Ramanujan J (2014) 35:21–110

DOI 10.1007/s11139-013-9528-5

Rediscovery of Malmsten’s integrals, their evaluation

by contour integration methods and some related

results

Iaroslav V. Blagouchine

Received: 18 November 2012 / Accepted: 5 October 2013 / Published online: 7 January 2014

© Springer Science+Business Media New York 2014

Abstract This article is devoted to a family of logarithmic integrals recently treated

in mathematical literature, as well as to some closely related results. First, it is shown

that the problem is much older than usually reported. In particular, the so-called

Vard i ’ s i n t egral, which is a particular case of the considered family of integrals, was

ﬁrst evaluated by Carl Malmsten and colleagues in 1842. Then, it is shown that un-

der some conditions, the contour integration method may be successfully used for

the evaluation of these integrals (they are called Malmsten’s integrals). Unlike most

modern methods, the proposed one does not require “heavy” special functions and is

based solely on the Euler’s Γ-function. A straightforward extension to an arctangent

family of integrals is treated as well. Some integrals containing polygamma functions

are also evaluated by a slight modiﬁcation of the proposed method. Malmsten’s inte-

grals usually depend on several parameters including discrete ones. It is shown that

Malmsten’s integrals of a discrete real parameter may be represented by a kind of

ﬁnite Fourier series whose coefﬁcients are given in terms of the Γ-function and its

logarithmic derivatives. By studying such orthogonal expansions, several interesting

theorems concerning the values of the Γ-function at rational arguments are proven.

In contrast, Malmsten’s integrals of a continuous complex parameter are found to be

connected with the generalized Stieltjes constants. This connection reveals to be use-

ful for the determination of the ﬁrst generalized Stieltjes constant at seven rational

arguments in the range (0,1)by means of elementary functions, the Euler’s con-

stant γ, the ﬁrst Stieltjes constant γ1and the Γ-function. However, it is not known

if any ﬁrst generalized Stieltjes constant at rational argument may be expressed in

the same way. Useful in this regard, the multiplication theorem, the recurrence rela-

tionship and the reﬂection formula for the Stieltjes constants are provided as well.

A part of the manuscript is devoted to certain logarithmic and trigonometric series

I.V. Blagouchine (B)

University of Toulon, La Valette du Var (Toulon), France

e-mail: iaroslav.blagouchine@univ-tln.fr

22 I.V. Blagouchine

related to Malmsten’s integrals. It is shown that comparatively simple logarithmico–

trigonometric series may be evaluated either via the Γ-function and its logarithmic

derivatives, or via the derivatives of the Hurwitz ζ-function, or via the antiderivative

of the ﬁrst generalized Stieltjes constant. In passing, it is found that the authorship of

the Fourier series expansion for the logarithm of the Γ-function is attributed to Ernst

Kummer erroneously: Malmsten and colleagues derived this expansion already in

1842, while Kummer obtained it only in 1847. Interestingly, a similar Fourier series

with the cosine instead of the sine leads to the second-order derivatives of the Hur-

witz ζ-function and to the antiderivatives of the ﬁrst generalized Stieltjes constant.

Finally, several errors and misprints related to logarithmic and arctangent integrals

were found in the famous Gradshteyn & Ryzhik’s table of integrals as well as in the

Prudnikov et al. tables.

Keywords Logarithmic integrals ·Logarithmic series ·Theory of functions of a

complex variable ·Contour integration ·Rediscoveries ·Malmsten ·Var d i ·Number

theory ·Gamma function ·Zeta function ·Rational arguments ·Special constants ·

Generalized Euler’s constants ·Stieltjes constants ·Otrhogonal expansions

Mathematics Subject Classiﬁcation 33B15 ·30-02 ·30D10 ·30D30 ·11-02 ·

11M35 ·11M06 ·01A55 ·97I30 ·97I80

1 Introduction

1.1 Introductory remarks and history of the problem

In an article which appeared in the American Mathematical Monthly at the end of

1980s, Vardi [67] treats several interesting logarithmic integrals found in Gradshteyn

and Ryzhik’s tables [28]. His exposition begins with the integrals

π/2

ˆ

π/4

lnln tg xdx =

1

ˆ

0

lnln 1

x

1+x2dx =∞

ˆ

1

lnln x

1+x2dx =1

2

∞

ˆ

0

lnx

chxdx

=π

2lnΓ(3/4)

Γ(1/4)√2π,(1)

which can be deduced one from another by a simple change of variable (these formu-

las are given in no. 4.229-7, 4.325-4 and 4.371-1 of [28]). Although the ﬁrst results

for such a kind of integrals may be found in the mathematical literature of the 19th

century (e.g., in famous tables [62]), they continue to attract the attention of modern

researchers and their evaluation still remains interesting and challenging. Vardi’s pa-

per [67] generated a new wave of interest to such logarithmic integrals and numerous

works on the subject, including very recent ones, appeared since [67], [2], [13], [7],

[47], [46], [68], [4], [8], [44]. On the other hand, since the subject is very old, it is

hard to avoid rediscoveries. For the computation of the above mentioned integral (1),

modern authors [2], [13, p. 237], [7, p. 160], [46], [4], [44], send the reader to the

Malmsten’s integrals and their evaluation by contour integration 23

Fig. 1 A fragment of p. 12 from the Malmsten et al.’s dissertation [40]

Vardi’s paper [67].1Vardi, failing to identify the author of formula (1) and failing

to locate its proof, proposed a method of proof based essentially on the use of the

Dirichlet L-function. However, formula (1), in all four forms, was already known to

David Bierens de Haan [62, Table 308-28, 148-1, 260-1], and if we go to a deeper

exploration of this question, we ﬁnd that integral (1) was ﬁrst evaluated by Carl Jo-

han Malmsten2and colleagues in 1842 in a dissertation written in Latin [40,p.12],

see Fig. 1. A part of this dissertation was later republished in the famous Journal für

die reine und angewandte Mathematik6[41], see, e.g., p. 7 for integral (1). Moreover,

two other logarithmic integrals,

1

ˆ

0

lnln 1

x

1+x+x2dx =∞

ˆ

1

lnln x

1+x+x2dx =π

√3lnΓ(2/3)

Γ(1/3)

3

√2π(2a)

and

1

ˆ

0

lnln 1

x

1−x+x2dx =∞

ˆ

1

ln ln x

1−x+x2dx =2π

√3ln6

√32π5

Γ(1/6)(2b)

mentioned in [2,67], [13, p. 238], [4,70], were also ﬁrst evaluated by Malmsten

et al. [40, pp. 12 and 43] and [41, formulas (12) and (72)].3Malmsten and his col-

leagues evaluated many other beautiful logarithmic integrals4and series as well, but

1Only Bassett [8], an undergraduate student, remarked that solutions for integrals (1)and(2a,2b)aremuch

older than Vardi’s paper [67].

2Carl Johan Malmsten, written also Karl Johan Malmsten (born April 9, 1814 in Uddetorp, died Febru-

ary 11, 1886 in Uppsala), was a Swedish mathematician and politician. He 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 (Kungliga Vetenskaps–akademien) in 1844. He was also a minister without

portfolio in 1859–1866 and Governor of Skaraborg County in 1866–1879. For further information, see

[36, vol. 17, pp. 657–658].

3Both results are presented here in the original form, as they appear in the given sources. In fact, both

formulas may be further simpliﬁed and written in terms of Γ(1/3)only, see (45)and(44), respectively

(see also exercise no. 32 where both integrals appear in a more general form). Surprisingly, the latter fact

escaped the attention of Malmsten, of his colleagues and of many other researchers. Moreover, Vardi [67,

p. 313] even wrote that “in (2a) the number 3 plays the ‘key role’ and in (2b) 6 is the ‘magic number”’.

A more detailed criticism of the latter statement is given in exercise no. 30.

4Many of which were independently evaluated in [2]andin[44].

24 I.V. Blagouchine

unfortunately, none of the above-mentioned contemporary authors mentioned them.

Moreover, the latter even named integral (1) after Vardi (they call it Vardi’s i n t egral),

and so did many well-known internet resources such as Wolfram MathWorld site [70]

or OEIS Foundation site [57].

