All language subtitles for What does the Laplace Transform really tell us？ A visual explanation (plus applications) [n2y7n6jw5d0].en

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this video was sponsored by brilliant
here we have a mass on a spring if I

here we have a mass on a spring if I

here we have a mass on a spring if I
pull it back and release one of four

pull it back and release one of four

pull it back and release one of four
things is going to happen first is

things is going to happen first is

things is going to happen first is
completely sinusoidal motion as a no

completely sinusoidal motion as a no

completely sinusoidal motion as a no
oscillate back and forth forever which

oscillate back and forth forever which

oscillate back and forth forever which
will happen if there's no air resistance

will happen if there's no air resistance

will happen if there's no air resistance
or damping present to is exponential

or damping present to is exponential

or damping present to is exponential
decay this happens if we put in some

decay this happens if we put in some

decay this happens if we put in some
thick or viscous fluid which will cause

thick or viscous fluid which will cause

thick or viscous fluid which will cause
the mass to exponentially decay to

the mass to exponentially decay to

the mass to exponentially decay to
equilibrium without overshooting yes

equilibrium without overshooting yes

equilibrium without overshooting yes
this entire video will assume damping

this entire video will assume damping

this entire video will assume damping
force is a multiple of velocity three is

force is a multiple of velocity three is

force is a multiple of velocity three is
a combination of the first two which

a combination of the first two which

a combination of the first two which
happens when the damping isn't as strong

happens when the damping isn't as strong

happens when the damping isn't as strong
or the spring itself is stronger in this

or the spring itself is stronger in this

or the spring itself is stronger in this
case the spring will oscillate but still

case the spring will oscillate but still

case the spring will oscillate but still
decay in the process until it eventually

decay in the process until it eventually

decay in the process until it eventually
settles and for literally anything else

settles and for literally anything else

settles and for literally anything else
can happen if we have some input force

can happen if we have some input force

can happen if we have some input force
or just non-ideal conditions

or just non-ideal conditions

or just non-ideal conditions
we're not directly as concerned with

we're not directly as concerned with

we're not directly as concerned with
this fourth case in this video but we

this fourth case in this video but we

this fourth case in this video but we
will have inputs later the main emphasis

will have inputs later the main emphasis

will have inputs later the main emphasis
though is these three functions which is

though is these three functions which is

though is these three functions which is
really two while the third is just a

really two while the third is just a

really two while the third is just a
combination now if I had to summarize

combination now if I had to summarize

combination now if I had to summarize
what the Laplace transform visually

what the Laplace transform visually

what the Laplace transform visually
tells us in just a few seconds it'd be

tells us in just a few seconds it'd be

tells us in just a few seconds it'd be
this well the Fourier transform tells us

this well the Fourier transform tells us

this well the Fourier transform tells us
which frequencies or sinusoids are

which frequencies or sinusoids are

which frequencies or sinusoids are
present in a function the Laplace

present in a function the Laplace

present in a function the Laplace
transform tells us which sinusoids and

transform tells us which sinusoids and

transform tells us which sinusoids and
Exponential's are present in a function

Exponential's are present in a function

Exponential's are present in a function
in fact we're soon going to see that the

in fact we're soon going to see that the

in fact we're soon going to see that the
Fourier transform is just a slice of the

Fourier transform is just a slice of the

Fourier transform is just a slice of the
Laplace transform now here's the Fourier

Laplace transform now here's the Fourier

Laplace transform now here's the Fourier
transform equation it takes in some

transform equation it takes in some

transform equation it takes in some
function of time and outputs a function

function of time and outputs a function

function of time and outputs a function
of Omega that tells you which sinusoids

of Omega that tells you which sinusoids

of Omega that tells you which sinusoids
are present in your signal if you put in

are present in your signal if you put in

are present in your signal if you put in
a pure cosine curve then out comes the

a pure cosine curve then out comes the

a pure cosine curve then out comes the
function with one spike since your

function with one spike since your

function with one spike since your
original curve was one sinusoidal well

original curve was one sinusoidal well

original curve was one sinusoidal well
for real functions these are symmetric

for real functions these are symmetric

for real functions these are symmetric
about the y-axis so you'll actually see

about the y-axis so you'll actually see

about the y-axis so you'll actually see
two spikes and the x-coordinates will

two spikes and the x-coordinates will

two spikes and the x-coordinates will
match the angular frequency of the

match the angular frequency of the

match the angular frequency of the
original signal if you put in a non

original signal if you put in a non

original signal if you put in a non
periodic function you know get out

periodic function you know get out

periodic function you know get out
something more complex which tells us

something more complex which tells us

something more complex which tells us
that takes infinitely many sinusoids to

that takes infinitely many sinusoids to

that takes infinitely many sinusoids to
make up this function on the right this

make up this function on the right this

make up this function on the right this
animation does a great job at showing

animation does a great job at showing

animation does a great job at showing
the original function being a sum of all

the original function being a sum of all

the original function being a sum of all
those sinusoids and fleam any

those sinusoids and fleam any

those sinusoids and fleam any
some is seen in yellow while the

some is seen in yellow while the

some is seen in yellow while the
magnitude the Fourier transform tells us

magnitude the Fourier transform tells us

magnitude the Fourier transform tells us
relatively how strong each of those

relatively how strong each of those

relatively how strong each of those
sinusoids is for any given frequency one

sinusoids is for any given frequency one

sinusoids is for any given frequency one
thing to note is that the y intercept of

