Carl Johan Malmsten

Carl Johan Malmsten (April 9, 1814, in Uddetorp, Skara County, Sweden – February 11, 1886, in Uppsala, Sweden) was a Swedish mathematician and politician. He is notable for early research into the theory of functions of a complex variable, for the evaluation of several important logarithmic integrals and series, for his studies in the theory of Zeta-function related series and integrals, as well as for helping Mittag-Leffler start the journal Acta Mathematica. Malmsten became Docent in 1840, and then, Professor of mathematics at the Uppsala University in 1842. He was elected a member of the Royal Swedish Academy of Sciences in 1844. He was also a minister without portfolio in 1859–1866 and Governor of Skaraborg County in 1866–1879.

Main contributions Usually, Malmsten is known for his earlier works in complex analysis. that Malmsten was first who evaluated several important logarithmic integrals and series, which are closely related to the gamma- and zeta-functions, and among which we can find the so-called Vardi's integral and the Kummer's series for the logarithm of the Gamma function. In particular, in 1842 he evaluated following lnln-logarithmic integrals.

The details and an interesting historical analysis are given in Blagouchine's paper. Adamchik, Medina and Moll. Moreover, some authors even named the first of these integrals after Vardi, who re-evaluated it in 1988 (they call it Vardi's integral), and so did many well-known internet resources such as Wolfram MathWorld site or OEIS Foundation site (taking into account the undoubted Malmsten priority in the evaluation of such a kind of logarithmic integrals, it seems that the name Malmsten's integrals would be more appropriate for them

In 1842, Malmsten also evaluated several important logarithmic series, among which we can find these two series

and

The latter series was later rediscovered in a slightly different form by Ernst Kummer, who derived a similar expression

in 1847 but it was Malmsten who proved it (Euler only suggested this formula and verified it for several integer and semi-integer values of s). The same formula for L(s) was unconsciously rediscovered by Oscar Schlömilch in 1849 (proof provided only in 1858). Four years later, Malmsten derived several other similar reflection formulae, which turn out to be particular cases of the Hurwitz's functional equation.

Speaking about the Malmsten's contribution into the theory of zeta-functions, we can not fail to mention the very recent discovery of his authorship of the reflection formula for the first generalized Stieltjes constant at rational argument

where m and n are positive integers such that m<n. This identity was derived, albeit in a slightly different form, by Malmsten already in 1846 and has been also discovered independently several times by various authors. In particular, in the literature devoted to Stieltjes constants, it is often attributed to Almkvist and Meurman who derived it in 1990s.