Carl Johan Malmsten (April 9, 1814, in Uddetorp, Skara County,
Sweden
Sweden, formally the Kingdom of Sweden, is a Nordic countries, Nordic country located on the Scandinavian Peninsula in Northern Europe. It borders Norway to the west and north, and Finland to the east. At , Sweden is the largest Nordic count ...
– February 11, 1886, in
Uppsala
Uppsala ( ; ; archaically spelled ''Upsala'') is the capital of Uppsala County and the List of urban areas in Sweden by population, fourth-largest city in Sweden, after Stockholm, Gothenburg, and Malmö. It had 177,074 inhabitants in 2019.
Loc ...
, 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 integral
In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
s 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
''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics.
According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journ ...
''.
Malmsten became
Docent
The term "docent" is derived from the Latin word , which is the third-person plural present active indicative of ('to teach, to lecture'). Becoming a docent is often referred to as habilitation or doctor of science and is an academic qualifi ...
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
The Royal Swedish Academy of Sciences () is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting nat ...
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.
However, he also greatly contributed in other branches of mathematics, but his results were undeservedly forgotten and many of them were erroneously attributed to other persons. Thus, it was comparatively recently that it was discovered by Iaroslav Blagouchine
[
]
PDF
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.
Many of these integrals were later rediscovered by various researchers, including Vardi,
[
]
Adamchik,
[V. Adamchik ''A class of logarithmic integrals.'' Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1-8, 1997.]
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
).
Malmsten derived the above formulae by making use of different series representations. At the same time, it has been shown that they can be also evaluated by
methods of contour integration,
by making use of the
Hurwitz Zeta function,
by employing
polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
s
and by using
L-functions.
More complicated forms of Malmsten's integrals appear in works of Adamchik
and Blagouchine
(more than 70 integrals). Below are several examples of such integrals
:
:
:
:
:
:
where ''m'' and ''n'' are positive integers such that ''m''<''n'', G is
Catalan's constant, ζ stands for the
Riemann zeta-function, Ψ is the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(z) = \frac\ln\Gamma(z) = \frac.
It is the first of the polygamma functions. This function is Monotonic function, strictly increasing a ...
, and Ψ
1 is the
trigamma function; see respectively eq. (43), (47) and (48) in Adamchik
for the first three integrals, and exercises no. 36-a, 36-b, 11-b and 13-b in Blagouchine
for the last four integrals respectively (the third integral being calculated in both works). It is curious that some of Malmsten's integrals lead to the
gamma- and
polygamma function
In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function:
:\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z).
Thus
:\psi^(z) ...
s of a complex argument, which are not often encountered in analysis. For instance, as shown by Iaroslav Blagouchine,
:
or,
:
see exercises 7-а and 37 respectively. By the way, Malmsten's integrals are also found to be closely connected to the
Stieltjes constants.
[
]
In 1842, Malmsten also evaluated several important logarithmic series, among which we can find these two series
:
and
:
already in 1749, but it was Malmsten who proved it (Euler only suggested this formula and verified it for several integer and semi-integer values of s). Curiously enough, the same formula for L(s) was unconsciously rediscovered by
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
.
Speaking about the Malmsten's contribution into the theory of zeta-functions, we can not fail to mentio
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
, it is often attributed to Almkvist and Meurman who derived it in 1990s.