## 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.