Reference request for a "well-known identity" in a paper of Shepp and Lloyd - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T00:57:27Z http://mathoverflow.net/feeds/question/19526 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19526/reference-request-for-a-well-known-identity-in-a-paper-of-shepp-and-lloyd Reference request for a "well-known identity" in a paper of Shepp and Lloyd John Jiang 2010-03-27T17:36:20Z 2010-03-28T22:24:16Z <p>I ran into a "well-known identity" on page 345 of Shepp and Lloyd's <a href="http://www.jstor.org/pss/1994483" rel="nofollow">On ordered cycle lengths in a random permutation</a>: $$\int_x^{\infty} \frac{\exp(-y)}y dy = \int_0^x \frac{1-\exp(-y)}y dy - \log x - \gamma,$$ where $\gamma$ is the Euler constant. I am clueless as to how it is derived. Any reference to the derivation of such formulae would suffice, but an explicit solution will also be appreciated.</p> http://mathoverflow.net/questions/19526/reference-request-for-a-well-known-identity-in-a-paper-of-shepp-and-lloyd/19539#19539 Answer by Ryan O'Donnell for Reference request for a "well-known identity" in a paper of Shepp and Lloyd Ryan O'Donnell 2010-03-27T19:27:04Z 2010-03-27T19:27:04Z <p>This identity appears on the Wikipedia page for the "exponential integral": <a href="http://en.wikipedia.org/wiki/Exponential_integral#Definition_by_Ein" rel="nofollow">http://en.wikipedia.org/wiki/Exponential_integral#Definition_by_Ein</a></p> <p>I imagine you can get it by integrating the Taylor series and playing around. Wikipedia, and several other places on the web, point to the book by Abramovitz and Stegun.</p> http://mathoverflow.net/questions/19526/reference-request-for-a-well-known-identity-in-a-paper-of-shepp-and-lloyd/19546#19546 Answer by Jacques Carette for Reference request for a "well-known identity" in a paper of Shepp and Lloyd Jacques Carette 2010-03-27T21:04:27Z 2010-03-28T02:32:34Z <p>You can apply WZ theory to such identities. In particular, both sides satisfy $$x*z''(x) + (x+1)z'(x)$$ Picking $x=1$ as the initial condition (since the DE is regular there, that helps), we see that both sides evaluate to $Ei(1,1)$ and their derivatives both evaluate to $-1/e$, so they are equal.</p> <p>I got that differential equation using Maple's <em>PDEtools[dpolyform]</em> function, which uses Groebner bases over differential polynomials to 'solve' this problem. All the rest is classical analysis (as in <a href="http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521588072" rel="nofollow">A course of modern analysis</a> by Whittaker and Watson, 1926 - which is unfortunately not material that is taught very much anymore, I certainly had to learn a lot of that 'on my own').</p> <p>[Edit: fixed an error in the evaluation of the derivative, I pasted in the wrong line]</p> http://mathoverflow.net/questions/19526/reference-request-for-a-well-known-identity-in-a-paper-of-shepp-and-lloyd/19646#19646 Answer by John Jiang for Reference request for a "well-known identity" in a paper of Shepp and Lloyd John Jiang 2010-03-28T18:42:36Z 2010-03-28T22:24:16Z <p>So one of the approaches to proving the equality in the question is via the following three steps: First differentiate both sides of the equation to see that they agree up to a constant. This reduces to showing the case of $x = 1$, for which $\log x = 0$.</p> <p>Next we apply integration by parts to get $$\int_1^{\infty} \exp(-y)/y dy - \int_0^1 \frac{1-\exp(-y)}{y}dy = \int_0^{\infty} \exp(-y) \log y dy$$</p> <p>Finally observe that $\Gamma'(1)$ equals the RHS, by differentiating under the integral sign, valid because things are decaying fast enough at infinity.</p> <p>So it remains to show $\Gamma'(1) =\gamma$. I saw a soft argument (i.e., without using infinite product) in the link scipp.ucsc.edu/~haber/ph116A/psifun_10.pdf This is re-exposed below:</p> <p>first we establish that for $\Psi(x) = \log \Gamma(x)$, $$\Psi'(x+1) = \Psi'(x) + 1/x$$ This is easy enough since we have we have the functional equation $\Gamma(x+1) = x\Gamma(x)$. Next using stirling approximation we get </p> <p>$$\Psi(x+1) = (x+1/2)\log x -x + 1/2 \log 2 \pi + O(1/x)$$ and then they differentiate this and claim that $O(1/x)' = O(1/x^2)$, which is clearly false (take $f(x) = 1/x cos(e^x)$). But I found in Wikipedia another formula that gives the precise error term in terms of an integral of the monotone function $arctan(1/x)$. So this is enough to establish $O(1/x^2)$ for the error term in the derivative of $\Psi$. So we get the asymptotics $\lim_{x \to \infty} \Psi'(x+1) = \log(x)$, from which we get $\Psi'(1) = \gamma$. Now notice $\Psi'(x) = \Gamma'(x)/ \Gamma(x)$, and $\Gamma(1) = 1$, so $\Gamma'(1) = \gamma$ also. </p>