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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