MATH201 Chapter 4: ch4-p1.pdf
16 Apr 2014
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The operation of di erentiation transforms a given function to another function: A transform is a mapping from one set of functions to another: For the di erential operator d, we have f1 = {all di erentiable functions}, whereas for the integral operator. I we might have f1 = {all continuous functions}. A useful tool for solving di erential equations is that of an integral transform, particularly on of the form: The function k(x, s) is called the kernel, and the transform is usually denoted as f = t [f ]. Several transforms have been found useful: laplace, fourier, mellin, hankel, . The laplace transform of a function f is de ned as provided the integral exists. This is an improper integral, so it doesn"t converge for just any old function f . We will use laplace transforms to solve odes in the following manner.
