthrough of calculating the LLas transform of an exponential function e^(Λt).
The result of the LLas transform of e^(Λt) is a function of s, specifically 1/(s - Λ), highlighting its significance.
Special case analysis when Lambda equals zero, revealing the LLas transform of one to be 1/s.
Emphasis on the importance of understanding the LLas transform for its wide applicability in real-world problems.
The LLas transform is positioned as a valuable tool for anyone in engineering or science due to its broad utility.
Encouragement to follow the course for mastering the LLas transform and its applications in solving differential equations.
Transcripts
hello I'm Jason welcome to the llas
transform tutor uh and here what we're
going to do in this lesson and in this
whole set of lessons is really build
your skills so that you can Master how
to use the llas transform now I expect
you to know absolutely nothing about the
llas transform going into this course
but at the end of it you'll not only
know what the transform is U but also
how to use it and how to use a table of
LL transforms and also the llas
transform properties to take lla
transforms and also inverse lla
transforms so it's a very practical
skill that we use in science and
engineering classes uh you know in order
to really solve some more complicated
problems and you'll find that lots and
lots of different classes that you'll
take as you go on in your engineering
sequence or in your science sequence
will use the LL transform so if you ever
get to a class in uh electrical
engineering or mechanics or uh uh pure
mathematics or differential equations
where LL transforms are used uh then
what you'll be able to do is come into
here and get that whole sequence of
information on how to use this very
powerful tool along with lots of
problems to practice your skills so uh
I'm going to obviously write some math
on the board in a second but in the uh
General scheme of things what the heck
is this thing called a lla transform uh
well first of all it is uh it is is
calculus so you have to know some
calculus in order to be able to
calculate the transform so I expect you
up until this point to know a little bit
of calculus really not much more than
calculus one to be honest with you will
enable you to be able to do the math
here but you will have to be proficient
with it also obviously your algebra
skills because I know that you all know
how to do basic algebra but when you get
into taking some of these transforms it
can um it can quickly explode into a
bunch of things to simplify none of it's
hard it's just a lot of little terms to
keep track of and so we'll be doing that
here and I'll show you as I go but it's
basically calculus and uh what you're
basically doing is you're using a
technique to take a function so here
we're going to be talking about
functions of time but really it's it's
any function you know F ofx you normally
study an algebra um but you know when
you get into engineering and studying
real systems you're usually talking
about functions of time things that
change with time for instance in
electrical engineering you might have an
input signal you know like a sign you
know wave or some kind of U random voice
uh recording of a voice might give you
an amplitude as a function of time that
would be a function of time right and so
the LL transform takes a function of
time and transforms it to a new function
that's uh in a new uh new way of
representing that original function and
you might say well why would you ever
care about doing that and the the truth
is the reason why it works is beyond the
scope of what I'm talking about right
now but basically llas transforms let
you solve a lot of different types of
differential equations in an easier way
than doing them by hand so in in the
original way so if you think back to my
differential equations tutor or any
class that you may have taken in
differential equations you should agree
with me that it's kind of a nightmare to
solve a lot of differential equations
there's a lot of math involved a lot of
theory and differential equations really
have a lot of different techniques so
you may do method of undetermined
coefficient coefficients in differential
equations you might do integrating
factors in different equations you might
use exact differential equations um
there's lots of different ways to solve
differential equations and a lot of it
boils down to recognizing what the
equation is figuring out the method that
works and then applying it and so it
seems to be disconnected though because
there's lots of different techniques for
different styles of differential
equations LL transform is a unified
method that really allows you to take a
differential equation and apply the
transform to it and change it into a new
form that's easier to solve
and then when you get the answer in the
LL domain or the S domain we'll talk
about in a minute then you can transform
the answer back into the time domain
that's why it's called a transform we
take the problem or the function or
whatever it is we transform it into the
llao domain which we're going to call
the S domain and a lot of times doing
the solution in that domain just becomes
algebra or some other easier Simple
Solution method than differential
equations brute force method so we solve
it over here in the llas domain or the S
Dom domain we get an answer and then we
can take that an inverse transform it
back into a function of time um so
that's kind of without getting into the
details that's basically why it's useful
and as you know differential equations
are used in all branches of science and
engineering we use them in electric
circuits we use them in mechanical
systems use them in control systems
