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**Basic Definitions and Results**

Let

*f*(

*t*) be a function defined on . The Laplace transform of

*f*(

*t*) is a new function defined as

The domain of is the set of , such that the improper integral converges.

**(1)**- We will say that the function
*f*(*t*) has an**exponential order**at infinity if, and only if, there exist and*M*such that

**(2)****Existence of Laplace transform**

Let*f*(*t*) be a function piecewise continuous on [0,*A*] (for every*A*>0) and have an exponential order at infinity with . Then, the Laplace transform is defined for , that is .**(3)****Uniqueness of Laplace transform**

Let*f*(*t*), and*g*(*t*), be two functions piecewise continuous with an exponential order at infinity. Assume that

then*f*(*t*)=*g*(*t*) for , for every*B*> 0, except maybe for a finite set of points.**(4)**- If , then

**(5)**- Suppose that
*f*(*t*), and its derivatives , for , are piecewise continuous and have an exponential order at infinity. Then we have

This is a very important formula because of its use in differential equations. **(6)**- Let
*f*(*t*) be a function piecewise continuous on [0,*A*] (for every*A*>0) and have an exponential order at infinity. Then we have

where is the derivative of order*n*of the function*F*. **(7)**- Let
*f*(*t*) be a function piecewise continuous on [0,*A*] (for every*A*>0) and have an exponential order at infinity. Suppose that the limit , is finite. Then we have

**(8)****Heaviside function**

The function

is called the Heaviside function at*c*. It plays a major role when discontinuous functions are involved. We have

When*c*=0, we write . The notation , is also used to denote the Heaviside function.**(9)**- Let
*f*(*t*) be a function which has a Laplace transform. Then,

and

Hence,

**Example:**Find

.

**Solution:**Since

,

we get

Hence,

In particular, we have