The Unit Step Function, Second Shifting Theorem, Dirac’s Delta Function
You probably have noticed (and most likely complained about) that of all the techniques we have learned so far almost nothing seemed to be applicable to "real" technical or scientific problems. After all, who wants to spend ones life with calculating harmonic oscillations or RCL circuits that are driven by perfect sine- or cosine-modulated forced only?
Well, in the following we will introduce the first tools that will enable you to study REAL problems professionally.
What is actually the first thing what we do, when we put some "juice" on an electric circuit (beg your pardon, I mean apply an "external electromotive force", of course…)?
We flip some kind of SWITCH. And the following function is exactly the same (mathematically): A SWITCH. To swithc anything off or on, as you will see. It is called the Unit Step Function.
Unit Step Function
Warning: 
is a
function, not a product! Sometimes, in the heat of a test, students tend to mix
this up - and are lost immediately!
Remember:
"u" is the function name,
"t" is the variable, and
"a" is a constant !
Now, the way our unit step function works is very simple but efficient:
The value of u is always zero as long as "t" is smaller than "a", and turns to one as soon as "t" is larger than "a":
Works as an "on" switch at time t=a:
The following graph shows the unit step function :
But this means if we multiply the unit step function
to
any arbitrary function f(x), the resulting product will be zero as
long as "t" is smaller than "a", and turns to f(x)
as soon as "t" is larger than "a" - we found an
"ON"-switch for our function!
Multiply any function f(x) with u(t-a):
Result will be 0 for t<a and f(x) for t>a only!
Furthermore, we can use
as an "OFF"-switch, too: Simply
multiply f(t) with "
":
The beauty of the unit-step function is that we now are able to express sets of quite complicated, and only piecewise defined functions, as one single function!
Example: 
is
such a function.
Without the unit step function, it must be written in three separate lines, one for each definition range. (And, if it becomes part of a D.E., you have to solve the D.E. three times, too!)
With the unit step function, we now can write f(t) as
Let's have a closer look at this:
The unit step function also can be very helpful when we are trying to "shift" a function in
order to solve an initial value problem. Shifting means to substitute the
variable "t" inside of the function with a new variable that is equal to
"t-a".
This will shift the graph of the function backwards along the t-axis by a step of the size "a".
We call this here t-shifting:
t-shifting: Replacing "t" by "t-a" in a function f(t)
Of course we want to know what will happen to the Laplace transform if we do such a t-shifting:
What happens to the Laplace transform?
Answer: 2nd shifting theorem; t-shifting:
(this is important, f must have a transform, of course
!!!)
("shifted
function")
has transform 
.
Means, if we shift a function 
then
in terms of the unit step function u, and
is given as 
Or, w.o.w.:
and, inverse,
And we can reverse this, too. Spare me the proof here, you can find it in the textbook.
Proof: see textbook.
And here comes one of our first "power tools" (by using the theorem above and setting
:
Laplace
transform of the unit step function
What is this good for ? We can find transforms of piecewise defined functions!
Let 's try this out immediately:
Example:
now can be written as
!!!!
(Keep in mind : u is a function, not a variable! And the content of the parenthesis is the argument of u!!!!)
Anyway, the most important thing here is that we are again able to take this piecewise defined function and rewrite it into one single solid function!
c
Which we now can transform like nothing:
And this transforms easily to
done.
And for an inverse transform:
And we can do this in both directions from "t"-space to "s"-space- as seen …
… and from "s"-space to "t"-space:
Let be 
And just transform it back term by term by term by term !
Then,
That easy !
Now let us try something more realistic (but still simple)
Next example: Excite an RC-circuit with a single square wave …
Single square wave: 
So we have a primitive electric circuit, just with some capacitance and resistance.
And we put some voltage on (DC) at t=a and switch it off at t=b.
A little modeling:
Kirchhoff’s law gives us
or, to be precise,
Ugly, isn't it? But now, with the help of the unit step function, we can formualte this differently!
Rewrite v(t): (using the unit step function u:)
and we get:
All the sudden we have one equation that unifies all pieces of v(t)! And now, after all what we learned about the Laplace transform, we don't even have to integrate in order to set up a D.E. but transform immediately into the "s"-space:
Without integrating we get for the transform:
solves to: 
simplify a little …
or, with 
For F(s) we find easily that 
(table
lookup!)
… and we transform back into the t-space (the world of the real circuit)
Then the solution I(t) is given as
We still have our solution in just one equation! Great!!!!!
But some people need to see it step by step, therefore ….
And we can rewrite this as
That's all!
Copyrights 1999, 2000 by Peter Dragovitsch and Ben A. Fusaro