**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

**3**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.