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I never found a text book where Laplace and fourier transform is described rigorously. The domain of laplace transform was never clear to me. There are null functions whose laplace transform is 0. Then how is Laplace transform invertible? What are the theorems that cover the topic of region of convergence? I want a book material or course that covers laplace transform rigouruosly using riemann integrals,Compleax analysis, measure theory, where the definitions are concrete and clear. Also it would be very helpful if u can suggest me a book describing delta function and z-transform rigorously. Thank you.

DEBANJAN DEBNATH
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    Engineering and physics textbooks often prioritise application over rigour; if you were to research these transforms from a pure-mathematical source, perhaps with a background of distribution theory, then you would have your rigour. – FShrike Feb 15 '22 at 13:31
  • Indeed, the Fourier-Transform is constant on equivalence classes. – Filippo Feb 15 '22 at 13:34
  • In first courses in differential equations, the integral transforms are there as tools. If you want to really develop the techniques then you will need a course in complex variables, as a start. – Sean Roberson Feb 15 '22 at 13:35

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The Fourier transform is a great tool for solving differential equations, which also means that it's often presented as a black box for that purpose without the underlying details. To some extent, that's fine; integration in general is another broadly useful operation that can be presented axiomatically or simply restricted to the continuous category, without going through all the infrastructure to develop it rigorously (and, at least for the Lebesgue integral, there's a substantial bit of infrastructure required). Still, there are quite a few technical issues in defining the Fourier transform, and getting (for example) the relationship between the smoothness of a function $f$ and the decay rate of $\hat f$, requires a bit of care. The sorts of questions in your post--- convergence issues, conditions for invertibility, etc.--- should have answers in any standard book that covers Fourier analysis at, say, the first-year grad student level.

As for specific references, take a look at Dudley's "Real Analysis and Probability," Rudin's "Real and Complex Analysis," or Adams and Guillemin's "Measure Theory and Probability." These books in particular discuss the characteristic function of a random variable, which is a bit different from the more elementary use of the transform to solve ODEs.

anomaly
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  • I want to specifically learn laplace transform rigouruosly in detail from begining to end, (not just utterly incorrect and vague description in control system books), can you recommend a book for that? ( I need the rigour, for example how is laplace transform invertible, and also the proof for laplace inversion theorem). Thank u. And i will surely check out the books you mentioned. – DEBANJAN DEBNATH Feb 15 '22 at 19:05
  • Those three books should cover it. Adams and Guillemin is the most elementary; then Rudin; then Dudley. – anomaly Feb 15 '22 at 19:55