Definition of Laplace Transform
Laplace Transform is a very useful mathematical tool, mainly used to solve differential equations. It converts the differential equations into algebraic form, which are then easy to be solved and then the solution is inversely converted, using Inverse Laplace Transform, to the form it should initially be.
The Laplace Transform of a function f(t) is given by the equation:
Note that before the transform, the function had as independent variable the variable t. After Laplace Transform, the independent variable of the function changed to s. In other words, Laplace Transform changes the domain of a function, from the domain of time (or whatever the independent variable is) to the domain of s. This “s” is a dummy domain (variable) used only for the calculations, it has no meaning for the real world. The result must be converted back to its original domain to have a meaningful result.
Laplace Transform is noted by a calligraphic L. For the resulting function we use the same letter of the initial function, but in capital. For example to note the Laplace Transform of the function f(x), we write it as follows:
Note again that before the transform we have a function of time and after the transform we have a function of s.
Let see how we can calculate the Laplace Transform of a constant number C.
For more examples about calculating Laplace Transform from the definition, have a look at the appropriate menu selection.
To solve a differential equation, we don’t have to transform it from the definition. We can use the transformation table to look up the answer. We do this for every factor of the differential equation, we solve the resulting algebraic equation and then, using the transformation tables again, we inversely transform the result to bring it back from the “s” world to the “real” world.
For more examples about solving differential equations, have a look at the appropriate menu selection.
Where does Laplace Transform come from? Why this integral with this exponent is used?
All begin from the power series. Power series is the infinite summation of a product of a coefficient by a variable raised to a power. If this infinite summation leads to a number, then we say that it converges and in some cases it can lead to a certain function. For example the following summation is equal to the Euler number raised to the variable:
In a more compact notation we would write it as:
More generally we can have the summation of the product of a coefficient, which is a function of n, by a power of x:
In a more compact notation we would write it as:
Up to now we have a discreet variable n. How would the previous equation look like in its continuous analog form? The first thing to change, in order to get its continuous analog, is the sigma summation symbol.
Instead of it we will use the integration symbol. Also we have to change the discreet variable n by a continuous one and since most of the problems we deal with refer to the time, we will change the variable n to t:
At this continuous analog form of power series, we will make two more changes to make it easier for calculations. The first change is to write C(t) as f(t), since this notation is the most common for functions. The second change has to do with the x to the t power. This is not practical for further calculations, so we will convert it to something with e as its base, to make integration easier:
Since time is always positive, this integration converges only if:
This assumption leads to an exponent that its value becomes smaller and smaller by the time, so the product of the function by this exponent also becomes smaller by the time making the integration to converge.
Otherwise the integration becomes infinite and it is impossible to calculate it.
To make the notation simpler, we define:
And we have:
So Laplace Transform is the continuous analog of the summation of power series.
In the general case, s can be a complex number (since x, the base of the power series, could be a complex number). So it is equal to:
Region of Convergence are the values of σ and ω for which the integration of the calculation of Laplace Transform from the definition formula, converges. The range of values can be plotted on the Cartesian system, which in this case is called the “s plane”.
Let’s calculate the Region of Convergence of the following function:
In this example the integration converges only when:
The Region of Convergence for this example is:
Figure 1: Example of RoC
Consider the following proper rational function:
The roots of the numerator are called zeros and the roots of the denominator are called poles. Obviously when s is equal to a zero, the value of the function becomes zero and when s is equal to a pole the value of the function becomes infinite. The values of s that are zeros or poles are important because they define the Region of Convergence. Zeros are noted by a “o” and poles are noted by a ”x”
For the above function plotting poles and zeros on the s plane would look like:
Figure 2: Example of Poles and Zeros