I would appreciate reference suggestions to learn how to deal with integrals involving $[x]$, $\{x\}$, $d[u]$, and $d\{u\}$. Thanks!
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Do you know what a Riemann-Stieljes integral is? – Pedro Aug 07 '13 at 01:53
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@PeterTamaroff I don't. – Aug 07 '13 at 01:55
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Well, that explains a lot. – Pedro Aug 07 '13 at 01:58
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@PeterTamaroff Any suggestions to pursue this? That would be much appreciated. Thanks – Aug 07 '13 at 01:59
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"Mathematical Analysis" by Tom Apostol has a great section on the Riemann Stieltjes integral. – Pedro Aug 07 '13 at 02:04
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@PeterTamaroff Thanks – Aug 07 '13 at 02:05
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Here are some related problems (I), (II). – Mhenni Benghorbal Aug 07 '13 at 10:39
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1@MhenniBenghorbal Thanks. Big help +1 x 2 – Aug 07 '13 at 11:00
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@Andrew: You are welcome. – Mhenni Benghorbal Aug 07 '13 at 11:15
1 Answers
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Well, you've tagged this as 'notation', so maybe I'm not getting your question correctly and I'm wrong, but if you're dealing with the case d[x] (I'm not sure what you mean by {x}, do you mean {x}=x-[x]?), You can convert such integrals to infinite series using Stieljes-Riemann integration. For [x], you can divide your interval and then sum the integrals. I hope that correctly answers your question.
user66733
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Thanks. Yes you are are correct in your assumption. Is there a text or set of notes that explain this in detail and gives some examples. That would be great. – Aug 07 '13 at 01:54
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Baby Rudin has a chapter on Riemann-Stieljes integration, if I remember it correctly, it's the subject of the 6th chapter of baby Rudin, but Rudin's book doesn't go into all the details in the text. Apostol's mathematical analysis explains Riemann-Stiejles integration in a more detailed way, also it has some good examples and ideas in there. If you want to go more into details, you can try to solve the exercises. Some of them are quite good and interesting. Search google and ask your questions on here too for more information. – user66733 Aug 07 '13 at 01:59