theory use them in chemical engineering
aerospace engineering differential
equations are everywhere and because of
that and because LL transform is kind of
a general method um to to make these
types of problems easier you're going to
find that LL transforms pop up in all
kinds of different situations and so
that's why I want to put this tutorial
in its own little course because you can
use it if you're mechanical engineer
electrical engineer uh physicist you
know anything so let's just dive into it
first of all uh and talk about it a
little bit so we have what we call the
llas
transform okay and I'm not going to beat
around the bush too much we're just
going to write it on the board and we'll
talk about it so the llas transform
gives you a function in the lass domain
which is the S domain that's why you're
getting a function capital f of x of s
that's what you're arriving at when you
perform this thing called the LL
transform and it's the integral from 0o
up to Infinity of e to the minus s t * F
of t d t so this ladies and gentlemen is
the famous L PLO transform all right so
what you're doing all right is if I give
you a function of time think about any
function of time could be a square wave
could be a s of T could be a saw too
could be any function of time you want
could be X squ or some function of time
squared you stick that function of time
in F of T then you multiply by this
exponential e to the minus St right and
so you're getting something inside of
the integral sign you integrate it over
time right and then you apply the limits
of integration to the answer that you
get which are obviously going into your
uh into your time spots because you're
integrating over time so the limits of
integration is from zero to Infinity in
terms of time right then when you do all
of that you're going to get just a
function of s back because if you
integrate over time and then you plug in
limits of integration over time there's
no T anymore left in what you've done
there's just the other uh uh aspects of
it here so there's no T anymore here
there's no T anymore here because you've
plugged in limits of integration only s
remains so when you do a ll transform
properly you get purely a function of s
and so that's the notation here you put
F of T little F of T is your input
function and out of this computation
comes another function we call it
capital F we use the same letter capital
F and little F to imply that little F
and capital F are related to one another
we we put a little F into the LL
transform and we get a capital f out
which is a function of s all right so we
can also
write we can also
write
as the following F capital F of s which
is the the pl transform is Curly L I
don't know how else you would say that
right F of T so so this is the shorthand
way of saying hey I'm going to take the
LL transform of some function of time or
some function and when I do that I'm
just going to get a pure function of s
so these two things are exactly the same
thing this is the shorthand way of
writing what you're doing you're taking
the the pl transform of a function of
time this is one of the details this is
what you actually do you have to
multiply this you have to integrate it
you have to plug in the limits of
integration so what we're doing is we're
transforming a function from the time
domain to the S domain
that's why I'm telling you that because
the answers that you get are just
functions of what we call S um and so we
get a new function as a result of that
and so I know that you've seen this this
kind of action before where you have
zero and infinity up on an integral you
probably saw that in calculus but just
to refresh your memory uh that's called
an improper integral right so when you
have an improper
integral and you've studied this once
probably a long time ago but basically
when you have any kind of uh 0 to
Infinity e to Theus St F of T DT what
you really are doing is you do the
integration but what you do is you take
the limit so what I'm going to do is
I'll say let me write it in a different
color what you're really doing here is
you're taking the
limit and you'll see what I mean in a
minute as H goes to
Infinity of the integral from 0 to h eus
s t uh F of T
DT I'm just mostly explaining what
you're doing with the calculus when you
see an integral like this with one of
the limits of integration is an Infinity
then what you're really doing is you're
taking the integral and you pretend that
this top number is is just some variable
you calculate the integral and then you
take the limit as H goes to Infinity
that's how you would write it
mathematically but in practice what you
really do is you enter you evaluate the
interal you plug Infinity in one limit
and zero in the other limit and then you
know how to evaluate Infinities you know
if they're on the denominator if they're
on the numerator they might drive your
answer a different way uh and so that's
basically what you're doing but
mathematically rigorously what you're
doing is you're integrating the guy and
then you're taking the limit as this uh
limit goes to
Infinity all right so what I would like
to do at this
point is um calculate a llas transform
basically what you have is we have the
definition that was on the board a
minute ago that's um really the the
bread and butter of it if you understood
how to how to apply that to all
situations then you wouldn't need me you
would just do the Theo transform all the
time but the truth is that as you do
some of these things um there's a little
bit of Tricks along the way to help make
it comprehensible to you and I'm going
to show you those here and what you're
going to find pretty quickly in any book
that deals with the llas transform is
they're going to give you a table of