thing to note is that the y intercept of

thing to note is that the y intercept of
the fourier transform is the area under

the fourier transform is the area under

the fourier transform is the area under
the curve of the original remember this

the curve of the original remember this

the curve of the original remember this
is the output when Omega equals zero and

is the output when Omega equals zero and

is the output when Omega equals zero and
when I'll make it equal zero then this

when I'll make it equal zero then this

when I'll make it equal zero then this
term goes to one and we have just the

term goes to one and we have just the

term goes to one and we have just the
integral of the original function aka

integral of the original function aka

integral of the original function aka
the area under the curve now for a lot

the area under the curve now for a lot

the area under the curve now for a lot
of this video I'll be working with this

of this video I'll be working with this

of this video I'll be working with this
equation or something similar because it

equation or something similar because it

equation or something similar because it
has a sinusoid and exponential component

has a sinusoid and exponential component

has a sinusoid and exponential component
however we will assume it's zero for all

however we will assume it's zero for all

however we will assume it's zero for all
negative values of T and it essentially

negative values of T and it essentially

negative values of T and it essentially
turns on at time equals zero which

turns on at time equals zero which

turns on at time equals zero which
avoids the function diverging to

avoids the function diverging to

avoids the function diverging to
infinity so when we put it into the

infinity so when we put it into the

infinity so when we put it into the
Fourier transform out comes a complex

Fourier transform out comes a complex

Fourier transform out comes a complex
function that I won't graph quite yet

function that I won't graph quite yet

function that I won't graph quite yet
I'm not worried about the calculus in

I'm not worried about the calculus in

I'm not worried about the calculus in
this video but if you just know

this video but if you just know

this video but if you just know
integration by parts you can do this

integration by parts you can do this

integration by parts you can do this
just treat I as a constant anyways I'll

just treat I as a constant anyways I'll

just treat I as a constant anyways I'll
foil the bond I'm here to get a new

foil the bond I'm here to get a new

foil the bond I'm here to get a new
equation just with a separated real and

equation just with a separated real and

equation just with a separated real and
imaginary component and this is what

imaginary component and this is what

imaginary component and this is what
will work with this function only has

will work with this function only has

will work with this function only has
one input Omega so I can just go on a

one input Omega so I can just go on a

one input Omega so I can just go on a
number line but out of the function will

number line but out of the function will

number line but out of the function will
come a complex number with the real and

come a complex number with the real and

come a complex number with the real and
imaginary component so we need two

imaginary component so we need two

imaginary component so we need two
dimensions to represent that so let's

dimensions to represent that so let's

dimensions to represent that so let's
put in some values now if Omega equals 1

put in some values now if Omega equals 1

put in some values now if Omega equals 1
then the output becomes 1 over 1 plus 2i

then the output becomes 1 over 1 plus 2i

then the output becomes 1 over 1 plus 2i
but that can also be rewritten as 0.2

but that can also be rewritten as 0.2

but that can also be rewritten as 0.2
minus 0.4 I which will be plotted on the

minus 0.4 I which will be plotted on the

minus 0.4 I which will be plotted on the
real and imaginary axis respectively for

real and imaginary axis respectively for

real and imaginary axis respectively for
Omega equals 1 now this is a distance of

Omega equals 1 now this is a distance of

Omega equals 1 now this is a distance of
roughly 0.45 from the origin known as

roughly 0.45 from the origin known as

roughly 0.45 from the origin known as
the magnitude and that we can plot

the magnitude and that we can plot

the magnitude and that we can plot
against the input at Omega equals 1 this

against the input at Omega equals 1 this

against the input at Omega equals 1 this
will be the magnitude of the Fourier

will be the magnitude of the Fourier

will be the magnitude of the Fourier
transform now for Omega equals 2 is the

transform now for Omega equals 2 is the

transform now for Omega equals 2 is the
input we get out 1 over negative 2 plus

input we get out 1 over negative 2 plus

input we get out 1 over negative 2 plus
4i which simplifies to negative 0.1-0.2

4i which simplifies to negative 0.1-0.2

4i which simplifies to negative 0.1-0.2
I and that will also go on the output

I and that will also go on the output

I and that will also go on the output
graph the magnitude for this is roughly

graph the magnitude for this is roughly

graph the magnitude for this is roughly
0.2 to 4 and that will also go on the

0.2 to 4 and that will also go on the

0.2 to 4 and that will also go on the
magnitude plot for Omega equals 2 and

magnitude plot for Omega equals 2 and

magnitude plot for Omega equals 2 and
lastly I'll plug in Omega equals 0

lastly I'll plug in Omega equals 0

lastly I'll plug in Omega equals 0
which outputs just 0.5 no imaginary

which outputs just 0.5 no imaginary

which outputs just 0.5 no imaginary
component and that will also go on the

component and that will also go on the

component and that will also go on the
magnet

magnet

magnet
plot remember that point five just the

plot remember that point five just the

plot remember that point five just the
area under the curve of our original

area under the curve of our original

area under the curve of our original
function

function

function
well accounting for the fact that areas

well accounting for the fact that areas

well accounting for the fact that areas
below the x-axis are negative if I were

below the x-axis are negative if I were

below the x-axis are negative if I were
to plot all the magnitudes for any input

to plot all the magnitudes for any input

to plot all the magnitudes for any input
Omega we'd get this the magnitude of the

Omega we'd get this the magnitude of the

Omega we'd get this the magnitude of the
Fourier transform keep this plot in mind

