Transformed of Laplace
1) Transformed of unilatera Laplace :
It is said transformed of Laplace of the variable function
given f(t) of real t one transformation that ago to correspond to the
function f(t) one F(p) function of the variable defined complex p from
the integral 
.
2) Index of the degree of crescenza of the function f(t):
It is the inferior end of the values of to for which the inequality has place |f(t)| £ Meat .
3) It originates them:
The function f(t) is said originates them of the T.d.L. F(p) on condition that you respect the 3 following conditions:
to) f he is locally sommabile that is 
it is convergent
b) f(t) = 0 for t<0
c) Exists constant M>0 and s0? " such that |f(t)| £ Mest
4) Condition of convergence of the integral of Laplace:
The 
integral converges in the
dominion King p > to, where to is the index of the degree of crescenza
of the function f(t) ; , for moreover every x0 > to, this integral uniform converges
in dominio the King p ³ x0 > to.
It is had p = x iy and of the rest to it is the index of
the degree of crescenza of the function f(t) that is is worth the
inequality |f(t)| £ Meat can therefore be maggiorare the integral with a n°
real and therefore the integral is convergent, in fact if to1 = to and 
. In analogous way
se to < x0 < x the uniform
convergence of the integral can be applied the theorem of Weierstrass
and be demonstrated.
5) Ulterior condition of convergence of the integral of Laplace:
F(t) it is defined for every t ³ 0 and exists a 0 number complessop such that the integral 
is satisfactory every convergent for p the condition King p >
0 Kingp the integral is convergent.
The absolute integrabilità of is had
if it is succeeded to demonstrate in particular 
that
it is convergent, placing King p = P0 q and integrating for parts 
is had where
and it is worth 0 for t=0 while for T®0 ¥
the term is worth 
therefore remains only
the integral to according to member which (t) < K can be increased
being the integral of a exponential in how much j.
6) the transformed one of Laplace of the function f(t) is a function analytics of the variable complex p in the dominion King p > to where to it is the index of the degree of crescenza of the function f(t).
7) Theorem of Omotetia:
For every to > 0
constant is had : 
8) Theorem of derivation of originates them:
If f ' (t), f ' ' (t)..., f(n)(t) is originates it them and 
allora 
e 
.
9) Theorem of derivation of the image:
The derivation of the image is reduced to the
multiplication of originates them for - t 
10) Theorem of integration of originates them:
Integration of originates them is reduced to the division
of the image for p 
11) Theorem of integration of the image:
Integration of the image gushes from the division for t of
originates them 
12) Theorem of translation in the dominion of Laplace:
The multiplication of originates them for a complex exponential from place to one translation of the image.
13) Theorem of the delay or translation in the dominion of the time:
A translation of originates them from place to the multiplication of the image for a complex exponential .
14) Definition of the delta of Dirac:
The delta of Dirac is one function defined from the 2
following : 
, it turns out to be the derivative of the
unitary step while transformed its is 1.
15) Theorem of the convoluzione:
The product of 2 functions images is the transformed one of the convoluzione of theirs originates them
This theorem is much profit in the calculation of the antitransformed ones.
16) Theorem of Mellin :
In the dominion King p > to the variable regular function
at times f(t) of the real one t with degree of crescenza to
x is F(p) transformed of one > a.
The function is defined 
and it
is demonstrated that for b®¥ it converges to f(t) in
particular replacing in it and taking advantage
of the uniform convergence a Pò of things from an integral is passed
to the other obtaining 
replacing therefore
p=a is has 
where the last one is the
integral of a exponential resolving which it is come to esplicitare a
breast expressed in terms of esponenziali, has therefore 
in which pu² to replace t = t x and to expand the integral from -¥ to ¥ to this point integrating for parts and using
the theorem of Riemann obtains that the integral stretches just to
f(t).
17) Conditions for the existence of the antitransformed one of Laplace:
We suppose that the F(p) function of variable p = the x iy satisfies the following conditions :
to) F(p) it is analytics in the Re(p) dominion > to
b) F(p) ® 0 for |p| ® ¥ in the dominion Re(p) > a in uniform way respect to arg p.
c) the integral 
x > a converges " Re(p) = x > to
the F(p) function for King p > a is the transformed one of defined the variable function f(t) of the real one t from the expression
The integral in the customary way is
demonstrated only to the convergence ofthe
improper integral maggiorando.
