Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results

Article (PDF Available)inThe Ramanujan Journal 35(1):21-110 · January 2014with 1,789 Reads
DOI: 10.1007/s11139-013-9528-5
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 Vardi’s integral, 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 under 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 integrals 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 useful 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 constant γ, the first Stieltjes constant γ 1 and 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 relationship 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 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 Hurwitz 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.
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
1x+x2dx =
ˆ
1
ln ln x
1x+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), 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. 5862). 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(1p) 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(1p) share the same property. An alternative
evaluation of integrals from exercises no. 6566 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 +2)/40.544 +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 cos1,sin
1,tg
1,.... We write Γ(z),Ψ(z)
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(s)=
n=1
(1)n+1
ns(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)=v1s
s1+vs
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 γnγn(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) =(121s(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
zs21zζz, v
2ζ(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
exz sin vz dz =v
x2+v2,x>0,v>0,
which, being integrated10 over vfrom 0 to u, becomes
2
ˆ
0
(1cosuz) exz
zdz =lnx2+u22lnx. (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
ezexz
zdz, (6)
which yields
lnx2+u2=2
ˆ
0
ezexz 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 πuezexz 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
sh au
sh πu cosuz d u =sin a
2(chz+cos a) ,
where π<a<πand −∞ <z<, expression (8) may be rewritten as follows:
ˆ
0
sh au
sh πu lnx2+u2du =
ˆ
0tg a
22exz sin a
1+2ezcosa+e2zezdz
z,(9)
π<a<π. Now make a change of variable in the last integral by putting y=ez.
This yields
ˆ
0
sh au
sh πu lnx2+u2du =
1
ˆ
0tg a
22yxsin a
1+2ycosa+y2dy
ln 1
yTa(x), (10)
π<a<π, where the last integral was designated by Ta(x) for brevity. It can be
easily demonstrated that if the parameter ais chosen so that a=πm/n, numbers m
and nbeing positive integers such that m<n(in other words, if ais a rational part of
π), then, the integral on the right part of the last equation may be always expressed
11These formulas may be also derived by contour integration methods, see e.g. [59, pp. 186, 197–198],
[69, p. 132],[23, pp. 276–277].
30 I.V. Blagouchine
in terms of the Γ-function. The differentiation of Ta(x) with respect to xgives
dT
dx =2
1
ˆ
0
yxsin a
1+2ycosa+y2dy. (11)
But if the parameter ais a rational part of π, the latter integral, in virtue of what was
established in [64, vol. II, pp. 163–165], is
1
ˆ
0
yxsin a
1+2ycosa+y2dy
=
n1
l=1
(1)l1sin(la ) Ψx+n+l
2nΨx+l
2n,if m+nis odd,
1
2(n1)
l=1
(1)l1sin(la ) Ψx+nl
nΨx+l
n,if m+nis even.
By substituting these formulas into the right part of (11), and by calculating the an-
tiderivative, we obtain
Ta(x) =
C1+2
n1
l=1
(1)l1sin(la ) lnΓx+n+l
2n
Γx+l
2n,if m+nis odd,
C2+2
1
2(n1)
l=1
(1)l1sin(la ) lnΓx+nl
n
Γx+l
n,if m+nis even,
(12)
where C1and C2are constants of integration. In order to find them, the following
procedure is adopted. Put in the last formula, first x=r, and then x=s. Subtracting
one from another and dividing by minus two, we have a(r, s) 1
2[Ta(s) Ta(r )]=
1
ˆ
0
yr(1ysr)sin a
1+2ycosa+y2·dy
ln 1
y
=
n1
l=1
(1)l1sin(la ) lnΓs+n+l
2nΓr+l
2n
Γr+n+l
2nΓs+l
2n,if m+nis odd,
1
2(n1)
l=1
(1)l1sin(la ) lnΓs+nl
nΓr+l
n
Γr+nl
ns+l
n,if m+nis even.
Malmsten’s integrals and their evaluation by contour integration 31
The difference between a(0,1)and a(1,2)yields
1
ˆ
0
(12y+y2)sin a
1+2ycosa+y2·dy
ln 1
y
=
n1
l=1
(1)l1sin(la ) ln
Γ2n+l+1
2nΓl+2
2nΓl
2n
Γ2l+1
2nΓn+l
2nΓn+l+2
2n
,
if m+nis odd,