Fourier transform keep this plot in mind

Fourier transform keep this plot in mind
because it will show up soon but now

because it will show up soon but now

because it will show up soon but now
let's get to the little pause transform

let's get to the little pause transform

let's get to the little pause transform
so you can see just how similar it is to

so you can see just how similar it is to

so you can see just how similar it is to
what we've done so far so here we have

what we've done so far so here we have

what we've done so far so here we have
the Fourier transform again and this is

the Fourier transform again and this is

the Fourier transform again and this is
the Laplace transform nearly identical

the Laplace transform nearly identical

the Laplace transform nearly identical
this could also be negative infinity

this could also be negative infinity

this could also be negative infinity
like above by the way but especially in

like above by the way but especially in

like above by the way but especially in
engineering we usually deal with signals

engineering we usually deal with signals

engineering we usually deal with signals
that like I said earlier turn on at time

that like I said earlier turn on at time

that like I said earlier turn on at time
equals zero so we can analyze the

equals zero so we can analyze the

equals zero so we can analyze the
transient response from there the only

transient response from there the only

transient response from there the only
other difference though is we have an S

other difference though is we have an S

other difference though is we have an S
here instead of an i Omega but s is

here instead of an i Omega but s is

here instead of an i Omega but s is
really alpha plus I Omega which I'll

really alpha plus I Omega which I'll

really alpha plus I Omega which I'll
substitute in because now we can

substitute in because now we can

substitute in because now we can
separate that exponent into two terms

separate that exponent into two terms

separate that exponent into two terms
now look at this the Fourier transform

now look at this the Fourier transform

now look at this the Fourier transform
of something is found by multiplying

of something is found by multiplying

of something is found by multiplying
that function by e to the minus I Omega

that function by e to the minus I Omega

that function by e to the minus I Omega
T and integrating well in the Laplace

T and integrating well in the Laplace

T and integrating well in the Laplace
transform we have the integral of

transform we have the integral of

transform we have the integral of
something times e to the minus I Omega T

something times e to the minus I Omega T

something times e to the minus I Omega T
meaning the Laplace transform of some

meaning the Laplace transform of some

meaning the Laplace transform of some
function is just the xlviii transform of

function is just the xlviii transform of

function is just the xlviii transform of
that function times an exponential term

that function times an exponential term

that function times an exponential term
doing this for all values of alpha all

doing this for all values of alpha all

doing this for all values of alpha all
real numbers gives you the entire

real numbers gives you the entire

real numbers gives you the entire
Laplace transform to see what I mean by

Laplace transform to see what I mean by

Laplace transform to see what I mean by
this let's look at the Laplace transform

this let's look at the Laplace transform

this let's look at the Laplace transform
of the same function as earlier which

of the same function as earlier which

of the same function as earlier which
again I'm not going to derive but it's

again I'm not going to derive but it's

again I'm not going to derive but it's
nearly identical to the Fourier

nearly identical to the Fourier

nearly identical to the Fourier
transform the only difference is s

transform the only difference is s

transform the only difference is s
represents a complex number which means

represents a complex number which means

represents a complex number which means
the input will require two dimensions

the input will require two dimensions

the input will require two dimensions
one for the real and one for the

one for the real and one for the

one for the real and one for the
imaginary component the output just like

imaginary component the output just like

imaginary component the output just like
before will be complex so we need four

before will be complex so we need four

before will be complex so we need four
dimensions for the Laplace transform in

dimensions for the Laplace transform in

dimensions for the Laplace transform in
total but notice that when alpha is zero

total but notice that when alpha is zero

total but notice that when alpha is zero
we get the exact same outputs as the

we get the exact same outputs as the

we get the exact same outputs as the
Fourier transform as in this alpha

Fourier transform as in this alpha

Fourier transform as in this alpha
equals zero line or the imaginary axis

equals zero line or the imaginary axis

equals zero line or the imaginary axis
of the Laplace transform is the Fourier

of the Laplace transform is the Fourier

of the Laplace transform is the Fourier
transform another way to see this is

transform another way to see this is

transform another way to see this is
with the integral because when alpha

with the integral because when alpha

with the integral because when alpha
equals zero the exponential goes to one

equals zero the exponential goes to one

equals zero the exponential goes to one
and we're left with the original Fourier

and we're left with the original Fourier

and we're left with the original Fourier
transform equation

transform equation

transform equation
so if I plug in something on that axis

so if I plug in something on that axis

so if I plug in something on that axis
we should get the same magnitudes as

we should get the same magnitudes as

we should get the same magnitudes as
before like if we put in 0 for alpha and

before like if we put in 0 for alpha and

before like if we put in 0 for alpha and
0 for Omega which goes here on the input

0 for Omega which goes here on the input

0 for Omega which goes here on the input
then out comes 0.5 that area under the

then out comes 0.5 that area under the

then out comes 0.5 that area under the
curve which goes here on the output and

curve which goes here on the output and

curve which goes here on the output and
since I don't really have another

since I don't really have another

since I don't really have another
dimension I'll just write that magnitude

dimension I'll just write that magnitude

dimension I'll just write that magnitude
above the corresponding input if I plug

above the corresponding input if I plug

above the corresponding input if I plug
in 1 for Omega and 0 for alpha which is

in 1 for Omega and 0 for alpha which is

in 1 for Omega and 0 for alpha which is
also on the same axis then out comes the

also on the same axis then out comes the

also on the same axis then out comes the
point 2 minus 0.4 I from before which on

