Properties and rules of Laplace transformation

1.) Laplace transformation of addition operation can be executed by element due to the linear property of Laplace transformation,

2.) Laplace transformation of derivative

Let function a general step function, where its Laplace transformation is .The question is: How is possible to derive the Laplace transformation of derived function.

where the improper integral has to carry out.

Let’s use the partial integral rule:
, where ’ denotes the derivative by „t”.

Consequently,

Then apply the rule by inductive way for higher (i.e. “n” order) derivative, by using the conditions, that general step function, and “n” ordered continuously differentiable in the interval.

, where are the higher (max. “n-1” order) ordered derivative value at point, i.e. at the entry point /these are the initial conditions/.

3.) Laplace transformation of integrals

Let be a general step function and .

Let’s determine the following Laplace transformation .
Definition of the Laplace transformation will be used as previously:

Let’s use the partial integral rule as happened above at the derivative case:

Similar to the “n” order derivative determination can be driven the Laplace transformation of “n” ordered integral.

4.) Theorem of similar

Let be a general step function and an optional real number.

This property of Laplace transformation is according to the property of linearity (see above the linear properties of Fourier and Laplace transformation).

5.) Attenuation theorem

Let be a general step function and an optional real number.

It is known, if general step function, then is the same.

Consequently,

6.) Shift theorem

Figure 27. and 28. show the graphs of general step function and shifted general step function.

Figure 27.

Figure 28.

If Laplace transformation of is , then Laplace transformation of is:

In order to proof the shift theorem, apply the original definition of Laplace transformation:

Apply the following new variable:
Consequently the new integral bounds are: