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

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.

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Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    Only Bassett [8], an undergraduate student, remarked that solutions for integrals (1) and (2a, 2b) are much older than Vardi’s paper [67].

  2. 2.

    Carl Johan Malmsten, written also Karl Johan Malmsten (born April 9, 1814 in Uddetorp, died February 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].

  3. 3.

    Both 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.

  4. 4.

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

  5. 5.

    Another 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.

  6. 6.

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

  7. 7.

    However, we were surprised to see that Malmsten in the article [41] did not even mention the aforementioned 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.

  8. 8.

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

  9. 9.

    Most of these notations come from Latin, e.g “\(\operatorname {\mathrm {ch}}\)” stands for cosinus hyperbolicus, “\(\operatorname {\mathrm {sh}}\)” stands for sinus hyperbolicus, etc.

  10. 10.

    This operation is permitted because the considered improper integral is uniformly convergent with respect to v [34, p. 44, § 1.12]*, [58, vol. II, pp. 262–269]*, [10, pp. 175–179]*.

  11. 11.

    These formulas may be also derived by contour integration methods, see e.g. [59, pp. 186, 197–198], [69, p. 132], [23, pp. 276–277].

  12. 12.

    From here, we shorten Malmsten’s proof since the result is almost straightforward. Malmsten’s proof was actually longer because he aimed for more general formulas.

  13. 13.

    There is a misprint in formula (13): the sign “−” in the denominator of the integrand should be replaced by “+”.

  14. 14.

    It seems, however, that there is a misprint in exercise no. 3.6 [2]. In the first line, the term x 5 should be removed from the denominator of the integrand.

  15. 15.

    A sketch of the Euler’s proof of formula (22) may be also found in [31, pp. 23–26].

  16. 16.

    We performed similar simplification for the Ψ-function in exercise no. 11, formula (48).

  17. 17.

    Gauss presented the proof of this theorem in January 1812 [26].

  18. 18.

    By the order of Malmsten’s integral we mean the order of poles of the corresponding integrands.

  19. 19.

    The use of divergent series was especially common in the 18th century, see, for instance, the excellent monograph [31].

  20. 20.

    One of these two errors comes from the Bierens de Haan’s tables (latter borrowed them in part from misprints in Malmsten’s work [41]; Bierens de Haan even complained about the number of misprints in this work, see [61, p. 265]). For example, integrals’ bounds are incorrect in [62, Table 148-1,2,3,4], [61, Table 191-1,2,3,4,5,6], [41, Eq. (10), (12) for both integrals]. The reader should be also careful with these sources since integrands in Gradshteyn and Ryzhik’s tables are presented in other form.

  21. 21.

    We will not consider here indirect methods, such as, for example, evaluation of logarithmic integrals based on the differentiation of the integrand (which does not contain a logarithm) with respect to a parameter.

  22. 22.

    The readers of this book should beware of misprints and of some incorrect results. Answers in exercises no. 84, 88, 91 are incorrect; on the p. 189, 2πi is forgotten in the right part of equation (1). Several errors were corrected in the recent second edition of this book, but the few ones are still present, e.g. answer in no. 7.91 is incorrect, the above-mentioned coefficient 2πi is absent.

  23. 23.

    An interesting and quite well-written historical overview on the Γ-function is given in [19]. Motivated readers who are not afraid of French and German are also invited to take the look at these classic books [15, 27, 49] and [6].

  24. 24.

    The non-asymptotic part of this formula (that containing an infinite integral) is also known as the second Binet’s expression for the logarithm of the Γ-function [12, pp. 335–336], [71, pp. 250–251], [9, vol. I, p. 22, Eq. 1.9(9)] (for more details, see also exercise no. 40 in the last section of this manuscript). As regards its asymptotic form, it was already known to Gauss [26, p. 33], and in a more simple form (for natural z), to Euler [21, part II, Chap. VI, p. 466], to Stirling and to de Moivre.

  25. 25.

    For the evaluation of the integral J R lazy readers may directly use formula 1.4.7-15 from [53, vol. I, p. 148]. Nevertheless, it is highly recommended that readers employ the proposed method rather than the ready formula, since the procedure for the calculation of the logarithmic integral is very similar.

  26. 26.

    For instance, we have not found these integrals in [28], neither in [53].

