Again this is quasi-mathematical imprecise thoughts about blah blah transforms.

I saw a book that has both Fourier and Laplace transforms in its title. The book says that Laplace transform is a generalisation of Fourier transform, in something like this way…

Let’s say is a function that has such and such nice properties. Then there is another function such that

,

which is called Fourier transform of .

Then letting , we have

,

which is called Lapalce trasnform. —I think should have been to make the things more straight.

Okay. The point is taken. is simpler than . I must admit Laplace transform is more general. This seemed to be all spoken about relation between Fourier and Laplace transforms in the book.

But wait a second. Usually Laplace transform is introduced to us as a means for solving initial value problems of differential equations, isn’t it? I thought Fourier transform also should be talked about as a similar method for solving IVPs, to make relationship between these trasnforms clearer…

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