Monday Math 102

The Laplace Transform
Part 3: Step and Delta Functions

The Laplace transform of the constant function f(t)=c, t≥0, can be computed easily from the definition of the transform; with (necessary for convergence of the integral),
.

Similarly, one can also find easily the Laplace transform of f(t)=H(tτ), where H(t) is the Heaviside step function, and τ a positive constant. [If τ≤0, then for t>0, f(t)=1, and the integral is identical to that above for the constant, and .]
Taking the integral, again with ,
.

Next, consider an impulse function f(t)=δ(tτ), where δ(t) is the Dirac delta function, and τ≥0.*.
Then we have
,
and letting τ=0, we see
.
Unlike the previous two Laplace transforms, this converges for all .

*The formal definition of the Laplace transform for acceptable distribution functions, such as the Dirac delta, is chosen so that any “point mass” at t=0 is entirely included in the transform; this is often written , where the lower limit 0 is shorthand for the limit as zero is approached from below:
.
Thus, the case of τ=0 above is valid.

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5 Responses to “Monday Math 102”

  1. Monday Math 103 « Twisted One 151's Weblog Says:

    […] integer n, then we have . Note also that r=0 gives a transform of 1/s, which is what we previously found for the Laplace transform of a constant. Setting , and using , we see . Similarly, since , […]

  2. Monday Math 105 « Twisted One 151's Weblog Says:

    […] H(t) is the Heaviside step function, and τ>0 is our time delay. Integrating as we did for the transform of the Heaviside step function, . Now, making u-substitution u=t-τ, we find . Thus, a time shift corresponds to multiplication […]

  3. Monday Math 108 « Twisted One 151's Weblog Says:

    […] is a discontinuity in f(t) at t=0, then we replace f(0) in the above formula with ; contrast to here.) Consider then the Laplace transform of the second derivative f”(t). We can apply the above […]

  4. Monday Math 109 « Twisted One 151's Weblog Says:

    […] of the s complex plane (and none in the right side), such as for the Laplace transform of the constant function, then exists and is […]

  5. Monday Math 110 « Twisted One 151's Weblog Says:

    […] 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 […]

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