Laplace's summation formula - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:03:33Z http://mathoverflow.net/feeds/question/43355 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43355/laplaces-summation-formula Laplace's summation formula Mike Spivey 2010-10-24T03:22:58Z 2010-11-20T16:06:58Z <p>I recently came across the following formula, which is apparently known as <em>Laplace's summation formula:</em></p> <p>$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{12} \left(\Delta f(b) - \Delta f(a)\right)$$ $$+ \frac{1}{24} \left( \Delta^2 f(b) - \Delta^2 f(a) \right) - \frac{19}{720} \left(\Delta^3 f(b) - \Delta^3 f(a) \right) + \cdots$$</p> <p>(Of course, the right-hand side isn't guaranteed to converge.) The coefficient on the term with $\Delta^{k-1}$ is $\frac{c_k}{k!}$, where $c_k$ is apparently called either a <em>Cauchy number of the first kind</em> or a <em>Bernoulli number of the second kind</em>. </p> <p>The formula looks to me like a finite calculus version of the <a href="http://en.wikipedia.org/wiki/Euler-Maclaurin_formula" rel="nofollow">Euler-Maclaurin summation formula</a>.</p> <p>I'm trying to find out more about Laplace's summation formula. However, the usual suspects (the arXiv, Wikipedia, MathWorld, Google) aren't turning up much. There was a little on MathSciNet, the most promising of which was a paper by Merlini, Sprugnoli, and Verri entitled "The Cauchy Numbers" (<em>Discrete Mathematics</em> 306(16): 1906-1920, 2006). The MathSciNet review says, "Application of the Laplace summation formula involving the harmonic numbers [is] also given." I've requested the paper through interlibrary loan, but it has not arrived yet.</p> <p>While I'm interested in the formula in general, I'm particularly interested in these two questions.</p> <ol> <li><p>What applications are there for the Laplace summation formula? (It seems like there ought to be a sufficient number of applications for it to deserve having Laplace's name attached to it. I suppose one could use it for asymptotic analysis, but I'm not sure what the advantage would be over Euler-Maclaurin.)</p></li> <li><p>What is the error bound on the formula when it is truncated after $n$ terms?</p></li> </ol> <p>I wasn't sure how to tag this; feel free to retag.</p> http://mathoverflow.net/questions/43355/laplaces-summation-formula/43361#43361 Answer by Gerry Myerson for Laplace's summation formula Gerry Myerson 2010-10-24T05:41:45Z 2010-10-24T05:41:45Z <p>Did you try the Online Encyclopedia of Integer Sequences? <a href="http://www.research.att.com/~njas/sequences/A006232" rel="nofollow">http://www.research.att.com/~njas/sequences/A006232</a> </p> <p>Perhaps some of the references there will get you where you want to go. </p> http://mathoverflow.net/questions/43355/laplaces-summation-formula/43518#43518 Answer by SandeepJ for Laplace's summation formula SandeepJ 2010-10-25T14:04:23Z 2010-10-25T14:04:23Z <blockquote> <p>However, the usual suspects (the arXiv, Wikipedia, MathWorld, Google) aren't turning up much.</p> </blockquote> <p>You forgot google books!</p> <p>There are references to the Laplace summation formula in two books. </p> <ol> <li><p>Page 248 in <em>The rise and development of the theory of series up to the early 1820s</em> by Giovanni Ferraro <a href="http://books.google.com/books?id=vLBJSmA9zgAC" rel="nofollow">http://books.google.com/books?id=vLBJSmA9zgAC</a></p></li> <li><p>Page 192 in <em>A history of numerical analysis from the 16th through the 19th century</em> by Herman Goldstine <a href="http://books.google.com/books?id=20csAQAAIAAJ" rel="nofollow">http://books.google.com/books?id=20csAQAAIAAJ</a></p></li> </ol> http://mathoverflow.net/questions/43355/laplaces-summation-formula/43967#43967 Answer by Anixx for Laplace's summation formula Anixx 2010-10-28T12:55:52Z 2010-10-28T13:17:33Z <p>You may be also interested in this formula for indefinite sum of $f(x)$:</p> <p>$$\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C$$</p> <p>where $(x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)}$ is a falling factorial.</p> http://mathoverflow.net/questions/43355/laplaces-summation-formula/46744#46744 Answer by Tyler Clark for Laplace's summation formula Tyler Clark 2010-11-20T16:06:58Z 2010-11-20T16:06:58Z <p>This is a bit late - I could be completely wrong, but I think the issue here is the domain being used.</p> <p>Laplace's summation formula should be used on the set of integers and will be used for calculations in discrete calculus. I believe that the Euler-Maclaurin summation formula is typically used on the reals though.</p> <p>I hope this helps.</p>