**The Laplace Transforn**

Part 11: Frequency Derivatives and Integrals

We found previously what happens when one takes the Laplace transform of the derivative of a function,

,

and what happens when we take the Laplace transform of the integral

.

But what happens if we take the derivative with respect to *s* of the Laplace transform? Namely, what is *F’*(*s*)?

Since *F*(*s*) is the Laplace transform of *f*(*f*),

.

Taking the derivative of both sides with respect to *s*,

.

This “frequency differentiation” formula is usually written

,

and gives us the transform of a function multiplied by *t* in terms of the function’s transform. For example, consider the ramp function, . From here, we know , and so the above tells us that

,

as expected from here.

In fact, we can apply the formula repeatedly

,

and in general,

;

compare to our earlier results for the transforms of positive integer powers of *t*.

From our previous discussion of limits, we found that for any Laplace transform *F*(*s*); thus the integral makes a more reasonable antiderivative of *F*(*s*) over the region of convergence. Note that if we define , then *G’*(*s*)=-*F*(*s*), and *G*(∞)=0 (as needed for it to be the Laplace transform of some *g*(*t*)).

Now, applying our frequency differentiation rule to *G*(*s*),

we have

,

and so dividing by time corresponds to a “frequency” integration.

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Tags: Laplace Transform, Math, Monday Math

This entry was posted on March 1, 2010 at 12:02 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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March 8, 2010 at 12:43 am |

[…] transform for cosine is , so Thus, via the frequency shift formula here, . Considering, then, our frequency differentiation formula, . Now consider the “sine cardinal” or “sinc” function . What is […]

April 26, 2010 at 12:08 am |

[…] initial conditions . Taking the Laplace transform of both sides, again with , . Now, recall our frequency differentiation formula: Thus, we see , and . Thus, our transformed equation becomes: , and we have turned our […]