point 2 minus 0.4 I from before which on

point 2 minus 0.4 I from before which on
the output is a magnitude of roughly

the output is a magnitude of roughly

the output is a magnitude of roughly
0.45 and I won't show it but if I did

0.45 and I won't show it but if I did

0.45 and I won't show it but if I did
plug in this point where Omega equals 2

plug in this point where Omega equals 2

plug in this point where Omega equals 2
the output would have a magnitude of

the output would have a magnitude of

the output would have a magnitude of
about 0.2 to 4 if I was using a third

about 0.2 to 4 if I was using a third

about 0.2 to 4 if I was using a third
dimension to plot those magnitudes then

dimension to plot those magnitudes then

dimension to plot those magnitudes then
we'd see the exact same function as

we'd see the exact same function as

we'd see the exact same function as
before above that imaginary axis we

before above that imaginary axis we

before above that imaginary axis we
could also make a contour plot though

could also make a contour plot though

could also make a contour plot though
like just imagine looking down on this

like just imagine looking down on this

like just imagine looking down on this
graph where colors are assigned to

graph where colors are assigned to

graph where colors are assigned to
different Y values because then those

different Y values because then those

different Y values because then those
colors can represent the magnitudes then

colors can represent the magnitudes then

colors can represent the magnitudes then
if I move the green line over to let's

if I move the green line over to let's

if I move the green line over to let's
say alpha equals negative 0.5 the

say alpha equals negative 0.5 the

say alpha equals negative 0.5 the
outputs will be the Fourier transform of

outputs will be the Fourier transform of

outputs will be the Fourier transform of
something slightly different let me just

something slightly different let me just

something slightly different let me just
put the original Laplace equation back

put the original Laplace equation back

put the original Laplace equation back
up top and you'll note that since

up top and you'll note that since

up top and you'll note that since
there's already a negative sign here

there's already a negative sign here

there's already a negative sign here
when I plug in negative 0.5 for alpha

when I plug in negative 0.5 for alpha

when I plug in negative 0.5 for alpha
this will just calculate the Fourier

this will just calculate the Fourier

this will just calculate the Fourier
transform of our original function times

transform of our original function times

transform of our original function times
e to the positive 0.5 T that Fourier

e to the positive 0.5 T that Fourier

e to the positive 0.5 T that Fourier
transform will look like this and again

transform will look like this and again

transform will look like this and again
I could write the magnitudes to show the

I could write the magnitudes to show the

I could write the magnitudes to show the
outputs or I could use colors if alpha

outputs or I could use colors if alpha

outputs or I could use colors if alpha
is swept through the plane we get the

is swept through the plane we get the

is swept through the plane we get the
entire Laplace transform plot and if we

entire Laplace transform plot and if we

entire Laplace transform plot and if we
actually use a third dimension for those

actually use a third dimension for those

actually use a third dimension for those
magnitudes it would look like this

this plane here shown in green which

this plane here shown in green which
outfit the plot a little so you can see

outfit the plot a little so you can see

outfit the plot a little so you can see
is all the inputs s X represents alpha

is all the inputs s X represents alpha

is all the inputs s X represents alpha
and y is really I Omega the 3d plot

and y is really I Omega the 3d plot

and y is really I Omega the 3d plot
itself is the magnitude of the complex

itself is the magnitude of the complex

itself is the magnitude of the complex
outputs this is the Laplace transform of

outputs this is the Laplace transform of

outputs this is the Laplace transform of
the original function e to the minus T

the original function e to the minus T

the original function e to the minus T
sine of T the thing is this doesn't tell

sine of T the thing is this doesn't tell

sine of T the thing is this doesn't tell
us much just by looking at it so what

us much just by looking at it so what

us much just by looking at it so what
I'm going to do is graph the equation

I'm going to do is graph the equation

I'm going to do is graph the equation
alpha or x equals 0 that'll give us a

alpha or x equals 0 that'll give us a

alpha or x equals 0 that'll give us a
plane as shown

plane as shown

plane as shown
because remember from before the alpha

because remember from before the alpha

because remember from before the alpha
equals zero line actually yields the

equals zero line actually yields the

equals zero line actually yields the
Fourier transform of the original

Fourier transform of the original

Fourier transform of the original
function which we can see with the

function which we can see with the

function which we can see with the
intersecting curve this is what I meant

intersecting curve this is what I meant

intersecting curve this is what I meant
by the Fourier transform is a slice of

by the Fourier transform is a slice of

by the Fourier transform is a slice of
the Laplace transform but now I'm going

the Laplace transform but now I'm going

the Laplace transform but now I'm going
to decrease alpha while also plotting

to decrease alpha while also plotting

to decrease alpha while also plotting
the original function times e to the

the original function times e to the

the original function times e to the
minus alpha T right now alpha is 0 so

minus alpha T right now alpha is 0 so

minus alpha T right now alpha is 0 so
that exponential is just 1 but as I

that exponential is just 1 but as I

that exponential is just 1 but as I
sweep alpha we start to see both plots

sweep alpha we start to see both plots

sweep alpha we start to see both plots
change and what you're seeing with that

change and what you're seeing with that

change and what you're seeing with that
intersection on the right is the Fourier

intersection on the right is the Fourier

intersection on the right is the Fourier
transform of the plot on the left at any

transform of the plot on the left at any

transform of the plot on the left at any
given time like here I'll pause it alpha

given time like here I'll pause it alpha

given time like here I'll pause it alpha
equals negative 0.5 because this is what

equals negative 0.5 because this is what

equals negative 0.5 because this is what
we just saw a minute ago where the

we just saw a minute ago where the

we just saw a minute ago where the
original curve times e to the positive

original curve times e to the positive

original curve times e to the positive
0.5 T because that double negative has a

0.5 T because that double negative has a

0.5 T because that double negative has a
Fourier transform with those two small

Fourier transform with those two small

Fourier transform with those two small
Peaks then I'll extend the range of Z

Peaks then I'll extend the range of Z

Peaks then I'll extend the range of Z
because as we get closer to alpha equals

because as we get closer to alpha equals

because as we get closer to alpha equals
negative 1 we get an intersection with

negative 1 we get an intersection with

negative 1 we get an intersection with
too much higher and narrower peaks we're

too much higher and narrower peaks we're

too much higher and narrower peaks we're
rendering also isn't looking so good but

