Chapter 18.1 Introduction
The  appropriate Laplace transform F(s) is assigned to every time function f(t).
F(s) is a quotient of 2 polynomians.

f(t) ==>F(s) (Fig. 18-1a)
And vice versa.
The  appropriate  time function f(t)  is assigned to every Laplace transform F(s).
F(s) ==>f(t) (Fig. 18-1b)
Conclusion
f(t)<==>F(s) (Rys. 18-1c)
*symbol s is a complex number. Don’t you know these numbers?  Don’ bother yourself.

Fig. 18-1
The particular example of the Laplace transform F(s)

Fig. 18-2
F(s) for f(t)=cos(t)
Note 1
Remember that f(t)=cos(t) for t>=0. For t<0 f(t) =0!!!! This rule involves all the time functions f(t).
Note 2:
The  expression f(t)=F(s) is a absolute nonsense.

Chapter. 18.2 F(s) formula

Fig. 18-3
It’s possible to calculate F(s) for easy functions f(t). For more complicated-no. I wondered to set this formula out even.  Never mind If you aren’t  math specialist.  But you have to realise only, that there is a very clever book with all pairs f(t)<==>F(s). The pair from Fig. 18-2c belongs to this book.
The Fig.18-4 represents some of the most popular time functions f(t) used in the automatics. The Fig.18-5 are the pairs f(t)<==>F(s). This is a small part of this clever book.

Fig. 18-4
Note that all these functions are null for t<0.  All begins in time t=0 in automatics usually.

Fig. 18-5
Number e=2,7182… used in formulas is the most known in the mathematics as  0, 1 ,Π…
There is pair δ(t)<==>1 with Dirac pulse too.

Chapter 18.3 Laplace transform and derivation

Fig. 18-6
The most important operational calculus formula.
This formula may be generalise for higher derivatives.

Fig. 18-7
These  formulas enable easy linear differrentiationd  equations solution. You will be convinced in the next chapter.