Errata for Emil Artin's 'The Gamma Function'? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:24:58Z http://mathoverflow.net/feeds/question/20272 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20272/errata-for-emil-artins-the-gamma-function Errata for Emil Artin's 'The Gamma Function'? Zavosh 2010-04-04T02:26:15Z 2010-04-04T15:54:45Z <p>In the English translation of <em>The Gamma Function</em> by Emil Artin (1964 - Holt, Rinehart and Winston) there appears to be a mistake in the formula given for the gamma function on page 24:</p> <p>$$\Gamma(x) = \sqrt{2\pi}x^{x-1/2}e^{-x+\mu(x)}$$ $$\mu(x)=\sum_{n=0}^\infty(x+n+\frac{1}{2})\text{log}(1+\frac{1}{x+n})-1=\frac{\theta}{12x},\ \ \ \ \ 0 &lt; \theta &lt; 1$$</p> <p>and on page 22 where this is derived, it is noted that '$\theta$ is a number independent of $x$ between 0 and 1'.</p> <p>This sounds incorrect, as $\theta$ does depend on $x$, but since the wording is a little ambiguous it may just be an unclear translation. The original German might have meant that $0&lt; \theta(x) &lt; 1$ for any $x$. That the variable $x$ is suppressed from $\theta$ could be just confusing notation, or someone's misunderstanding (possibly mine.)</p> <p>The preface does mention that a (different) formula had to be corrected for the English reprint.</p> <p>I would like to know if there are mistakes in this book, and if so, whether they exist in the German edition. Is there an available list of errata? </p> http://mathoverflow.net/questions/20272/errata-for-emil-artins-the-gamma-function/20305#20305 Answer by Jim Humphreys for Errata for Emil Artin's 'The Gamma Function'? Jim Humphreys 2010-04-04T14:16:24Z 2010-04-04T15:54:45Z <p>It seems clear that <code>$\theta$</code> can indeed be chosen to be a number independent of <code>$x$</code> as stated, to get Stirling's formulas for the gamma function when <code>$x$</code> is <em>large</em>. The wording, at least in English, is not too helpful in this section. But I'm less clear about where in the formula on page 24 there is supposed to be a mistake. Here as in any mathematics book (especially a translation) one has to be wary about misprints or errors. Probably there is no publicly available list of errata for this small monograph published originally in 1931 in German and later republished in 1964 in an English translation by Michael Butler. This English version is included in the 2007 AMS softcover book <em>Exposition by Emil Artin: A Selection</em> edited by Michael Rosen. (There is an older 1965 book <em>The Collected Papers of Emil Artin</em> published by Addison-Wesley and edited by Lang &amp; Tate. This contains Artin's research papers, in the original German or English.) As Zavosh observes, the 1964 preface by Edwin Hewitt reprinted here does indicate one formula corrected in the translation: " ... a small error following formula (59) (this edition) was corrected..." However, the formula seems to be the one actually numbered (5.9). Caveat lector. </p>