rendering also isn't looking so good but

rendering also isn't looking so good but
anyways this happens because the left

anyways this happens because the left

anyways this happens because the left
plot is approaching just a regular

plot is approaching just a regular

plot is approaching just a regular
sinusoidal ZAR about to cancel once we

sinusoidal ZAR about to cancel once we

sinusoidal ZAR about to cancel once we
get to alpha equals negative 1 and we

get to alpha equals negative 1 and we

get to alpha equals negative 1 and we
saw before that a pure sinusoid as a

saw before that a pure sinusoid as a

saw before that a pure sinusoid as a
Fourier transform of 2 infinite spikes

Fourier transform of 2 infinite spikes

Fourier transform of 2 infinite spikes
which we can also see on the 3d plot so

which we can also see on the 3d plot so

which we can also see on the 3d plot so
the quick summary to what we've seen

the quick summary to what we've seen

the quick summary to what we've seen
already is that to construct a Laplace

already is that to construct a Laplace

already is that to construct a Laplace
transform take whatever function you

transform take whatever function you

transform take whatever function you
want to work with multiplied by e to the

want to work with multiplied by e to the

want to work with multiplied by e to the
minus alpha T for some alpha let's just

minus alpha T for some alpha let's just

minus alpha T for some alpha let's just
change it something random like negative

change it something random like negative

change it something random like negative
0.93 and graph the 2d Fourier transform

0.93 and graph the 2d Fourier transform

0.93 and graph the 2d Fourier transform
of that function then keep doing that as

of that function then keep doing that as

of that function then keep doing that as
you sweep through all values of alpha

you sweep through all values of alpha

you sweep through all values of alpha
stacking two-dimensional 48

stacking two-dimensional 48

stacking two-dimensional 48
plot side by side until you get your 3d

plot side by side until you get your 3d

plot side by side until you get your 3d
Laplace transform you will likely not be

Laplace transform you will likely not be

Laplace transform you will likely not be
shown this in a classroom setting and

shown this in a classroom setting and

shown this in a classroom setting and
that's because most of it pretty much

that's because most of it pretty much

that's because most of it pretty much
doesn't matter all we care about are

doesn't matter all we care about are

doesn't matter all we care about are
these two peaks which do go to infinity

these two peaks which do go to infinity

these two peaks which do go to infinity
known as the poles now if we go back to

known as the poles now if we go back to

known as the poles now if we go back to
the 2d plot those poles are located at

the 2d plot those poles are located at

the 2d plot those poles are located at
negative 1 plus and minus I which we can

negative 1 plus and minus I which we can

negative 1 plus and minus I which we can
represent with an X poles and also zeros