  27. 27.

    Integer powers only.

  28. 28.

    Strictly speaking, Kummer’s result [35, p. 4] is obtained from the Malmsten et al.’s one [40, formula (74), p. 62] by putting a=π(2x−1).

  29. 29.

    Malmsten originally wrote a/2 instead of φ.

  30. 30.

    The series being uniformly convergent.

  31. 31.

    Prudnikov et al.’ tables provides, however, several formulas for the series (c) and (e) when a is rational [53, vol. I, § 5.4.3].

  32. 32.

    This integral was first evaluated by Poisson in 1813 in the famous work [51, pp. 214–220], see also [12, p. 240]. The work [51] was later republished in several volumes of the Journal de l’École Polytechnique, see namely [52].

  33. 33.

    In fact, Vardi was not very clear in defining his idea of the relationship between the poles of the integrand and the argument of the Γ-function with the help of which Malmsten’s integrals are expressed. The statement “in Eq. (2a) the number 3 plays the ‘key role’ and in Eq. (2b) 6 is the ‘magic number’’’ [67, p. 313] may be also interpreted in the sense that the least possible integer in the denominator of the argument of the Γ-function (and not the inverse multiplicative of the argument of the Γ-function) should be equal to the degree in which the poles of the integrand are the roots of unity. However, the fallacy of this statement is also evident from the proof given above.

  34. 34.

    To assure the convergence, we should take in the denominator \(1+2x\operatorname {\mathrm {ch}}t +x^{2}\) rather than \(1-2x\operatorname {\mathrm {ch}}t +x^{2}\).

  35. 35.

    More precisely, the last term in the right-hand side of [53, vol. I, no. 2.7.5-10] is incorrect: the argument of the square root should be multiplied by 2. Curiously, in the original Russian edition of [53], this integral is neither correctly evaluated, but the error is not the same: the coefficient 2 in the argument of the logarithm must be placed under the square root sign.

  36. 36.

    However, it seems fair to remark that an integral quite similar to (b) was also evaluated by Malmsten et al. [40, p. 55, Eq. (67)].

  37. 37.

    Alternatively, it is also possible to consider only the upper integral and then study two cases: (m+n) is odd and (m+n) is even. This method will lead to the same formula, albeit the calculation might seem more tedious.

  38. 38.

    With the help of Maple 12.

  39. 39.

    For more details of the finite Fourier series, see e.g. [30, Chap. 6].

  40. 40.

    According to [2] formula (c) was first proved by Almkvist and Meurman in a private communication.

  41. 41.

    This expansion for the Riemann ζ-function was first given by Stieltjes, and therefore, was written in terms of constants γ n γ n (1), which were later called the Stieltjes constants. Constants γ n (v) with arbitrary v represent a more general case and occur when expanding the Hurwitz ζ-function instead of the Riemann ζ-function; such constants are called generalized Stieltjes constants. For more information, see [9, vol. I, p. 26, Eq. 1.10(9), and vol. III, §17.7, p. 189], [14], [66, p. 16], [11, 17, 18, 38, 48].

  42. 42.

    This formula appears with an error in [18, p. 1836, Eq. (3.54)]: in the right part \(\frac{1}{2}\) should be replaced by \(\frac{1}{2}\ln q\).

  43. 43.

    Note that these relationships are quite similar to those for the logarithm of the Γ-function.

  44. 44.

    An attentive analysis shows that equations no. 63-b.2, (58) and (63) for n=2 and v=m/(2n) [variable n corresponds to (58)] are linearly dependent.

  45. 45.

    Moreover, the exact determination of γ 1(1/3) and γ 1(2/3) is based on a particular case of (3.28) at k=1; such a case is given by (3.54) [18]. The latter equation, as we already noticed in footnote 42, contains an error.

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Correspondence to Iaroslav V. Blagouchine.

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An erratum to this article is available at http://dx.doi.org/10.1007/s11139-015-9763-z.

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Blagouchine, I.V. Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results. Ramanujan J 35, 21–110 (2014). https://doi.org/10.1007/s11139-013-9528-5

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Keywords

  • Logarithmic integrals
  • Logarithmic series
  • Theory of functions of a complex variable
  • Contour integration
  • Rediscoveries
  • Malmsten
  • Vardi
  • 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