Main theorems of the Laplace transform

a)

Superposition theorem:

For arbitrary constants and it follows that

The Laplace transformation is a linear integral transformation.

b)

Similarity theorem:

For an arbitrary constant

is valid. This follows from Eq. (2.1) by the substitution of .

c)

Real Shifting theorem:

For an arbitrary constant

is valid. This follows directly from Eq. (2.1) by the substitution of .

d)

Complex Shifting theorem:

For an arbitrary constant

is valid. This follows directly from Eq. (2.1).

e)

Derivative theorem:

For a causal function of time, , for which the derivative for exists, then as shown in section A.1.3, one obtains

and in the case of multiple differentiation

f)

Complex differentiation theorem:

This theorem shows that a differentiation of the mapped function corresponds to a multiplication with the time in the time domain:

g)

Integral theorem:

The integral of a function is mapped by

as shown in section A.1.3.

h)

Convolution in the time domain:

The convolution of two functions of time and , presented by the symbolic notation , is defined as

In section A.1.3 it is shown that the convolution of the two original functions corresponds to the multiplication of the related mapped functions, that is

i)

Convolution in the frequency domain:

Whereas in h) the convolution of two functions of time was given, a similar result for the convolution of two functions in the frequency domain exists and is given by

Here and is valid. Furthermore, is the complex variable of integration. According to this theorem the Laplace transform of the product of two functions of time is equal to the convolution of and in the mapped domain. This is shown in detail in section A.1.3.

j)

Initial and final value theorems:

The theorem of the initial condition allows the direct calculation of the function value of a causal function of time from the Laplace transform . If the Laplace transform of and exist, then

is valid if the exists, see section A.1.3.

Using the theorem of the final value the value of for can be determined from , if the Laplace transform of and exist and the limit also exists. Then it follows from section A.1.3 that

One has to observe that

can be calculated only from the corresponding Laplace transform by application of the theorems of the initial or final value, if the existence of the related limit in the time domain is a priori assured. The following two examples should explain this:

Example 2.3.1  

The limit does not exist so that the final value theorem may not be applied.

Example 2.3.2  

The limit does not exist and therefore the final value theorem may not be applied.

It can be concluded from the last two examples that the following general statement is valid: If the Laplace transform has, apart from a single pole at the origin , poles on the imaginary axis or in the right-half plane, then the initial or final value theorems cannot be applied.