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 thatthen

*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 haveThis 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 havewhere 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 functionis called the Heaviside function at

*c*. It plays a major role when discontinuous functions are involved. We haveWhen

*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

The next example deals with the Gamma Function.

**
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