Archive for May 13th, 2008

Lorenzo Mascheroni and the Euler-Mascheroni Constant

May 13, 2008 May 13th is Lorenzo Mascheroni’s birthday. According to the MacTutor History of Mathematics website, he was born in 1750 in Bergamo, Lombardo-Veneto. After training for the priesthood (and ordination at age 17), he taught rhetoric, physics and mathematics.

Mascheroni is primarily remembered for proving that every ruler and compass construction can in fact be done using compasses alone (a result that was found independently by Georg Mohr in 1672). Mascheroni’s name is also associated with one of the fundamental constants of mathematics….

The story of the Euler-Mascheroni Constant

In 1734, Leonhard Euler presented a paper to the Academy of Sciences in St. Petersburg on properties of harmonic progressions. In that paper, he notes that the harmonic series diverges, and indeed that $1 + \frac12 + \frac13 + \cdots + \frac1n$ approaches the function $C + \ln(1+n)$ (for a particular constant $C$) as $n$ goes to infinity. That constant $C$ is now known as the Euler-Mascheroni Constant, also known as gamma.

The relevant portion of Euler’s paper goes as follows:

Step 1. We’ll define a function $s(n) = 1 + \frac12 + \frac13 + \cdots + \frac1{n}$, and note that if $n$ increases by 1, then $s(n)$ will increase by $\frac1{n+1}$. From this, Euler concludes that as $n$ goes to infinity the function $s(n)$ satisfies the differential equation $\frac{ds}{dn} = \frac1{n+1}$, and thus $1+\frac12+\frac13+\cdots+\frac1n = C + \ln(1+n)$ for some constant $C$.

Step 2. Since (via Taylor Series) we know that $\ln(1+\frac1k) = \frac1k - \frac12 (\frac1k)^2 + \frac13 (\frac1k)^3 - \cdots$, if we solve for $\frac1k$ we find that $\frac1k = \ln( \frac{k+1}{k} ) + \frac{1}{2k^2} - \frac{1}{3k^3} + \frac{1}{4k^4} - \cdots$

Step 3. We will substitute the identity found in Step 2 for $\frac1k$ for each term of the harmonic series in the identity from Step 1. This gives $1 = \ln 2 + \frac12 - \frac13 + \frac14 - \frac15 + \cdots$ $\frac12 = \ln\frac32 + \frac1{2\cdot4} - \frac1{3\cdot8} + \frac1{4\cdot16} - \frac1{5\cdot32} + \cdots$ $\frac13 = \ln\frac43 + \frac1{2\cdot9} - \frac1{3\cdot27} + \frac1{4\cdot81} - \frac1{5\cdot243} + \cdots$ $\frac14 = \ln\frac54 + \frac1{2\cdot16} - \frac1{3\cdot64} + \frac1{4\cdot256} - \frac1{5\cdot1024} + \cdots$

etcetera, concluding with $\frac1n = \ln\frac{n+1}n + \frac1{2\cdot n^2} - \frac1{3\cdot n^3} + \frac1{4\cdot n^4} - \frac1{5\cdot n^5} + \cdots$

Adding in columns, and using the fact that $\ln(2) + \ln(\frac32) + \ln(\frac43) + \cdots + \ln(\frac{n+1}{n}) = \ln(2 \cdot \frac32 \cdot \frac43 \cdots \frac{n+1}{n}) = \ln(n+1)$ we find that $1 + \frac12 + \frac13 + \cdots +\frac1n = \ln(n+1)$ $\qquad \mbox{.} \qquad\qquad\qquad\qquad\qquad + \frac12 (1 + \frac14 + \frac19 + \frac1{16} + \cdots )$ $\qquad \mbox{.} \qquad\qquad\qquad\qquad\qquad - \frac13 (1+\frac18 + \frac1{27} + \frac1{64} + \cdots )$ $\qquad\mbox{.} \qquad\qquad\qquad\qquad\qquad + \frac14 (1+\frac1{16} + \frac1{81} + \frac1{256} + \cdots)$ $\qquad\mbox{.} \qquad\qquad\qquad\qquad\qquad - \cdots$

Thus Euler has found a way to approximate $C$, using the alternating sum on the right hand side of this identity, expressed in terms of series whose values Euler knew approximations for (values of the zeta function). Euler does this calculation, and publishes the approximation $C \approx 0.577218$.

Euler returned to this constant throughout his career, and using an approach similar to that outlined above was able to compute its first sixteen decimal places.

In 1790, Lorenzo Mascheroni published a commentary on Euler’s integral calculus texts, in which he gave a 32 digit approximation to the constant. Mascheroni’s record was eclipsed in 1809 when Johann von Soldner redid the calculation to 40 decimal places, and got a different value in the 19th place. Much consternation ensued, until 1812 when Gauss and F. Nicolai evaluated $C$ to 40 decimal places, confirming von Soldner’s result. Throughout the subsequent decades, both estimates to $C$ remained in circulation, and a number of mathematicians undertook the task of recomputing its value.

Mascheroni, through his digit error, may very well have generated extra buzz for this constant. It is also Mascheroni who coined the name “gamma” for it, and henceforth gamma has been known as the Euler-Mascheroni constant.

In honor of Mascheroni’s 258th birthday, a little musical offering….