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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
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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. :\int_0^1 \!\frac\,dx\, = \,\int_1^\infty \!\frac\,dx\, = \,\frac\ln\left\ :\int_0^\frac\,dx = \int\limits_1^ \!\frac\,dx = \frac \bigl(\ln\pi - \ln2 -\gamma\bigr), :\int\limits_0^\! \frac\,dx = \int_1^\! \frac\,dx = \frac\ln \biggl\ :\int\limits_0^\! \frac\,dx = \int\limits_1^\! \frac\,dx = \frac\ln \biggl\ : \int\limits_0^1 \!\frac \,dx \,=\int\limits_1^\!\frac\,dx = \frac\ln \left\ , \qquad -\pi<\varphi<\pi :\int\limits_0^ \!\frac\,dx\, = \int\limits_1^\!\frac\,dx = :\quad =\, \frac\sec\frac\!\cdot\ln \pi + \frac\cdot\!\!\!\!\!\!\sum_^ \!\!\!\! (-1)^ \cos\frac\cdot \ln\left\ ,\qquad n=3,5,7,\ldots :\int\limits_0^ \!\frac\,dx \, = \int\limits_1^\!\frac\,dx = : \qquad =\begin \displaystyle \frac\tan\frac\ln2\pi + \frac\sum_^ (-1)^ \sin\frac\cdot \ln\left\ ,\quad n=2,4,6,\ldots \\ 0mm\displaystyle \frac\tan\frac\ln\pi + \frac\!\!\!\!\! \sum_^ \!\!\!\! (-1)^ \sin\frac\cdot \ln\left\ ,\qquad n=3,5,7,\ldots \end 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 : \int\limits_0^1 \frac\,dx =\int\limits_1^\infty \frac\,dx =\frac\ln\frac-\frac \left\ : \int\limits_0^1 \!\frac\,dx =\int\limits_1^\infty \!\frac\,dx =-\frac-\frac\ln\frac + \frac \left\ : \int\limits_0^1 \frac\, dx= \int\limits_1^\infty \frac\, dx = \frac : \int\limits_0^1 \frac\, dx = \int\limits_1^\infty \frac\, dx = \frac : \begin \displaystyle \int\limits_0^1 \frac\, dx = \int\limits_1^\infty \frac\, dx = \!\!\!&\displaystyle \frac \sum_^ \sin\dfrac\cdot\ln\Gamma\!\left(\!\frac\!\right) - \,\frac\cot\frac\cdot\ln\pi n \\ mm &\displaystyle - \,\frac\ln\!\left(\!\frac\sin\frac\!\right) - \,\frac \end : \begin \displaystyle \int\limits_0^1 \frac\, dx = \int\limits_1^\infty \frac\, dx = -\frac\!\sum_^ \! (-1)^l \cos\dfrac \cdot\ln\Gamma\!\left(\!\frac\right) \\ mm\displaystyle \,\, +\frac\! \sum_^ \! (-1)^l \sin\dfrac\cdot \Psi\!\left(\!\frac\right) -\frac \!\sum_^(-1)^l \cos\dfrac\cdot \Psi_1\!\left(\!\frac\right) + \,\frac\sec\dfrac\cdot\ln2\pi n \end 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 Ψ1 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, : \int\limits_0^1 \!\frac\,dx =\int\limits_1^\!\frac\,dx = \frac \mathrm\!\left ln\Gamma\!\left(\!\frac-\frac\right)\!\right+\, \frac\ln\pi or, : \int\limits_^ \!\frac\,dx = \int\limits_^ \!\frac\,dx =-\frac \mathrm\!\left ln\Gamma\!\left(\!\frac\right) - \ln\Gamma\!\left(\!\frac-\frac\right)\!\right-\frac-\frac 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 : \sum_^(-1)^\frac \,=\,\frac\big(\ln\pi - \gamma) -\pi\ln\Gamma\left(\frac\right) and : \sum_^(-1)^ \frac \,=\,\pi\ln\left\ - \frac\big(\gamma+\ln2 \big) -\frac\ln\cos\frac\,, \qquad -\pi The latter series was later rediscovered in a slightly different form by
Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...
, who derived a similar expression : \frac\sum_^\frac = \ln\Gamma(x) - \frac\ln(2\pi) + \frac\ln(2\sin\pi x) - \frac(\gamma+\ln2\pi)(1-2x)\,, \qquad 0 in 1847 (strictly speaking, the Kummer's result is obtained from the Malmsten's one by putting a=π(2x-1)). Moreover, this series is even known in analysis as ''Kummer's series'' for the logarithm of the
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
, although Malmsten derived it 5 years before Kummer. Malsmten also notably contributed into the theory of zeta-function related series and integrals. In 1842 he proved following important functional relationship for the L-function :L(s)\equiv\sum_^\frac \qquad\qquad L(1-s)=L(s)\Gamma(s) 2^s \pi^\sin\frac, as well as for the M-function :M(s)\equiv\frac\sum_^\frac \sin\frac \qquad\qquad M(1-s)=\displaystyle\frac \, M(s)\Gamma(s) 3^s (2\pi)^\sin\frac, where in both formulae 0Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
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 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 mentio
the very recent discovery
of his authorship of the reflection formula for the first generalized Stieltjes constant at rational argument : \gamma_1 \biggl(\frac\biggr)- \gamma_1 \biggl(1-\frac \biggr) =2\pi\sum_^ \sin\frac \cdot\ln\Gamma \biggl(\frac \biggr) -\pi(\gamma+\ln2\pi n)\cot\frac 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.


References

{{DEFAULTSORT:Malmsten, Carl Johan 19th-century Swedish mathematicians Members of the Första kammaren Members of the Royal Swedish Academy of Sciences 1814 births 1886 deaths Members of the Göttingen Academy of Sciences and Humanities Members of the Royal Society of Sciences in Uppsala People from Skara Municipality