transforms in other words there are some
pretty basic functions that are pretty
easy to apply this definition to that
you get the answer and that answer is
very useful going forward um because
some functions pop up in nature all the
time you know exponentials pop up all
the time um for instance so we want to
learn how to take the LL transform of
things like exponentials and things like
cosiness and signs because those things
pop up all the time so what we're going
to do in this lesson in the next few
lessons is we're going to apply that
definition of the laas transform this
this full-blown definition of the LL
transform to a few core um functions and
then we're going to assemble our own
table ofl transforms which in many cases
you'll just find them listed in a book
but I'm going to derive how they get how
we get there so that you'll understand
how we're applying this once you have a
basic table of La transforms of common
functions then what you typically do is
you use that table to explore more
complicated functions so we're kind of
getting our footing we first learn what
the real definition of laass transform
is then we're going to apply it to some
simple functions we're going to assemble
our table of common functions that we
are useful to know the lla transform and
then we'll apply that to many problems
going forward so the most important uh
function that you could probably know
how to or or want to know how to take
the LL transform of would be the LL
transform of the function e to the
Lambda t e to the Lambda T now this is
just an exponential it's e to the T it's
a function of time right but there's a
constant in front we're calling it
Lambda in your book it might say e to
the a t or it might say e to the BT or I
might say e to the alpha t e to the beta
T it doesn't matter but there is some
number in front of t t it could be one
it could be two it's left open-ended
because a lot of times exponentials pop
up in solutions to differential
equations they pop up in lots of
physical systems right so it's an
exponential if it's e to the T the ual
value of Lambda just changes the slight
shape of what it looks like so we're
leaving that as a constant left open in
there but we want to find out in general
what would this LL transform be so the
way you do it is you just apply it
directly 0 to infinity and you apply
that LL transform which if you remember
was e to Theus s t f of T DT this was
the General equation uh for the LL
transform so then what we do is we say 0
to Infinity eus S T then we put our
function of time in here which is e to
the Lambda t d t and this is what we
want to integrate and we suspect and we
claim and I'm telling you that once you
do this properly all you get is a
function of s and we say that that LL
transform function of s is inexorably
tied to the function of time through
this thing called the transform and you
have to trust me on faith that once you
know how to do these transforms and once
you get proficient at them that they
help you solve real problems you're
going to have to take that part on faith
we're going to get to that part a little
bit later all right so what we're going
to do is begin to evaluate this interval
I could just give you the answer of
course I could do that but I want to
walk you through it so that you can
really feel like you understand what's
really going on so notice this is uh two
exponentials so we can simply combine
the exponents at the top that's
something we can do because these
exponentials are multiplied together
right
so we can say that we'll
have s- Lambda DT make sure you agree
that there's nothing different here if
you distribute the negative n you get
Negative s which is what we have here uh
we have a t here as well right neg s t
and then here this will be positive
Lambda T positive Lambda T so basically
uh we are adding these guys together in
the exponent we're basically adding
negative St plus Lambda T we're adding
that together but I pull the negative
sign out so but it still becomes St plus
Lambda T in the exponential uh there all
right so what you need to do is is
integrate this right integrate this this
is an exponential integrals of
exponentials are relatively simple the
only thing is um notice we're
integrating over time here is your time
variable Lambda is just a constant s is
going to end up being a variable that we
are going to have in our transform in
the end but since we're not integrating
over s we're integrating over T you
basically treat S as a constant and
that's really important for you to
understand you know anytime you take
derivatives or integrals you have to
look at what variable you're taking it
with respect to if I give you an
expression I say take derivative with
respect to X if you're talking about
derivatives then you pretend everything
else besides X is a constant in that uh
in that function that's how you take
partial derivatives right well in
integrals it's the same thing if we're
integrating over time then this is the
variable we care about every other
letter or symbol in there you pretend as
a constant so for the purposes of this
integral you pretend that s is a
constant uh and Lambda is also a
constant which means that everything in
front of the T is really just a constant
so it's like a giant number here so this
integral looks intimidating but really
since it's all just an constant it's not
that intimidating so many of you can
look at this and write the answer to the
integral down but a lot of times we have
to make a substitution to make it
absolutely clear and so what you would
do is you would say U is equal to minus
um s minus
Lambda right T so we want to make a
substitution because we want to make it
an easy integral to solve so when I do
