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Carl Johan Malmsten (April 9, 1814 in Uddetorp, Skara County, Sweden – February 11, 1886 in Uppsala, Sweden) was a Swedish mathematician. 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. 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 Blagouchine that Malmsten was first who evaluated several important logarithmic integrals and series, 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

$\underset{0}{\overset{1}{\int }}\frac{\ln \ln \frac{1}{x}}{1+x^2}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln \ln x}{1+x^2}dx=\frac{\pi }{2}\ln \left\{\frac{\mathrm{\Gamma }(3/4)}{\mathrm{\Gamma }(1/4)}\sqrt{2\pi }\right\}$

$\underset{0}{\overset{1}{\int }}\frac{\ln \ln \frac{1}{x}}{(1+x{)}^2}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln \ln x}{(1+x{)}^2}dx=\frac{1}{2}(\ln \pi -\ln 2-\gamma ),$

$\underset{0}{\overset{1}{\int }}\frac{\ln \ln \frac{1}{x}}{1-x+x^2}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln \ln x}{1-x+x^2}dx=\frac{2\pi }{\sqrt{3}}\ln \left\{\frac{\sqrt[6]{32{\pi }^5}}{\mathrm{\Gamma }(1/6)}\right\}$

$\underset{0}{\overset{1}{\int }}\frac{\ln \ln \frac{1}{x}}{1+x+x^2}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln \ln x}{1+x+x^2}dx=\frac{\pi }{\sqrt{3}}\ln \left\{\frac{\mathrm{\Gamma }(2/3)}{\mathrm{\Gamma }(1/3)}\sqrt[3]{2\pi }\right\}$

$\underset{0}{\overset{1}{\int }}\frac{\ln \ln \frac{1}{x}}{1+2x\cos \phi +x^2}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\ln \ln x}{1+2x\cos \phi +x^2}dx=\frac{\pi }{2\sin \phi }\ln \left\{\frac{(2\pi {)}^{\frac{\phi }{\pi }}\mathrm{\Gamma }(\frac{1}{2}+\frac{\phi }{2\pi })}{\mathrm{\Gamma }(\frac{1}{2}-\frac{\phi }{2\pi })}\right\},\left|\mathrm{\Re }\phi \right|<\pi .$

$\underset{0}{\overset{1}{\int }}\frac{x^{n-2}\ln \ln \frac{1}{x}}{1-x^2+x^4-\cdots +x^{2n-2}}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{x^{n-2}\ln \ln x}{1-x^2+x^4-\cdots +x^{2n-2}}dx=$

$=\frac{\pi }{2n}\sec \frac{\pi }{2n}\cdot \ln \pi +\frac{\pi }{n}\cdot \sum _{l=1}^{\frac{1}{2}(n-1)}(-1{)}^{l-1}\cos \frac{(2l-1)\pi }{2n}\cdot \ln \left\{\frac{\mathrm{\Gamma }(1-\frac{2l-1}{2n})}{\mathrm{\Gamma }\left(\frac{2l-1}{2n}\right)}\right\},n=3,5,7,\dots$

$\underset{0}{\overset{1}{\int }}\frac{x^{n-2}\ln \ln \frac{1}{x}}{1+x^2+x^4+\cdots +x^{2n-2}}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{x^{n-2}\ln \ln x}{1+x^2+x^4+\cdots +x^{2n-2}}dx=$

$=\{\begin{array}{l}\frac{\pi }{2n}\tan \frac{\pi }{2n}\ln 2\pi +\frac{\pi }{n}\sum _{l=1}^{n-1}(-1{)}^{l-1}\sin \frac{\pi l}{n}\cdot \ln \left\{\frac{\mathrm{\Gamma }(\frac{1}{2}+\frac{l}{2n})}{\mathrm{\Gamma }\left(\frac{l}{2n}\right)}\right\},n=2,4,6,\dots \\ \frac{\pi }{2n}\tan \frac{\pi }{2n}\ln \pi +\frac{\pi }{n}\sum _{l=1}^{\frac{1}{2}(n-1)}(-1{)}^{l-1}\sin \frac{\pi l}{n}\cdot \ln \left\{\frac{\mathrm{\Gamma }(1-\frac{l}{n})}{\mathrm{\Gamma }\left(\frac{l}{n}\right)}\right\},n=3,5,7,\dots \end{array}$

Many of these integrals were later rediscovered by various researchers, including Vardi, 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). Malmsten derived 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 polylogarithms and by using L-functions. By the way, Malmsten's integrals are also found to be connected with the Stieltjes constants,.

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

$\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{\ln (2n+1)}{2n+1}=\frac{\pi }{4}(\ln \pi -\gamma )-\pi \ln \mathrm{\Gamma }\left(\frac{3}{4}\right)$

and

$\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n-1}\frac{\sin an\cdot \ln n}{n}=\pi \ln \left\{\frac{{\pi }^{\frac{1}{2}-\frac{a}{2\pi }}}{\mathrm{\Gamma }(\frac{1}{2}+\frac{a}{2\pi })}\right\}-\frac{a}{2}(\gamma +\ln 2)-\frac{\pi }{2}\ln \cos \frac{a}{2},-\pi <a<\pi .$

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

$\frac{1}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{\sin 2\pi nx\cdot \ln n}{n}=\ln \mathrm{\Gamma }(x)-\frac{1}{2}\ln \pi +\frac{1}{2}\ln \sin \pi x-(\gamma +\ln 2\pi )(1-2x),0<x<1,$

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, although Malmsten derived it 5 years before Kummer.

Malsmten also contributed into the theory of zeta-function related series and integrals. In 1842 he proved following important functional relationship for the L-function

$L(s)\equiv \sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1{)}^s}L(1-s)=L(s)\mathrm{\Gamma }(s)2^s{\pi }^{-s}\sin \frac{\pi s}{2},$

as well as for the M-function

$M(s)\equiv \frac{2}{\sqrt{3}}\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n^s}\sin \frac{\pi n}{3}M(1-s)=\frac{2}{\sqrt{3}}M(s)\mathrm{\Gamma }(s)3^s(2\pi {)}^{-s}\sin \frac{\pi s}{2},$

where in both formulae 0<s<1. First of these formulae was proposed by Leonhard Euler already in 1749, but it was Malmsten who proved it (Euler only suggested this formula and verified it for several integer and demi-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.

References

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