Re: how to explain this weird effect? Integrate
- To: mathgroup at smc.vnet.net
- Subject: [mg46257] Re: [mg46245] how to explain this weird effect? Integrate
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Thu, 12 Feb 2004 22:46:17 -0500 (EST)
- References: <200402121216.HAA12039@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 12 Feb 2004, at 13:16, steve_H wrote: > I type: > > r = Integrate[Sin[m x] Sin[n x], {x, 0, 2 Pi}] > > then I type > > r /. {n -> 2, m -> 2} > > I get error (1/0 expression encountered) and no result. > > but when I let m=2 and n=2 right into the integral first, it works: > > r = Integrate[Sin[2 x] Sin[2 x], {x, 0, 2 Pi}] > > and I get Pi as expected. > > I wanted to integrate this once, and try the output for different n,m. > > I did not think it will make a difference as to when I replace m and n > by their numerical values, but Mathematica disagrees. > > I know Mathematica is correct in this, since it is clear from the > result > of the integration why I get 1/0. But it seems to me I should > get the same result if I replace m,n inside the integral before > the integration operation starts, or replace them afterwords. > > For example, when I type > > Integrate[Sin[m x], {x, 0, Pi}] > % /. m -> 4 > > I get zero. > > and when I replace m with 4 inside the integral first, I get the same > result as above: > > Integrate[Sin[4 x], {x, 0, Pi}] > 0 > > > So, what do you think? is there something I am missing here? > > thanks > Steve > > > It's not hard to explain if you actually look at the output you get before substituting values for n and m. r = Integrate[Sin[m*x]*Sin[n*x], {x, 0, 2*Pi}] (n*Cos[2*n*Pi]*Sin[2*m*Pi] - m*Cos[2*m*Pi]*Sin[2*n*Pi])/(m^2 - n^2) So as you see Mathematica returned an answer with (m^2 - n^2) in the denominator. The numerator also becomes 0 when you sent n and m to the same value, but because of the way Mathematica evaluates fractions it sees 1/0 and complains. In any case, it returns Indeterminate. However, Limit[r, n -> m] /. m -> 2 Pi which, I hope, answers your question. Andrzej Kozlowski Chiba, Japan http://www.mimuw.edu.pl/~akoz/