this off to the side I'm doing a
substitution I say du
DT is equal to now notice if I'm taking
a derivative with respect to time this
whole thing is a constant so the
derivative is just going to be - s minus
Lambda because it's almost like this is
just the number three or the number five
or the number seven out in front of the
time the T just disappears for our first
derivative now since we're going to end
up substituting it back here we want us
we want to solve for DT and so what
we're going to get is -1 over s minus
Lambda duu all we've done here is move
the DT over here and then we've taken
all of this stuff and moved it over here
so we could solve for DT getting 1 / s-
Lambda
du all right so now that we have that we
want to take this and stick it back in
and substitute into our integral so
we'll change colors again and for now
we're going to leave the limits of
integration 0 to Infinity we're not
going to change them right now because
we're going to you know we'll see how to
handle that in a minute it becomes e to
the power of U because this is exactly
coming from that and DT just becomes
what we have found here so it's going to
be NE 1/ s- Lambda du U right so that's
sub substitution just goes in now
remember this is an integral over du now
right so everything in here is again
just a constant it's just a constant so
what we're going to have here we pull
the whole thing out- 1/ s minus Lambda
comes out we integrate from 0 to
Infinity e to the U du now we did this
whole thing so that we could get an
integral into a form that we know how to
solve very
easily right and
so uh what we're going to do then is say
we have -1/ s- Lambda the integral of e
to the U Still Remains e to the U and
again I'm leaving my limits of
integration 0 to Infinity for now
because what I'm going to do before I
apply the limits of integration is I'm
going to substitute back in for U so
what I'm going to have is -1/ s minus
Lambda and then U is going to come right
back substituted in as it was before E
minus s minus Lambda
T now goes from zero to Infinity so you
see I I realize um that as you have the
limits of integration here this really
If This Were a true statement you would
have to transform the limits of
integration into limits of U and this
would have to be limits of U but I'm not
going to substitute the limits in until
after I've plugged in for U so I end up
having a function of time here's a
limits in terms of T and so then I can
substitute everything in as exactly as I
want to and then what I'm going to get
is -1/ s- Lambda right and then we have
to apply this in so what this basically
becomes if you think about it this is s
minus Lambda time T and I'm putting
Infinity in for T so really it becomes e
to the neg negative Infinity it doesn't
matter what s and t are if I put
Infinity here it's going to be negative
infinity and then you do a subtraction
because you're evaluating the limits e
to the 0 because 0 goes in there now
this becomes very tractable because we
know that e to the infinity it's like
taking that
limit what you're going to have e to the
negative Infinity is like 1 over e to
the infinity so what you're going to
have this guy is going to be zero and
this guy anything to the power of zero
is just a one so you have a negative one
here and this whole thing evaluates to
negative 1 which makes a positive one
when you multiply this whole thing here
so you get 1 / s - Lambda 1 / s-
Lambda all right so what I'm going to do
then is show you that through this whole
situation all we did was we found the
laage transform of this guy and said
that it was equal to that so let me go
to the next board and summarize that
because it's a very important result so
what we found that is that the LL
transform of e to the lamb * T which is
just e uh to the power of some number
time T is just equal to 1 s minus Lambda
so for instance if it was e to 2T then
it would be 1 / s - 2 if it were e to
the 7t it would be 1 / s - 7 if it were
e to the 4T it would be 1/ S - 4 so you
see what we get as a result is just a
function of s Lambda is just a constant
it's going to be locked down by whatever
we start with right so the answer that
we get F of S capital F of s remember we
said let me go back we said that when
you do this LL transform you should get
just a function of s and that's why we
call it capital F of s and that's
exactly what we got we got a single
function pretty simple looking function
just a function of s this guy is just a
constant uh there now since it's in the
denominator and we don't want any zeros
to be in our denominator because then
you get undefined you know Port parts of
the transform or Infinity parts then
what you can also say is that this is
valid for S greater than Lambda that
just ensures that the denominator is not
going to be zero it ensures the
denominator is going to be positive this
is something that mathematically you
write down just to lock it down you
don't have any zeros in the in the
bottom but realistically you don't
really use this fact all that terribly
much when you're solving essential basic
problems with a ll transform now this is
an important result and it's an
important enough that I'm going to
circle it for you right and circle the
whole thing in fact and it's also
important enough that we want to draw
your attention to something else let me
change to Blue here in the special
case let's say Lambda is equal to zero
so let's
say
that Lambda is just equal to zero
then you would have e to the 0 * T right
which would be e to the 0 that would be
my function and of uh time if Lambda
were 0 be e 0 t and e to the 0 is just
one because anything to the power of
zero is just one so because of this we
can draw kind of another conclusion
