Definition talk:Laplace Transform

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Perhaps the "help" template wasn't the right one. It's just that my sources gloss over details involving complex functions. I don't want to only cite pages I find online on lecture notes and the like, so I was asking for people with better physical books than I have. --GFauxPas (talk) 10:05, 7 May 2014 (UTC)

I will get to this, in due course, but for the moment if your sources are inadequate, I would recommend steering clear of this area until complex analysis has been placed on a much more secure footing. As it stands there is too little rigour.
I don't understand what is meant by "depending on context" -- the Laplace transform converts a real function in $t$ to a complex function in $s$, that's what it is, that's what it does. So its restriction to the real number line is puzzling. I will have to go away and study it all again.
I understand that for applications (and I guess that your source work is such as this) it is not necessary to prove the basics of L.T's -- you just read the tables and plug in the values -- but for $\mathsf{Pr} \infty \mathsf{fWiki}$ we need a higher level of rigor. --prime mover (talk) 11:19, 7 May 2014 (UTC)
I suppose you're right. I'll stick to characteristic equations for the moment. --GFauxPas (talk) 11:23, 7 May 2014 (UTC)
Seriously, is this how the Laplace Transform is defined in that source work of yours? I can't make head or tail of the definition of the domain and range of $f$. --prime mover (talk) 12:10, 8 May 2014 (UTC)
I consulted other sources as advised. I'm trying to say that $f(t)$ takes positive real $t$ and that $f$ can be $\R$ valued or $\C$ valued, the definition is the same. I'll think of how to present it more clearly. As to piecewise continuity, looking at the sources, it's mentioned in only one source and only in passing, so I guess its not the standard defn. I'll put up more sources later. --GFauxPas (talk) 18:29, 8 May 2014 (UTC)
It's the fact that $t$ can be $\C$ valued which is puzzling me ... maybe it's just that I've only ever seen these things in a particular context. And (now I've gone back to check some of them) some of my sources are somewhat non-rigorous themselves (1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables for example using the $s > a$ condition and not bothering to mention the domain or range of either $f$ or $F$). Tell you what, I'll let you get on with it the best way you can, and we can revisit it later when we find a rigorous source. I may well have the appropriate literature, but it's buried somewhere within a teetering pile at the moment. --prime mover (talk) 19:08, 8 May 2014 (UTC)

Incidentally, since I spent quite some time tidying up the sources, which edition of Boyce and DiPrima are you using? I see a 2009 date in it, is that the edition date or the date of the printing? If the latter, please amend it to the date the edition was published, if the former, then a page will need to be set up for that edition. --prime mover (talk) 19:12, 8 May 2014 (UTC)

I'm using the 9th edition, and the date is the copyright date. I only see one date mentioned so I'm not sure what you mean by edition date vs printing date. --GFauxPas (talk) 19:55, 8 May 2014 (UTC)
You get books that say: first published (year), second printing, third printing etc. etc. then second edition (some other year) and then so on. --prime mover (talk) 20:05, 8 May 2014 (UTC)
Okay, so I've started the page in question, I'll leave it to you to finish it off. Thx. --prime mover (talk) 20:09, 8 May 2014 (UTC)

... and it turns out I do have Edition 5 of the Boyer/DiPrima work -- which explains why it is so carefully documented already. I'm going to have to unearth it from the bottom of its pile and crack it open -- but that won't happen at least this next couple of weeks, I have my teeth in something else again. --prime mover (talk) 22:24, 8 May 2014 (UTC)

Domain

I don't know how to present it clearly: The domain of $f$ must be a superset of all intervals otf $\left[{t..\to}\right)$. Theres probably a very concise and clear way to get down that description to a few words. --GFauxPas (talk) 17:59, 14 November 2016 (EST)

If $S$ must be a superset of all those intervals then $S = \R_{\ge 0}$, yes? Or am I missing something subtle? --prime mover (talk) 18:09, 14 November 2016 (EST)
Okay, after reading up on Vretblad, that's what he does for this, but he has a separate definition for integrands with $\delta$ that require the left endpoint of the interval to be $-\epsilon, \epsilon > 0$. But since he has that as a separate definition, I'll follow his lead and keep this definition simple, as it works fine for L.T. on functions. --GFauxPas (talk) 18:13, 14 November 2016 (EST)
Much more sense. I was afraid I was losing it! --prime mover (talk) 18:29, 14 November 2016 (EST)