**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.

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Tags: Laplace Transform, Math, Monday Math

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