Talk:Euler-Maclaurin formula
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[edit] Motivation for the existence
I don't understand why Δ = eD − I, or even what I is (the integral used earlier in the article?). Can someone explain? Also, is there a reference for this argument (other than "Legendre")? Fredrik Johansson 20:05, 17 May 2006 (UTC)
- No, I is not that integral. I is the identity operator on functions, i.e.
- D is the differentiation operator, i.e.
- and Δ is the forward difference operator, i.e.
- Using the power series
- we have
- Therefore
- If f happens to be a polynomial function, then all but finitely many of these terms vanish, and a bit of algebra shows that this sum adds up to
- i.e. it adds up to
- To what extent this works when f is not a polynomial, is a more difficult question. Michael Hardy 22:58, 17 May 2006 (UTC)
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- Thanks, Michael. Fredrik Johansson 23:02, 17 May 2006 (UTC)
---I should add that the Differentiation operator acting on that space of polynomials is just an emulation of the differentiation operation but uses no calculus since is just defined as ::D[a0xn + a1xn − 1 + a2xn − 2 + ... + an − 1x + an] = n * a0xn − 1 + (n − 1) * a1xn − 2 + ... + 2 * an − 2x + an − 1 and isn't necessarily defined for an arbitrary function.
[edit] Some Formula
The article doesn't say exactly how to find the following Sn, here is the process. The idea is from Trapezium rule
Sn=Sum i^2 = n*(n+1)*(2n+1)/6
- from b^3-a^3=(b-a)*(b^2+ab+a^2) b=i, a=i-1,
- i^3-(i-1)^3=(i^2+i*(i-1)+(i-1)^2)=2*i^2-i+(i-1)^2
- Sum (i^3-(i-1)^3)=n^3=3Sn-n^2-n*(n+1)/2, you find Sn
Sn=Sum i^3 = n^2*(n+1)^2/4
- from b^4-a^4=(b^2-a^2)(b^2+a^2)=(b-a)(b+a)(b^2+a^2)
- b=i, a=i-1, i^4-(i-1)^4=4*i^3-6*i^2+4^i-1
- Sum (i^4-(i-1)^4)=n^4=4Sn-n*(n+1)*(2n+1)+2n*(n+1)-n, you find Sn
With the same method, we can find all others. Please help put the text in a nice format.
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- RESPONSE
- I'd be happy to help put that in a nice format, but it's a bit unclear. What are the sums ranging from, and what does "from" refer to? If you want to do it yourself, use the simple commands like
<math> and </math> \sum_{0}^{\infty} would give sum from 0 to infinity
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- Lavaka 19:34, 6 September 2006 (UTC)
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[edit] unclear variable
In the "Remainder Term" section, where did the variable N come from? I can only assume it's supposed to be p? Lavaka 19:34, 6 September 2006 (UTC)