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(t-τ), 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(t-τ), 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.
Tags: Laplace Transform, Math, Monday Math
January 14, 2010 at 11:42 pm |
[...] 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 , [...]
January 25, 2010 at 12:03 am |
[...] 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 [...]
February 15, 2010 at 12:40 am |
[...] 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 [...]
February 22, 2010 at 12:38 am |
[...] 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 [...]
March 1, 2010 at 12:08 am |
[...] 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 [...]