§3 Laplace transform 

 

    Laplace transform of [ Laplace transform and its inversion formula ] 

               ( s is a complex number, s = )

    Inversion formula of Laplace transform

                 

The integral is taken along any straight line Res= , which is the growth index, and at the same time, the integral is understood in the sense of the principal value .

    [ Conditions for the existence of Laplace transform ]   If the following three conditions are satisfied, then its Laplace transform exists .

(i)          The complex-valued function sum of real variables is continuous except for the discontinuous points of the first type (there are at most a finite number in any finite interval);

(ii)        when t < 0 , =0;

(iii)      is of finite order, that is to say it is possible to find constants and A > 0 such that

                            

The number here is called the growth index, and when it is a bounded function, it can take =0.

    If the above three conditions are satisfied, then L ( s ) is an analytic function on the half-plane Res> . The inversion formula holds at the continuous points of .

[ Properties of Laplace Transform ]

            ( a is a constant )

               ( a , b are constants )

        

in the formula

              

is called the convolution (or convolution) of the function and g ( t ) .

 

[ Main formula table of Laplace transform]

 

original function	Laplace transformed function
	( the nth derivative)
	( n -fold integration)
f ( n ) ( t )
original function	Laplace transformed function
	( ) m s n L ( m ) ( s )
( n -fold integration)
f ( t 2 )
t v- 1 f ( t )             (Re v > )
	L (ln s )

 

[Laplace transformation table]

              

          

 

 

 

Laplace Transformation Table I

 

(It is convenient to use this table to find the Laplace transform of a known function)

f ( t )	L ( s )
	1
( c > 0 )	e – cs
1
t
t n
t v                ( Rev > )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( > 0 )
( a > 0)
( 2 t + t 2 ) v ( a >0, Rev > )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
e a t
te a t
t n e a t
t v e a t (             Rev > )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
(Rev > -1)
( Rev > )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
ln t	( for Euler's constant)
erf ( a t )       ( a >0 )
( a > 0 )
( a > 0 )
( a > 0 )
( a > 0 )
( Rev > -1 )
(Rev > 0)
( Rev > -2 )
( Rev > -1 )
( Rev > -1 )
( Rev > -1 )

 

Laplace Transformation Table II

 

(The Laplace transform of a known function is convenient to use this table to check its original function)

L ( s )	f ( t )
( varies)
	( )

 

    [ Double Laplace transform and its inversion formula ]

    The double Laplace transform of the function f ( x,y ) is

    The inversion formula of the double Laplace transform is:

Among them .