On the other hand, it is understandable that, sometimes, it can be quite difﬁcult

to ﬁnd the original source of a formula, especially because of the oldness of the

result, because the chain of references may be too long and confusing, and because it

could be published in many different languages. For example, Gradshteyn and Ryzhik

wrote originally in Russian (their book [28] is a translation from Russian), Bierens de

Haan published usually in Dutch or in French, Malmsten wrote in Swedish, French

and Latin, and we now use mostly English. As the reference for (1), Gradshteyn and

Ryzhik [28] as well as Vardi [67], cite the famous Bierens de Haan’s tables [62]. In

the latter, on the p. 207, Table 148, the reference for the integral (1) is given as “(IV,

265)”.5This means that this result comes from the 4th volume of the Memoirs of the

Royal Academy of Sciences of Amsterdam, which is entirely composed of the Tables

d’intégrales déﬁnies by Bierens de Haan [61], and which is an old version of the well-

known Nouvelles tables d’intégrales déﬁnies [62]. The old version [61] is essentially

the same as the new one [62], except that it provides original sources (the new version

[62] contains much less misprints and errors, but original references given in the

old edition were removed). Thus, we may ﬁnd in the old edition [61, pp. 264–265,

Table 191-1] that integral (1) was evaluated by Malmsten in the work referenced as

“Cr. 38. 1.” This is often the most difﬁcult part of the work, to understand what an old

abbreviation may stand for. Bierens de Haan does not explain it, and unfortunately,

neither do modern dictionaries nor encyclopedia. After several hours of search, we

ﬁnally found that “Cr.” stands for “Crelle’s Journal”, which is a jargon name for

the Journal für die reine und angewandte Mathematik.6The number 38 stands for

the volume’s number, and 1 is not the issue’s number but the number of the page

from which the manuscript starts. Furthermore, a deeper study of Malmsten’s works,

see e.g. [63, p. 31], shows that this article is a concise and updated version of the

collective dissertation [40] which was presented at the Uppsala University in April–

June 1842.7Therefore, taking into account the undoubted Malmsten and colleagues’

priority in the evaluation of the logarithmic integrals of the type (1) and (2a), (2b),

we think that integral (1) should be called Malmsten’s integral rather than Vard i ’s

integral. Throughout the manuscript, integrals of kind (1) and (2a), (2b) are called

Malmsten’s integrals.

The aim of the present work is multifold; accordingly, the article is divided in three

parts. In the ﬁrst part (Sect. 2), we present Malmsten’s original proof that Vardi and

other modern researchers missed. The presentation of this proof may be of interest

5Another frequently encountered notation for references in Bierens de Haan’s tables [61]and[62]—which

may be not easy to understand for English-speaking readers—is “V. T.” which stands for voir tableau,

i.e. “see table” in English.

6English translation: “Journal for Pure and Applied Mathematics”.

7However, we were surprised to see that Malmsten in the article [41] did not even mention the afore-

mentioned dissertation [40]. In fact, integrals (1)and(2a), (2b) were very probably derived by one of his

students or colleagues, but now it is almost impossible to know who exactly did it.

Malmsten’s integrals and their evaluation by contour integration 25

for a large audience of readers for multiple reasons. First, it may be quite difﬁcult

to ﬁnd references [40] and [41], as well as cited works. Second, the manuscript was

written in Latin, and references are in French. Latin, being discarded from the study

program of most mathematical faculties, may be difﬁcult to understand for many

researchers. Third, Malmsten does not make use of special functions other than Γ-

function; instead, he smartly employs elementary transformations, so that his proof

may be understood even by a ﬁrst-year student. Fourth, the work [41] contains numer-

ous misprints in formulas and a proper presentation might be quite useful as well. At

the end of the presentation, we brieﬂy discuss further Malmsten et al.’s contributions,

such as, for example, the Fourier series expansion for the logarithm of the Γ-function

(obtained 5 years before Kummer) or the derivation of the reﬂection formula for two

series closely related to ζ-functions (obtained 17 years before the famous Riemann’s

functional relationship for the ζ-function). Also, connections between the logarithm

of the Γ-function and the digamma function, written by Malmsten as a kind of dis-

crete cosine transform, are interesting and provide some further ideas that we later

re-used in Sect. 4.5. At the end of this part, we remark that several widely known

tables of integrals, such as Gradshteyn and Ryzhik’s tables [28], Bierens de Haan’s

tables [61,62], and probably, Prudnikov et al.’s tables [53], borrowed a large amount

of Malmsten’s results, butin most of them, original references to Malmsten were lost.