that's already here but we can kind of
write it ourself we can say that the LL
transform of the number one
right would be 1 / s - 0 right because
the way you would come up with that is
you would
say uh which would be 1 / s right the
way you would come up with that is you
would say all right I can take the
special case when Lambda is zero in that
case this exponential just becomes a one
so it's the same as taking the lla
transform of one and then we' be putting
Lambda equals 0 in here getting an
answer of 1 / s so the laage transform
form of the number one is just 1/ s and
that's important enough while I will
also Circle it and of course it's for S
greater than zero because you don't want
any denominator uh driving the whole
thing to Infinity so this is the first
important conclusion of what we have
have done here I'll go quickly through a
a
um a brief history of what we've done we
basically said there's this thing called
A Plus transform it's an integral notice
it's not a double integral or a triple
integral or spherical coordinates or
anything crazy like you get into
calculus 3 territory it's really a
calculus one maybe a calculus 2 type of
thing but the implications of how you
use it is really why it's interesting
and we're going to get to that later but
basically you just stick a function of
time in here you evaluate the integral
evaluate the limits of integration and
if you do it correctly you should get
just a a function of s and we label it
capital F of s because capital F and
little f are related to one another by
this thing we call a transform we say
that cap that little F yields capital F
of s and later on we'll find out that
you can go in the reverse Direction and
start with a ll transform function of s
and get the corresponding function of
time so these things are kind of linked
by an invisible chain and that's the
chain which being is is the transform
that's why we can take a problem which
has which has functions of time
transform it into the LL domain which is
function of s solve it usually
algebraically so you don't you don't
have to deal with differential equations
you deal with algebra and then you get a
function of s and then we're going to
learn how to transform that back to the
time domain and you get your answer in
terms of time and if you do it right it
should be simpler we also say that the
llas transform F of s is squiggly L uh
operating on the function of time and we
just talked about improper integral
saying that it's basically like taking
the limit but really what you're doing
is plugging in the limits of integration
uh there and then we do real problem so
this is how a lot of these problems are
going to go you take your function of
time you put it in and you simplify and
then you realize the critical step here
is that since you're integrating over T
everything here is a constant right so
I've done the details here we did the
substitution of U putting it all in
there and pulling this junk out because
it's a constant giving us an integral
that everyone watching this should know
how to solve right so we do that we plug
the limits of integration in the limits
of ation greatly simplify what's
happening because this becomes a zero
this becomes a one uh because it's
negative Infinity remember e to the
negative Infinity is like 1 over e to
the positive Infinity which means 1 over
infinity giving you zero so all this
stuff drops away giving you negative 1
giving you positive 1 / s minus Lambda
so the conclusion block here is the
appli transform of e to the Lambda T is
just a function of s this Lambda is lock
down with whatever your specific
function is all right and then as a
special case of that we say hey what
happens if Lambda is zero then what you
should get if Lambda is zero is 1 / s
right and if Lambda is zero then it's
the whole thing just goes to one so what
we're saying is the L transform of the
number one is 1 / s if you remember back
to calculus one however many years ago
that was for you first thing we talked
about besides limits but for as far as
derivatives the first thing we talked
about was how to take simple derivatives
how to take the derivative of a constant
remember you had to learn that at one
point in the past how to take the
derivative of x how to take the
derivative of x s then you learn how to
take derivatives of pols then you learn
how to do it when there's a giant
fraction then you learn about signs and
cosines you build those skills learning
how to take those derivatives we're
doing the same thing with lass
transforms we're taking very simple LL
transforms first and we're recording
these answers um which are useful then
as we go on we're going to take more and
more complicated LL transforms to the
point where you can actually uh get
pretty proficient then I'm going to show
you how to use it to solve real problem
so right now you look at this and you're
like what's it for you know well I can't
get into that until you know a little
bit so just trust me follow me on to the
next lesson we'll build our skills
deriving these essential transforms uh
and then we'll we'll get a a repertoire
going so that you have some skills that
we can apply to real problems and
whenever you see how how um much simpler
it makes solving certain kinds of
differential equations you'll understand
that it's worth its weight in gold just
for that application but also since
differential equations are used in all
branches of science and engineering um
they really lend the llas transform to
lots and lots of different uh uh uh
situations in real Math Science and
Engineering so follow me on to the next
lesson in mastering the llas transform
here in the llas transform
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