Monday Math 110

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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2 Responses to “Monday Math 110”

  1. Monday Math 111 « Twisted One 151's Weblog Says:

    […] 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 […]

  2. Monday Math 117 « Twisted One 151's Weblog Says:

    […] 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 […]

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