represent with an X poles and also zeros

represent with an X poles and also zeros
which doesn't apply to our function are

which doesn't apply to our function are

which doesn't apply to our function are
pretty much all you'll ever see for

pretty much all you'll ever see for

pretty much all you'll ever see for
these plots the reason the poles are

these plots the reason the poles are

these plots the reason the poles are
there though is because if we plug in

there though is because if we plug in

there though is because if we plug in
negative 1 plus or minus I into the

negative 1 plus or minus I into the

negative 1 plus or minus I into the
Laplace equation we get 1 over 0 which

Laplace equation we get 1 over 0 which

Laplace equation we get 1 over 0 which
I'll just write as infinity because on

I'll just write as infinity because on

I'll just write as infinity because on
the plots those are represented with an

the plots those are represented with an

the plots those are represented with an
X and one more thing you'll notice when

X and one more thing you'll notice when

X and one more thing you'll notice when
I was sweeping the Alpha plane I stopped

I was sweeping the Alpha plane I stopped

I was sweeping the Alpha plane I stopped
at those poles or alpha equals negative

at those poles or alpha equals negative

at those poles or alpha equals negative
1 and never went behind that that's

1 and never went behind that that's

1 and never went behind that that's
because once alpha goes below negative 1

because once alpha goes below negative 1

because once alpha goes below negative 1
the function we were plotting diverges

the function we were plotting diverges

the function we were plotting diverges
which means the Fourier transform of

which means the Fourier transform of

which means the Fourier transform of
this does as well so the Laplace

this does as well so the Laplace

this does as well so the Laplace
transform doesn't actually exist in that

transform doesn't actually exist in that

transform doesn't actually exist in that
region whereas this is the good region

region whereas this is the good region

region whereas this is the good region
which we give a name to the region of

which we give a name to the region of

which we give a name to the region of
convergence just think of this region is

convergence just think of this region is

convergence just think of this region is
all the Alpha is such that this part of

all the Alpha is such that this part of

all the Alpha is such that this part of
the Laplace transform eventually

the Laplace transform eventually

the Laplace transform eventually
converges to zero this means everything

converges to zero this means everything

converges to zero this means everything
I said earlier with taking the Fourier

I said earlier with taking the Fourier

I said earlier with taking the Fourier
transform of this function and at being

transform of this function and at being

transform of this function and at being
a slice of our plot is true if that

a slice of our plot is true if that

a slice of our plot is true if that
slice is in the region of convergence

slice is in the region of convergence

slice is in the region of convergence
this part is so we're good there however

this part is so we're good there however

this part is so we're good there however
this part is not which is why again the

this part is not which is why again the

this part is not which is why again the
function diverges for those alpha values

function diverges for those alpha values

function diverges for those alpha values
essentially this exponential term has

essentially this exponential term has

essentially this exponential term has
now one making Laplace undefined in that

now one making Laplace undefined in that

now one making Laplace undefined in that
region anyways going back notice that

region anyways going back notice that

region anyways going back notice that
for our pull the imaginary coordinate is

for our pull the imaginary coordinate is

for our pull the imaginary coordinate is
1 and negative 1 which matches the

1 and negative 1 which matches the

1 and negative 1 which matches the
coefficient or angular frequency of our

coefficient or angular frequency of our

coefficient or angular frequency of our
sinusoid the real component of negative

sinusoid the real component of negative

sinusoid the real component of negative
1 matches the coefficient in the

1 matches the coefficient in the

1 matches the coefficient in the
exponential term this is finally what

exponential term this is finally what

exponential term this is finally what
the Laplace transform visually tells us

the Laplace transform visually tells us

the Laplace transform visually tells us
the imaginary axis represents the

the imaginary axis represents the

the imaginary axis represents the
sinusoids and if my poles are on that

sinusoids and if my poles are on that

sinusoids and if my poles are on that
axis it means my original function is

axis it means my original function is

axis it means my original function is
just sinusoidal and the further from the

just sinusoidal and the further from the

just sinusoidal and the further from the
origin the poles are the higher

origin the poles are the higher

origin the poles are the higher
frequency that signal is if my poles are

frequency that signal is if my poles are

frequency that signal is if my poles are
on the real axis then there

on the real axis then there

on the real axis then there
only exponentials in the original signal

only exponentials in the original signal

only exponentials in the original signal
which DK faster as we move further from

which DK faster as we move further from

which DK faster as we move further from
the origin when the pool has real and

the origin when the pool has real and

the origin when the pool has real and
imaginary components like here then we

imaginary components like here then we

imaginary components like here then we
have a combination since this graph is

have a combination since this graph is

have a combination since this graph is
symmetric about the real axis I'm going

symmetric about the real axis I'm going

symmetric about the real axis I'm going
to move it down and to help with

to move it down and to help with

to move it down and to help with
visualizations I'm going to change the

visualizations I'm going to change the

visualizations I'm going to change the
x-axis so every tick mark is point 2

x-axis so every tick mark is point 2

x-axis so every tick mark is point 2
units apart but anyways now the pole

units apart but anyways now the pole

units apart but anyways now the pole
represents this function with that

represents this function with that

represents this function with that
exponential and sinusoidal component

exponential and sinusoidal component

exponential and sinusoidal component
whose graph looks like this if I move

whose graph looks like this if I move

whose graph looks like this if I move
the pole away from the x-axis then the

the pole away from the x-axis then the

the pole away from the x-axis then the
sinusoidal frequency increases causing

sinusoidal frequency increases causing

sinusoidal frequency increases causing
faster oscillations and the resulting

faster oscillations and the resulting

faster oscillations and the resulting
equation will have an angular frequency

equation will have an angular frequency

equation will have an angular frequency
that matches what we see on the

that matches what we see on the

that matches what we see on the
imaginary axis if I move to the left

imaginary axis if I move to the left

imaginary axis if I move to the left
then the number and the exponent gets

then the number and the exponent gets

then the number and the exponent gets
more negative causing for a faster decay

more negative causing for a faster decay

more negative causing for a faster decay
rate the equation for this function

rate the equation for this function

rate the equation for this function
still has coefficients that match what

still has coefficients that match what

still has coefficients that match what
we see on the axes and if I move to the

we see on the axes and if I move to the

we see on the axes and if I move to the
right then the decay rate slows until we

right then the decay rate slows until we

right then the decay rate slows until we
reach the imaginary axis where we get up

reach the imaginary axis where we get up

reach the imaginary axis where we get up
pure sinusoid and pulls on the right

pure sinusoid and pulls on the right

pure sinusoid and pulls on the right
hand side correspond to functions with

hand side correspond to functions with

hand side correspond to functions with
exponential growth so now the Laplace

exponential growth so now the Laplace

exponential growth so now the Laplace
transform equation should make way more

transform equation should make way more

transform equation should make way more
sense like sine of 3t has the Laplace

sense like sine of 3t has the Laplace

sense like sine of 3t has the Laplace
transform of 3 over s squared plus 9 if

transform of 3 over s squared plus 9 if

transform of 3 over s squared plus 9 if
we find the poles or on the denominator

we find the poles or on the denominator

we find the poles or on the denominator
0 we get plus and minus 3i and this

0 we get plus and minus 3i and this

0 we get plus and minus 3i and this
tells us the original function has no

tells us the original function has no

tells us the original function has no
exponential term but it does have a

exponential term but it does have a

exponential term but it does have a
sinusoid with an angular frequency of 3

sinusoid with an angular frequency of 3

sinusoid with an angular frequency of 3
which we already knew however the most

which we already knew however the most

which we already knew however the most
common talking point you here with

common talking point you here with

common talking point you here with
Laplace is not anything we've seen so

Laplace is not anything we've seen so

Laplace is not anything we've seen so
far but rather its ability to turn

far but rather its ability to turn

far but rather its ability to turn
calculus into algebra if we take some

calculus into algebra if we take some

calculus into algebra if we take some
arbitrary function X of T it will have a

arbitrary function X of T it will have a

arbitrary function X of T it will have a
corresponding Laplace transform X of s

corresponding Laplace transform X of s

corresponding Laplace transform X of s
but if you take the derivative of that

but if you take the derivative of that

but if you take the derivative of that
same function the Laplace transform will

same function the Laplace transform will

same function the Laplace transform will
be the exact same thing times s there is

