**The Laplace Transform**

Part 10: Initial and Final Value Theorems

Last time, we found the formula for the Laplace transform of the derivative of a function; if we have *f*(*t*), *t*>0, with Laplace transform *F*(*s*), then

.

Now, let us consider what happens to Laplace transforms as *s*→∞. By the definition of the Laplace transform, . Noting that as for all *t*>0, we see that in the limit as *s*→∞, the integrand goes to zero throughout the entire region of integration, except the point *t*=0; so long as f(t) does not have a delta function spike at *t*=0, the integral has to go to zero in the limit. Thus,

.

So, let us take the limit as *s*→∞ of our derivative formula. So long as *f*(*t*) has no step discontinuities at *t*=0, *f’*(*t*) will not have a delta function spike at *t*=0, and the above applies to its transform, so we see that

.

This is called the initial value theorem (since it involves the initial value of *f*).

To find the version for when there is a discontinuity at *t*=0, we have to remove the point at zero from the integral (replace lower limit 0 with 0^{+}, and so we get

.

Next, let us take the limit as *s*→0 of the Laplace transform of *f’*(*t*). When *s*=0, , and so

,

and so, if we take the limit as *s*→0 of our derivative formula,

,

this last being the final value theorem. This tells us that the behavior of *f*(*t*) as *t*→∞ is reflected in the location of the poles of *F*(*s*); specifically, we see four cases:

I. If all poles are on the left side () of the *s* complex plane, then *F*(0) exists and is finite, and so ; our function *f*(*t*) decays to zero exponentially (or faster) at large *t*.

II. If there are poles are on the right side () of the *s* complex plane, such as for the Laplace transforms of the hyperbolic sine and cosine, then *f*(*t*) contains terms growing exponentially, and *f*(∞) does not exist.

III. If there are complex conjugate poles on the imaginary axis () of the *s* complex plane, such as for the Laplace transforms of the sine and cosine, then *f*(*t*) contains sinusoidal terms, and *f*(∞) is undefined.

IV. If there is a simple pole at the origin 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 nonzero.

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