Moreover, some of these results appear with misprints.

In the second part of the manuscript (Sect. 3), we introduce a family of logarith-

mic integrals of which integral (1) and many other Malmsten’s integrals are simple

particular cases. We propose an alternative method for the analytical evaluation of

such a kind of integral. Unlike most modern methods, the proposed one does not

require “heavy” special functions and is based on the methods of contour integra-

tion. A non-exhaustive condition under which considered family of integrals may be

always expressed in terms of the Γ-function is provided. A straightforward exten-

sion to an arctangent family of integrals is treated as well. At the end of this part,

we consider in detail examples of application of the proposed method to four most

frequently encountered Malmsten’s integrals.

The third part of this work (Sect. 4) is designed as a collection of original exercises

containing new formulas and theorems, which can be derived directly or indirectly by

the proposed method. The exercises and theorems have been grouped thematically:

•Logarithmic Malmsten’s integrals containing hyperbolic functions and some

closely related results are treated in Sect. 4.1. In particular, integrals, which can be

evaluated by the direct application of the proposed method, are given in Sect. 4.1.1.

These may be roughly divided in two parts: relatively simple Malmsten’s integrals

containing two or three parameters (e.g. exercises no. 1,2,4,5,6-a, 7,8,17,

etc.) and complete Malmsten’s integrals depending on three or more parameters,

including discrete ones (e.g. exercises no. 3,6-b,c,d, 9,13,11,14). Simple Malm-

sten’s integrals, some of which are evaluated up to order 20,18 often lead to various

special constants such as Euler’s constant γ,Γ(1/3),Γ(1/4),Γ(1/π), Catalan’s

constant G, Apéry’s constant ζ(3)and others. As regards complete Malmsten’s

integrals, whose evaluation is carried out up to order 4, it is found that such inte-

grals, when depending on a discrete real parameter, may be represented by a kind

of ﬁnite Fourier series whose coefﬁcients are given in terms of the Γ-function and

26 I.V. Blagouchine

its logarithmic derivatives. In contrast, when the considered discrete real parameter

becomes continuous and complex, such integrals may be expressed by means of

the ﬁrst generalized Stieltjes constants (such exercises are placed in Sect. 4.5).

•The results closely related to logarithmic Malmsten’s integral are placed in

Sect. 4.1.2. These include the evaluation of integrals similar to Malmsten’s ones

and that of certain closely connected series. Most of these series are logarithmico–

trigonometric and may be evaluated either via the Γ-function and its logarithmic

derivatives, or via the derivatives of the Hurwitz ζ-function, or via the antideriva-

tive of the ﬁrst generalized Stieltjes constant (conversely, such series may be re-

garded as Fourier series expansions of the above-mentioned functions).

•In Sect. 4.2, we treat ln ln-integrals, most of which are obtained by a simple change

of variable of integrals from Sect. 4.1. In the same section, we also show that

Vardi’s hypothesis about the relationship between the argument of the Γ-function

and the degree in which the poles of the corresponding integrand are the roots of

unity is not true in general (exercise no. 30).

•Section 4.3 is devoted to arctangent integrals. Similarly to logarithmic integrals,

arctangent integrals can be roughly classiﬁed into several categories: compara-

tively simple, complete and closely related (such as, e.g., exercise no. 40 where

an analog of the second Binet’s formula for the logarithm of the Γ-function is

derived).

•In Sect. 4.4, we show that some slight modiﬁcations of the method developed in

Sect. 3may be quite fruitful for the evaluation of certain integrals containing log-

arithm of the Γ-function and the polygamma functions.