be the exact same thing times s there is

be the exact same thing times s there is
an extra term that has to do with

an extra term that has to do with

an extra term that has to do with
initial conditions but I will assume

initial conditions but I will assume

initial conditions but I will assume
those are 0 from here on then the

those are 0 from here on then the

those are 0 from here on then the
Laplace transform of the second

Laplace transform of the second

Laplace transform of the second
derivative is again the same X of s

derivative is again the same X of s

derivative is again the same X of s
times s squared this time and some more

times s squared this time and some more

times s squared this time and some more
initial condition terms that will ignore

initial condition terms that will ignore

initial condition terms that will ignore
this pattern would continue and this is

this pattern would continue and this is

this pattern would continue and this is
where the Laplace is really useful for

where the Laplace is really useful for

where the Laplace is really useful for
example this is the differential

example this is the differential

example this is the differential
equation that describes a mass on a

equation that describes a mass on a

equation that describes a mass on a
spring we've got the force from the

spring we've got the force from the

spring we've got the force from the
spring itself the damping force which is

spring itself the damping force which is

spring itself the damping force which is
a multiple velocity some arbitrary

a multiple velocity some arbitrary

a multiple velocity some arbitrary
put X of T and all forces sum to mass

put X of T and all forces sum to mass

put X of T and all forces sum to mass
times acceleration here the grouping of

times acceleration here the grouping of

times acceleration here the grouping of
the terms just came from some

the terms just came from some

the terms just came from some
rearranging so what we can do is take

rearranging so what we can do is take

rearranging so what we can do is take
the Laplace transform of both sides

the Laplace transform of both sides

the Laplace transform of both sides
however due to linearity this is the

however due to linearity this is the

however due to linearity this is the
same as taking the Laplace transform of

same as taking the Laplace transform of

same as taking the Laplace transform of
each individual term the transform of

each individual term the transform of

each individual term the transform of
some X of T is X of s and for K times

some X of T is X of s and for K times

some X of T is X of s and for K times
some Y of T it becomes KY of s for be Y

some Y of T it becomes KY of s for be Y

some Y of T it becomes KY of s for be Y
prime from the rule above we got the

prime from the rule above we got the

prime from the rule above we got the
same Y of s times s and the B is there

same Y of s times s and the B is there

same Y of s times s and the B is there
as well then the second derivative

as well then the second derivative

as well then the second derivative
outputs M s squared Y of s since all the

outputs M s squared Y of s since all the

outputs M s squared Y of s since all the
terms on the left have a Y of s I can

terms on the left have a Y of s I can

terms on the left have a Y of s I can
factor that out to get this here some of

factor that out to get this here some of

factor that out to get this here some of
you may know this part is the auxilary

you may know this part is the auxilary

you may know this part is the auxilary
or characteristic equation will then

or characteristic equation will then

or characteristic equation will then
isolate y of s and we're left with this

isolate y of s and we're left with this

isolate y of s and we're left with this
here so we may not know the actual

here so we may not know the actual

here so we may not know the actual
output Y of T but like we've already

output Y of T but like we've already

output Y of T but like we've already
seen if I know when the denominator or

seen if I know when the denominator or

seen if I know when the denominator or
the Laplace transform is 0 aka the poles

the Laplace transform is 0 aka the poles

the Laplace transform is 0 aka the poles
then I can tell you a lot about your

then I can tell you a lot about your

then I can tell you a lot about your
output function let's assume the input

output function let's assume the input

output function let's assume the input
is a constant force because this is the

is a constant force because this is the

is a constant force because this is the
same as having a vertical spring or the

same as having a vertical spring or the

same as having a vertical spring or the
mass and subject to gravity like good

mass and subject to gravity like good

mass and subject to gravity like good
engineers we'll say the mass is 1 thus

engineers we'll say the mass is 1 thus

engineers we'll say the mass is 1 thus
force is 10 Newtons and when the block

force is 10 Newtons and when the block

force is 10 Newtons and when the block
is released at T equals 0 or the forces

is released at T equals 0 or the forces

is released at T equals 0 or the forces
immediately turned on so to speak this

immediately turned on so to speak this

immediately turned on so to speak this
is known as a step function written U of

is known as a step function written U of

is known as a step function written U of
T and its Laplace transform is 1 over s

T and its Laplace transform is 1 over s

T and its Laplace transform is 1 over s
meaning our force 10 U of T becomes 10

meaning our force 10 U of T becomes 10

meaning our force 10 U of T becomes 10
over s which will plug in for X of s

over s which will plug in for X of s

over s which will plug in for X of s
then again mass is 1 and let's say the

then again mass is 1 and let's say the

then again mass is 1 and let's say the
damping coefficient is 0 while the

damping coefficient is 0 while the

damping coefficient is 0 while the
spring constant is 1 then I'll just move

spring constant is 1 then I'll just move

spring constant is 1 then I'll just move
the s down to the denominator since

the s down to the denominator since

the s down to the denominator since
there's no damping the spring will

there's no damping the spring will

there's no damping the spring will
oscillate forever around in equilibrium

oscillate forever around in equilibrium

oscillate forever around in equilibrium
but let's check that this just show

but let's check that this just show

but let's check that this just show
without a Laplace equation as well we

without a Laplace equation as well we

without a Laplace equation as well we
have one pull up positive I and another

have one pull up positive I and another

have one pull up positive I and another
and negative I which is what makes this

and negative I which is what makes this

and negative I which is what makes this
part zero then we also have a pole at 0

part zero then we also have a pole at 0

part zero then we also have a pole at 0
from the s out here and we'll put

from the s out here and we'll put

from the s out here and we'll put
everything on the pole-zero plot again

everything on the pole-zero plot again

everything on the pole-zero plot again
I'm hiding the bottom to make room but

I'm hiding the bottom to make room but

I'm hiding the bottom to make room but
it always looks identical to the top

it always looks identical to the top

it always looks identical to the top
half and now we have everything we need

half and now we have everything we need

half and now we have everything we need
these poles say that the output will

these poles say that the output will

these poles say that the output will
have a sinusoid with an angular

have a sinusoid with an angular

have a sinusoid with an angular
frequency of 1

frequency of 1

frequency of 1
and we haven't seen a pole at the origin

