§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 .