•Lastly, in Sect. 4.5 we put exercises and theorems related to the values of the Γ-

function at rational arguments and to the Stieltjes constants. Mostly, these results

are deduced from precedent exercises. For instance, by means of ﬁnite orthogonal

representations obtained for the Malmsten’ integrals in Sect. 4.1, we prove several

interesting theorems concerning the logarithm of the Γ-function at rational argu-

ments, including some variants of Parseval’s theorem (exercises no. 58–62). By the

way, with the help of the same technique one can derive similar theorems implying

polygamma functions. In the second part of Sect. 4.5, we show that some com-

plete Malmsten’s integrals, which were previously evaluated in Sect. 4.1, may be

also expressed by means of the ﬁrst generalized Stieltjes constants. This connec-

tion between Malmsten’s integrals of a real discrete and of a continuous complex

parameters is not only interesting in itself, but also permits evaluation of the ﬁrst

generalized Stieltjes constant γ1(p) at p=1

2,1

3,1

4,1

6,2

3,3

4,5

6by means of el-

ementary functions, the Euler’s constant γ, the ﬁrst Stieltjes constant γ1and the

Γ-function (see exercise no. 64). However, it is still unknown if any ﬁrst gener-

alized Stieltjes constant at rational argument may be expressed in the same way

(from this point of view, the evaluation of γ1(1/5)could be of special interest). In

this framework, we also discovered that the sum of the ﬁrst generalized Stieltjes

constant γ1(p),p∈(0,1), with its reﬂected version γ1(1−p) may be expressed,

at least for seven different rational values of p, in terms of elementary functions,

the Euler’s constant γand the ﬁrst Stieltjes constant γ1. At the same time, it is

not known if other sums γ1(p) +γ1(1−p) share the same property. An alternative

evaluation of integrals from exercises no. 65–66 could probably provide some light

on this problem.

Malmsten’s integrals and their evaluation by contour integration 27

Finally, answers for all exercises were carefully veriﬁed numerically with Maple

12 (except exercises with Stieltjes constants which were veriﬁed with Wolfram Al-

pha Pro). By default, if nothing is explicitly said, the presented result coincides with

the numerical one. Actually, only in few cases Maple 12 fails to correctly evaluate

integrals. For instance, it fails both numerically and symbolically to evaluate the ﬁrst

integral on the left in (1). Maple 12.0 gives (−πln2 +iπ2)/4≈0.544 +i2.467,

while it is clear that this integral has no imaginary part at all, and the real one is nei-

ther correctly evaluated. By the way, authors of [44] also reported incorrect numerical

and symbolical evaluation of this integral by Mathematica 6.0. However, unlike [44],

we will not specify wherever Maple 12 is able or unable to evaluate integrals analyt-

ically, because in almost all cases Maple 12 was unable to do it.

1.2 Notations

Throughout the manuscript, following abbreviated notations are used: γ=

0.5772156649... for the Euler’s constant, γnfor the nth Stieltjes constant, γn(p) for

the nth generalized Stieltjes constant at point p,8G=0.9159655941 ... for Cata-

lan’s constant, xfor the integer part of x,tgzfor the tangent of z,ctgzfor the cotan-

gent of z,chzfor the hyperbolic cosine of z,shzfor the hyperbolic sine of z,thzfor

the hyperbolic tangent of z,cthzfor the hyperbolic cotangent of z.9In order to avoid

any confusion between compositional inverse and multiplicative inverse, inverse

trigonometric and hyperbolic functions are denoted as arccos, arcsin, arctg,... and

not as cos−1,sin

−1,tg

−1,.... We write Γ(z),Ψ(z),Ψ

1(z), Ψ2(z), Ψ3(z), . . . , Ψn(z)

to denote, respectively, gamma, digamma, trigamma, tetragamma, pentagamma,...,

(n −2)th polygamma functions of argument z. The Riemann ζ-function, the η-

function (known also as the alternating Riemann ζ-function) and the Hurwitz ζ-

function are, respectively, deﬁned as

ζ(s)=∞

n=1

1

ns,η(s)=∞

n=1

(−1)n+1

ns,ζ(s,v)=∞

n=0

1

(n +v)s,

v=0,−1,−2,..., with Re s>1fortheζ-functions and Res>0fortheη-function.

