The Laplace Transform plays a critical role in engineering, physics, and mathematics.
The Laplace Transform allows us to evaluate
the stability and frequency response of a system.
And the Laplace Transform gives us a method for easily solving differential equations.
Throughout this video, the color signifies the phase of a function.
The phase of a complex number is the angle that the
rotating white line makes with the positive real axis.
The magnitude of a complex number is the length of the rotating white line.
In this function, time does not exist.
The input to our function is a complex number
with the real and imaginary values along the two axes.
The output of our function is also a complex number.
It is represented by the height and by the color of the graph.
The height of the graph at each point represents the magnitude of the output.
The color of the graph at each point represents the phase of the output.
This is the function where the output is exactly equal to the input, "s."
This is the function where the output is equal to one divided by the input.
We can specify for what values of the input, “s”, the function exists.
Here, the function exists where
the real component of “s” is greater than zero.
This is an example of what we call a “frequency domain” function,
and this example is associated with the following “time domain” function.
A time domain function is simply a function of time.
Every time domain function has an associated frequency domain function,
which we call its “Laplace Transform.”
There is a lot of beauty in this transformation, but to understand it,
we need to first talk about the sums of exponentials of complex numbers.
For functions of time, the variable “s” is always a constant.
If we increase the imaginary component of “s”, the white line will rotate faster.
If we increase the real component of “s”, the magnitude will increase more rapidly.
If the real component of “s” is zero, then the magnitude will stay constant.
We can multiply our function by a constant.
In this example, the constant is the real number 3,
but the constant can also be a complex number.
Let’s now consider what happens when we add exponential functions like these together.
All waveforms can be created by adding together exponential functions in this way.
The Laplace Transform of a waveform tells us
how much of each exponential function to add.
Let’s draw an infinite line as shown.
The line can be at any one of these locations.
With the line fixed at one location, each of these points represents one of
the exponential functions that we are going to add together to create our waveform.
The complex number “s” is where each point is located along the real and imaginary axes.
For each of these points, we are going to add the following exponential function of time,
where “s” is the constant represented by that point.
Each of the exponential functions will be multiplied by a complex number
that is the value of the Laplace Transform at that point,
as was represented by the height and color of the graph at that location.
Since there are an infinite number of points,
adding a function for each of these points will add up to infinity,
unless we also multiply by the extremely small number
that represents the interval between each of these points.
As the interval between each of the points approaches zero,
the sum of all these exponential functions can be represented as shown.
We multiply the result by the following constant.
We have now created our original function of time.
The real number constant “c” indicates where the line is located.
Since this line can be at an infinite number of different locations,
there are an infinite number of different ways to create our original waveform.
If we move the line over here where the real component of “s” is zero,
then this is a special case where we will be generating the waveform
out of sine waves, as in the case of a Fourier Transform.
The Laplace Transform is more general than the Fourier Transform in that
with the Laplace Transform, we can also add together
“sine waves” whose magnitudes change with time.
What if we already know what our waveform is, and we want to find its Laplace Transform?
Consider the following expression,
where we now have a negative sign in front of the constant “s.”
Multiply this expression by the function of the waveform
who’s Laplace Transform we wish to find.
In the result of the multiplication,
each moment in time is a complex number represented by an arrow.
Add all the arrows together through the following equation.
The result is a complex number described by the white arrow, and this is
the value of the Laplace Transform of our waveform for this particular value of “s.”
If the result of this integral is infinite, then we say that
the Laplace Transform does not exist for this waveform at this value of “s.”
This may happen, for example, if the real component of “s” is a large negative number.
This is why in our example,
the Laplace Transform did not exist where the real component of “s” was negative.
When deciding where to draw our line to recover our original waveform, we need to
always draw it in a location where the Laplace Transform exists for that waveform.
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