**The Laplace Transform**

Part 6: Frequency shifting, time shifting, and scaling properties

Expanding upon last time’s work on the Laplace transform of the exponential, we can prove quickly the “frequency shifting” property of the Laplace transform. Let real function *f*(*t*), *t*≥0, have Laplace transform . Then, for real number *a*, we see that

,

and thus multiplication by an exponential in the time domain equals a translation in the *s* domain.

This can be combined with our results for sine and cosine to get

.

And our results for powers of *t* to get

.

Note that for all of these, the shift in the “frequency” domain entails an equal shift in the region of convergence; if for , then for .

Next, let’s consider what happens when we scale our time variable. Again, we have *f*(*t*), *t*≥0, with Laplace transform . Now, let us find the Laplace transform for *f*(*at*), with *a*>0.

.

Making u-substitution *u*=*at*, then

.

As with our other result, the region of convergence will be transformed along with the scaling.

We found what happens in the time domain when the transform is shifted in the “frequency” domain, but what happens when we shift in the time domain? Recall that our function *f*(*t*) is defined for *t*≥0; when we shift our function forward, we must set those portions that are at *t*<0 before our shift to zero, so that our translated function is *f*(*t*–*τ*)*H*(*t*–*τ*), where *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 by an exponential in the “frequency” domain.

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Tags: Laplace Transform, Math, Monday Math

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