and we haven't seen a pole at the origin

and we haven't seen a pole at the origin
yet but remember this is the area under

yet but remember this is the area under

yet but remember this is the area under
the curve the intercept of the Fourier

the curve the intercept of the Fourier

the curve the intercept of the Fourier
transform the pull this represents an

transform the pull this represents an

transform the pull this represents an
infinite area which just means there's

infinite area which just means there's

infinite area which just means there's
some offset in our output or something

some offset in our output or something

some offset in our output or something
that doesn't converge to zero to get

that doesn't converge to zero to get

that doesn't converge to zero to get
that infinite area since we're

that infinite area since we're

that infinite area since we're
considering the top to be y equals 0

considering the top to be y equals 0

considering the top to be y equals 0
then the output is exactly what we

then the output is exactly what we

then the output is exactly what we
expected if I increase the value of K or

expected if I increase the value of K or

expected if I increase the value of K or
make this bring stronger and the poles

make this bring stronger and the poles

make this bring stronger and the poles
start to separate more and more which

start to separate more and more which

start to separate more and more which
represents faster oscillations about an

represents faster oscillations about an

represents faster oscillations about an
equilibrium if I were to add some

equilibrium if I were to add some

equilibrium if I were to add some
damping now or increase B then we expect

damping now or increase B then we expect

damping now or increase B then we expect
slow exponential decay I'll just stop

slow exponential decay I'll just stop

slow exponential decay I'll just stop
here B equals 2 for example so we can

here B equals 2 for example so we can

here B equals 2 for example so we can
see there's now a sinusoidal and

see there's now a sinusoidal and

see there's now a sinusoidal and
exponential component to our equation

exponential component to our equation

exponential component to our equation
which matches what was expected even

which matches what was expected even

which matches what was expected even
though no I'm not graphing the exact

though no I'm not graphing the exact

though no I'm not graphing the exact
output if we make the damping much

output if we make the damping much

output if we make the damping much
stronger we get to a point of critical

stronger we get to a point of critical

stronger we get to a point of critical
damping

damping

damping
we're finally oscillations go away and

we're finally oscillations go away and

we're finally oscillations go away and
it's only exponential decay then moving

it's only exponential decay then moving

it's only exponential decay then moving
from there the exponential decay just

from there the exponential decay just

from there the exponential decay just
slows down from this term not decaying

slows down from this term not decaying

slows down from this term not decaying
as fast plotting the path those pulses

as fast plotting the path those pulses

as fast plotting the path those pulses
took is the idea behind a root locus

took is the idea behind a root locus

took is the idea behind a root locus
plot by the way for those in the

plot by the way for those in the

plot by the way for those in the
controlls class but this is where the

controlls class but this is where the

controlls class but this is where the
design part comes in because by

design part comes in because by

design part comes in because by
analyzing the locations or the poles we

analyzing the locations or the poles we

analyzing the locations or the poles we
can determine how a system will respond

can determine how a system will respond

can determine how a system will respond
to different inputs many systems out

to different inputs many systems out

to different inputs many systems out
there would be extremely difficult to

there would be extremely difficult to

there would be extremely difficult to
solve using only functions of time

solve using only functions of time

solve using only functions of time
plenty of you probably know how not fun

plenty of you probably know how not fun

plenty of you probably know how not fun
this would be to solve using only

this would be to solve using only

this would be to solve using only
differential equations but moving to the

differential equations but moving to the

differential equations but moving to the
S domain makes it an algebra problem

S domain makes it an algebra problem

S domain makes it an algebra problem
that's much more doable and especially

that's much more doable and especially

that's much more doable and especially
in control systems Laplace is crucial I

in control systems Laplace is crucial I

in control systems Laplace is crucial I
mean we just did a problem where an

mean we just did a problem where an

mean we just did a problem where an
input was multiplied by the system

input was multiplied by the system

input was multiplied by the system
transfer function and we got the output

transfer function and we got the output

transfer function and we got the output
transform it wasn't that bad but even

transform it wasn't that bad but even

transform it wasn't that bad but even
when there's more going on between the

when there's more going on between the

when there's more going on between the
input and output it simplifies fairly

input and output it simplifies fairly

input and output it simplifies fairly
nicely when using the S domain as we can

nicely when using the S domain as we can

nicely when using the S domain as we can
still just multiply the input by a more

still just multiply the input by a more

still just multiply the input by a more
complex but still manageable transfer

complex but still manageable transfer

complex but still manageable transfer
function to find the corresponding

function to find the corresponding

function to find the corresponding
output of course there's plenty more to

output of course there's plenty more to

output of course there's plenty more to
all this and if you want to continue

all this and if you want to continue

all this and if you want to continue
your learning I highly recommend

your learning I highly recommend

your learning I highly recommend
checking out brilliant

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on differential equations this includes

on differential equations this includes

on differential equations this includes
two full courses which started the

two full courses which started the

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basics for those who may need a

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refresher or just haven't learned this

refresher or just haven't learned this
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information yet but by the second course
they go through topics I never even came

they go through topics I never even came

they go through topics I never even came
across in college as an engineer so

across in college as an engineer so

across in college as an engineer so
there's a lot more for anyone to learn

there's a lot more for anyone to learn

there's a lot more for anyone to learn
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brillian cludes very hands-on exercises

brillian cludes very hands-on exercises
intuitive animations and in-depth

intuitive animations and in-depth

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basics to the more advanced concepts as

basics to the more advanced concepts as

basics to the more advanced concepts as
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you move through their courses

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