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 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 the Blagouchine's paper. 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, and so did many well-known internet resources such as Wolfram MathWorld site or OEIS Foundation site. 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. More complicated forms of Malmsten's integrals appear in works of Adamchik and Blagouchine. Below are several examples of such integrals where m and n are positive integers such that m<n, G - is the Catalan's constant, ζ - stands for the Riemann zeta-function, Ψ - is the digamma function, Ψ1 - is the trigamma function; see respectively eq., and in for the first three integrals, and exercises no. 36-a, 36-b, 11-b and 13-b in for the last four integrals respectively. It is curious that some of Malmsten's integrals lead to the gamma- and polygamma functions 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 The latter series was later rediscovered in a slightly different form by Ernst Kummer, who derived a similar expression in 1847. 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 notably contributed into the theory of zeta-function related series and integrals. In 1842 he proved following important functional relationship for the L-function as well as for the M-function where in both formulae 0Leonhard Euler already in 1749, but it was Malmsten who proved it. Curiously enough, the same formula for L was unconsciously rediscovered by Oscar Schlömilch in 1849. 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 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.
