University Mathematics Handbook (2015)

XIII. Fourier Series and Integral Transforms

Chapter 3. Laplace Transformation Formulas

3.1  Definition

a.  Let be a piecewise continuous function : →. Function of real variable is calledLaplace transform of .

b.  If there exist real constants and such that , then is defined for every .

3.2  Formulas of Laplace Transform

a.  Linearity

b.  Differential formula:

c.  n-th order differential:

d.  

e.  

f.  

g.  If is a periodic function with period , then

3.3  Heaviside Step Function

For all real positive , the function is Heaviside step function.

a.  

b.  If is Laplace transform of and , then .

c.  If , then .

3.4  Dirac Delta Function

a.  Let be a given real number. "Function" , holding for every function continuous in the neighborhood of , and for all set containing a neighborhood of , is called the Dirac delta function on .

b.   only exists as a function of functions. One common description of is a "limit of a process". For every , let us define function .

Properties of :

1.

2. , for every

3. , for every

Function can be perceived as equal to the so-called limit .

c.  

d.  

3.5  Convolution (see 2.3 h,j)

a.  

b.  

c.  If there exist constants , , and such that and for every , then

, and there holds

.

3.6  Table of Laplace Transforms

are real numbers and