**The Laplace Transform**

Part 12: Combining Rules

We can combine many of the previous rules we’ve found to determine the Laplace transform of some more complicated functions.

For example, what is the Laplace transform of ? We know from here that the Laplace transform for cosine is , so Thus, via the frequency shift formula here, .

Considering, then, our frequency differentiation formula,

.

Now consider the “sine cardinal” or “sinc” function . What is the Laplace transform of sinc(*kt*)?

We recall from here that the Laplace transform for the sine is

. We apply to this our frequency integration formula to get the transform of sinc(*kt*)

,

.

Lastly, let’s consider the square wave with period 2π given by . As this is a periodic function, we use our periodic function Laplace transform

,

where *T* is the period.

For our current function, this gives us:

.

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Tags: Hyperbolic Tangent, Laplace Transform, Math, Monday Math, Periodic Function

This entry was posted on March 8, 2010 at 12:43 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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April 19, 2010 at 12:13 am |

[…] for t<0. Taking the Laplace transform of our differential equation, letting , Recall from here that we found, using the periodic function rule, that the square wave has Laplace transform , so […]