Main theorems of the Laplace transform

Derivative theorem

For a causal function of time , for which the derivative for exists, and take into account any steps at , the value of will be chosen for the lower limit of integration for Eq. (A.1). This is necessary to eliminate the case from the integration interval. This has no influence on the value of the integral as far as we restrict on classical functions (no distributions). So one obtains by partial integration

or

In the case of multiple differentiation it follows that

Integral theorem

From

Convolution in the time domain

For the convolution of two functions of time and

I can easily be shown by permutation of the variables that the convolution is a symmetrical operation, so that

or

In the following it will be shown that the convolution of two original functions corresponds to multiplication of the related mapped functions, that is

The Laplace transform of Eq. (A.6) is given by

Substituting and , respectively and using the valid extension of the upper bounds of integration to yields

As both functions and have zero values for , it follows with respect to the lower limit of integration that

The right-hand side of this equation is just the product .

Convolution in the frequency domain

Whereas in section A.1.3 the convolution of two functions of time was the focus of interest, here the convolution of two functions in the frequency domain is of concern and it can be shown that

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.

For the product of two causal functions of time

with Laplace transforms and and areas of convergence and , respectively, the expression

follows after taking the Laplace transform of . Using the inverse integral according to Eq. (A.2)

and by substituting this relationship into Eq. (A.10) it follows that

Permuting the sequence of integration (as far as the integrals fulfil the conditions of convergence) one obtains

where for the second integral one can make the substitution

This integral converges for . By substituting Eq. (A.14) into Eq. (A.13) the validity of Eq. (A.8) is shown.

Initial value theorem

It is required to show that

One has that

which as can be written

As the integration is independent of , the calculation of the limit and the integration can be permuted provided that the integral converges uniformly. If exists, then

is valid. Therefore one gets

Final value theorem

The final value theorem states that

To prove this one evaluates the limit

Again one can permute the sequence of determining the limit and the integration provided the integral converges. The result is

and after integration it follows that