Many differential equations textbooks (“First course” books) limit themselves to taking Laplace transforms of functions of exponential order. That is a reasonable thing to do. However I’ll present an example of a function NOT of exponential order that has a valid (if not very useful) Laplace transform.

Consider the following function:

Now note the following: is unbounded on , does not exist and

One can think of the graph of as a series of disjoint “rectangles”, each of width and height The rectangles get skinnier and taller as goes to infinity and there is a LOT of zero height in between the rectangles.

Needless to say, the “boxes” would be taller and skinnier.

Note: this ~~is an example~~ can be easily modified to provide an example of a function which is (square integrable) which is unbounded on . Hat tip to Ariel who caught the error.

It is easy to compute the Laplace transform of :

. The transform exists if, say, by routine comparison test as for that range of and the calculation is easy:

Note: if one wants to, one can see that the given series representation converges for by using the ratio test and L’Hoptial’s rule.