integration limit notation
in [Math]
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From: Bill Dubuque on 25 Feb 2010 09:52 rob(a)trash.whim.org (Rob Johnson) wrote: >"cronusf(a)gmail.com" <cronusf(a)gmail.com> wrote: >> >>I'm looking at a book that solves a differential equation with >> integrating factor. In the end that get something like: >> >> L(x) = exp( int_^x()dx ) >> >> Note that they omit the lower limit of integration and just keep >> the upper limit of integration. What does this mean? Is this a >> shortcut where the lower integration limit is implied somehow >> (it would be 0 in the context of the problem). > > My guess is that > > |\x > L(x) = exp( | () dx ) > \| > > was really intended to use the indefinite integral > > |\ > L(x) = exp( | () dx ) > \| > > Unfortunately, the extra limit, perhaps intended to make something > more explicit, is not only non-standard, but also confusing. The OP misread: the integral has the form int^x f(t) dt (not dx) so it's just a change of variable t -> x in an indefinite integral cf. bottom of p. 2 in: http://www.cs.cornell.edu/courses/cs667/2005sp/notes/09wang.pdf
From: Rob Johnson on 25 Feb 2010 10:43 In article <l2chbp5nz9f.fsf(a)shaggy.csail.mit.edu>, Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >rob(a)trash.whim.org (Rob Johnson) wrote: >>"cronusf(a)gmail.com" <cronusf(a)gmail.com> wrote: >>> >>>I'm looking at a book that solves a differential equation with >>> integrating factor. In the end that get something like: >>> >>> L(x) = exp( int_^x()dx ) >>> >>> Note that they omit the lower limit of integration and just keep >>> the upper limit of integration. What does this mean? Is this a >>> shortcut where the lower integration limit is implied somehow >>> (it would be 0 in the context of the problem). >> >> My guess is that >> >> |\x >> L(x) = exp( | () dx ) >> \| >> >> was really intended to use the indefinite integral >> >> |\ >> L(x) = exp( | () dx ) >> \| >> >> Unfortunately, the extra limit, perhaps intended to make something >> more explicit, is not only non-standard, but also confusing. > >The OP misread: the integral has the form int^x f(t) dt (not dx) >so it's just a change of variable t -> x in an indefinite integral >cf. bottom of p. 2 in: >http://www.cs.cornell.edu/courses/cs667/2005sp/notes/09wang.pdf Actually, on page 4, section 3.1, the second equation reads |\t - | \sigma_a(t) dt L(t) = C e \| Where t is used as limit and integrand. However, this is the only place I see. As Robert Israel mentions, the lower limit could be set to anything as that only means using a different constant C. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font
From: Bill Dubuque on 25 Feb 2010 12:32 rob(a)trash.whim.org (Rob Johnson) wrotw: >Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >>rob(a)trash.whim.org (Rob Johnson) wrote: >>>"cronusf(a)gmail.com" <cronusf(a)gmail.com> wrote: >>>> >>>>I'm looking at a book that solves a differential equation with >>>> integrating factor. In the end that get something like: >>>> >>>> L(x) = exp( int_^x()dx ) >>>> >>>> Note that they omit the lower limit of integration and just keep >>>> the upper limit of integration. What does this mean? Is this a >>>> shortcut where the lower integration limit is implied somehow >>>> (it would be 0 in the context of the problem). >>> >>> My guess is that >>> >>> |\x >>> L(x) = exp( | () dx ) >>> \| >>> >>> was really intended to use the indefinite integral >>> >>> |\ >>> L(x) = exp( | () dx ) >>> \| >>> >>> Unfortunately, the extra limit, perhaps intended to make something >>> more explicit, is not only non-standard, but also confusing. >> >> The OP misread: the integral has the form int^x f(t) dt (not dx) >> so it's just a change of variable t -> x in an indefinite integral >> cf. bottom of p. 2 in: >> http://www.cs.cornell.edu/courses/cs667/2005sp/notes/09wang.pdf > > Actually, on page 4, section 3.1 [...] That's not relevant since the OP asked about the integral on (bottom of) p. 2, which has the form that I said above.
From: Rob Johnson on 25 Feb 2010 14:49
In article <l2cd3ztnrvp.fsf(a)shaggy.csail.mit.edu>, Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >rob(a)trash.whim.org (Rob Johnson) wrotw: >>Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >>>rob(a)trash.whim.org (Rob Johnson) wrote: >>>>"cronusf(a)gmail.com" <cronusf(a)gmail.com> wrote: >>>>> >>>>>I'm looking at a book that solves a differential equation with >>>>> integrating factor. In the end that get something like: >>>>> >>>>> L(x) = exp( int_^x()dx ) >>>>> >>>>> Note that they omit the lower limit of integration and just keep >>>>> the upper limit of integration. What does this mean? Is this a >>>>> shortcut where the lower integration limit is implied somehow >>>>> (it would be 0 in the context of the problem). >>>> >>>> My guess is that >>>> >>>> |\x >>>> L(x) = exp( | () dx ) >>>> \| >>>> >>>> was really intended to use the indefinite integral >>>> >>>> |\ >>>> L(x) = exp( | () dx ) >>>> \| >>>> >>>> Unfortunately, the extra limit, perhaps intended to make something >>>> more explicit, is not only non-standard, but also confusing. >>> >>> The OP misread: the integral has the form int^x f(t) dt (not dx) >>> so it's just a change of variable t -> x in an indefinite integral >>> cf. bottom of p. 2 in: >>> http://www.cs.cornell.edu/courses/cs667/2005sp/notes/09wang.pdf >> >> Actually, on page 4, section 3.1 [...] > >That's not relevant since the OP asked about the integral >on (bottom of) p. 2, which has the form that I said above. Ah, now that I look back to an offshoot of the thread, I see that the OP mentions page 2. As you say, the form he asks about is not on page 2. However, the thread to which I was replying did not mention page 2, and his form does exist on page 4, so it seemed relevant. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font |