Where necessary, these deﬁnitions may be extended to other domains by the principle

of analytic continuation. For example, one of the most known analytic continuations

for the Hurwitz ζ-function is the so-called Hermite representation

ζ(s,v)=v1−s

s−1+v−s

2+2∞

ˆ

0

sinsarctg x

v

(e2πx −1)v2+x2s/2dx , Re v>0,(3)

which extends ζ(s,v) to the entire complex plane except at s=1, see e.g. [9,vol.I,

p. 26, Eq. 1.10(7)]. Note also that the η-function may be easily reduced to the Rie-

8We remark, in passing, that by convention γn≡γn(1)for any natural n.

9Most of these notations come from Latin, e.g “ch” stands for cosinus hyperbolicus, “sh” stands for sinus

hyperbolicus,etc.

28 I.V. Blagouchine

mann ζ-function η(s) =(1−21−s)ζ(s), while the Hurwitz ζ-function is an indepen-

dent transcendent (except some particular values). Moreover, the alternating Hurwitz

ζ-function η(s,v) may be similarly reduced to the ordinary Hurwitz ζ-function

η(s,v) =∞

n=0

(−1)n

(n +v)s=lim

z→s21−zζz, v

2−ζ(z,v),(4)

v= 0,−1,−2,...,Res>0. Rezand Im zdenote, respectively, real and imaginary

parts of z. Natural numbers are deﬁned in a traditional way as a set of positive inte-

gers, which is denoted by N. Kronecker symbol of arguments land kis denoted by

δl,k. Letter iis never used as index and is √−1. Complex integration over region A

Im zBmeans that the complex line integral is taken around an inﬁnitely long hor-

izontal strip delimited by inequality AIm zB, where (A, B ) ∈R2(i.e. the inte-

gration contour is a rectangle with vertices at [R+iA,R+iB,−R+iB,−R+iA]

with R→∞. The notation resz=af(z) stands for the residue of the function f(z) at

the point z=a. Other notations are standard. Finally, we remark that the references

to the formulas are given between parentheses “( )”, those to the number of exercise

from Sect. 4are preceded by “no.”; the bibliographic references are given in square

brackets “[ ]”.

2 Malmsten’s method and its results

2.1 Malmsten’s original proof of the integral formula (1)

The proof is presented in a way closest to the original [41]; we have only replaced the

old notations by the new ones, as well as correcting numerous misprints in formulas

(by the way, [40] contains much less misprints). In the effort to make it more accessi-

ble for the readers, several modern references were also added, but, of course, the old

ones are also left. These references are marked with an ∗(only in this subsection).

Malmsten begins with the elementary integral

∞

ˆ

0

e−xz sin vz dz =v

x2+v2,x>0,v>0,

which, being integrated10 over vfrom 0 to u, becomes

2∞

ˆ

0

(1−cosuz) e−xz

zdz =lnx2+u2−2lnx. (5)

10This operation is permitted because the considered improper integral is uniformly convergent with re-

spect to v[34, p. 44, § 1.12]*, [58, vol. II, pp. 262–269]*, [10, pp. 175–179]*.

Malmsten’s integrals and their evaluation by contour integration 29

The latter logarithm is then replaced by one of Frullani’s integrals [34, pp. 406–407,

§ 12.16]∗,[50]∗,[24, p. 455]∗,

lnx=∞

ˆ

0

e−z−e−xz

zdz, (6)

which yields

lnx2+u2=2∞

ˆ

0

e−z−e−xz cosuz

zdz. (7)

Multiply the last equality by sh au

sh πu, parameter abeing in the range (−π, +π), and

then integrate it over all values of ufrom 0 to ∞

∞

ˆ

0

sh au

shπu lnx2+u2du =2∞

ˆ

0∞

ˆ

0

sh au

sh πue−z−e−xz cos uzdudz

z.(8)

In virtue of formulas (b) and (a) from [64, vol. II, p. 186]11

∞

ˆ

0

sh au

sh πu du =1

2tg a

2and ∞

ˆ

0