**The Laplace Transform**

Part 5: Exponential and trigonometric functions

Next ia a very simple Laplace transform: that of exponential functions. If , we use the definition of the Laplace transform to find that

,

with for convergence.

From this, and the linearity of the Laplace transform, we see

,

and

,

both with .

Now, in the integral we used for the exponential, we see that the integration still holds, for , if we replace *a* with *ıω*; we get

.

Thus, we use and to get

,

and

,

both with .

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