The Ramanujan Journal

, Volume 35, Issue 1, pp 21–110 | Cite as

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



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 γ1 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.


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 


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formula, Graphs and Mathematical Tables. Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (1961) MATHGoogle Scholar
  2. 2.
    Adamchik, V.: A class of logarithmic integrals. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 1–8 (1997) CrossRefGoogle Scholar
  3. 3.
    Ahlfors, L.: Complex Analysis, 3rd edn. McGraw-Hill Science, New York (1979) MATHGoogle Scholar
  4. 4.
    Amdeberhan, T., Espinosa, O., Moll, V.H.: The Laplace transform of the digamma function: an integral due to Glasser, Manna and Oloa. Proc. Am. Math. Soc. 136(9), 3211–3221 (2008) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976) MATHGoogle Scholar
  6. 6.
    Artin, E.: Einführung in die Theorie der Gammafunktion. Teubner, Leipzig (1931) MATHGoogle Scholar
  7. 7.
    Bailey, D.H., Borwein, J.M., Calkin, N.J., Girgensohn, R., Luke, R.D., Moll, V.H.: Experimental Mathematics in Action. AK Peters, Wellesley (2007) MATHGoogle Scholar
  8. 8.
    Bassett, M.E.: Integrals, L functions, prime numbers: a short tour of analytic number theory. Report on a talk given at UCL (University College London) Undergrad Maths Colloquium on Analytic Number Theory, London’s Global University (2010) Google Scholar
  9. 9.
    Bateman, H., Erdélyi, A.: Higher Transcendental Functions. McGraw-Hill Book, New York (1955) [in 3 volumes] MATHGoogle Scholar
  10. 10.
    Bermant, A.F.: In: Course of Mathematical Analysis. Part II, 5th edn., Gosudarstvennoe izdatel’stvo tehniko–teoreticheskoj literatury, Moscow (1954) [in Russian] Google Scholar
  11. 11.
    Berndt, B.C.: The Gamma function and the Hurwitz Zeta-function. Am. Math. Mon. 92(2), 126–130 (1985) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Binet, M.J.: Mémoire sur les intégrales définies eulériennes et sur leur application à la théorie des suites, ainsi qu’à l’évaluation des fonctions des grands nombres. J. Éc. Polytech. XVI(27), 123–343 (1839) Google Scholar
  13. 13.
    Boros, G., Moll, V.: Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, Cambridge (2004) CrossRefMATHGoogle Scholar
  14. 14.
    Briggs, W.E.: Some constants associated with the Riemann Zeta-function. Mich. Math. J. 3(2), 117–121 (1955) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Campbell, R.: Les intégrales eulériennes et leurs applications. Dunod, Paris (1966) MATHGoogle Scholar
  16. 16.
    Carathéodory, C.: Theory of Functions of a Complex Variable. Chelsea, New York (1954) [in 2 volumes] MATHGoogle Scholar
  17. 17.
    Coffey, M.W.: Series representations for the Stieltjes constants. arXiv:0905.1111v2 (2009)
  18. 18.
    Coffey, M.W.: On representations and differences of Stieltjes coefficients, and other relations. Rocky Mt. J. Math. 41(6), 1815–1846 (2011) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Davis, P.J.: Leonhard Euler’s integral: a historical profile of the Gamma function. Am. Math. Mon. 66, 849–869 (1959) CrossRefMATHGoogle Scholar
  20. 20.
    Euler, L.: Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques. Histoire de l’Académie Royale des Sciences et Belles-Lettres, année MDCCLXI, Tome 17, pp. 83–106, A Berlin, chez Haude et Spener, Libraires de la Cour et de l’Académie Royale (1768 [read in 1749]) Google Scholar
  21. 21.
    Eulero, L.: Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. Academiæ Imperialis Scientiarum Petropolitanæ, Saint-Petersburg (1755) Google Scholar
  22. 22.
    Evgrafov, M.A.: Analytic Functions. Saunders, Philadelphia (1966) MATHGoogle Scholar
  23. 23.
    Evgrafov, M.A., Sidorov, Y.V., Fedoriuk, M.V., Shabunin, M.I., Bezhanov, K.A.: A Collection of Problems in the Theory of Analytic Functions. Nauka, Moscow (1969) [in Russian] Google Scholar
  24. 24.
    Frullani, G.: Sopra gli integrali definiti. Mem. Soc. Ital. Sci. XX(II), 448–467 (1829) Google Scholar
  25. 25.
    Fuchs, B.A., Shabat, B.V.: Functions of a Complex Variable and Some of Their Applications. International Series of Monographs in Pure and Applied Mathematics. Pergamon, Oxford (1961/1964) [in 2 volumes] MATHGoogle Scholar
  26. 26.
    Gauss, C.F.: Disquisitiones generales circa seriem infinitam \(1+\frac{\alpha\beta}{1\cdot\gamma}x+ \frac{\alpha(\alpha+1)\beta(\beta+1)}{1\cdot2\cdot\gamma(\gamma+1)}xx+ \frac{\alpha(\alpha+1)(\alpha+2)\beta(\beta+1)(\beta+2)}{1\cdot2\cdot3\cdot\gamma(\gamma+1)(\gamma+2)}x^{3} +\mathrm{etc}\). Commentationes Societatis Regiae Scientiarum Gottingensis recentiores, Classis Mathematicæ, vol. II, pp. 3–46 [republished later in “Carl Friedrich Gauss Werke”, vol. 3, pp. 265–327, Königliche Gesellschaft der Wissenschaften, Göttingen, 1866] (1813) Google Scholar
  27. 27.
    Godefroy, M.: La fonction Gamma; Théorie, Histoire, Bibliographie. Gauthier-Villars, Imprimeur Libraire du Bureau des Longitudes, de l’École Polytechnique, Quai des Grands-Augustins, 55, Paris (1901) Google Scholar
  28. 28.
    Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products, 4th edn. Academic Press, New York (1980) MATHGoogle Scholar
  29. 29.
    Gunther, N.M., Kuzmin, R.O.: A Collection of Problems on Higher Mathematics, vol. 3, 4th edn. Gosudarstvennoe izdatel’stvo tehniko–teoreticheskoj literatury, Leningrad (1951) [in Russian] Google Scholar
  30. 30.
    Hamming, R.W.: Numerical Methods for Scientists and Engineers. McGraw-Hill, New York (1962) MATHGoogle Scholar
  31. 31.
    Hardy, G.H.: Divergent Series. Clarendan, Oxford (1949) MATHGoogle Scholar
  32. 32.
    Henrici, P.: Applied and Computational Complex Analysis. Wiley, New York (1974/1977/1986) [in 3 volumes] MATHGoogle Scholar
  33. 33.
    Hurwitz, A., Courant, R.: Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie, 4th edn. Springer, Berlin (1964) CrossRefMATHGoogle Scholar
  34. 34.
    Jeffreys, H., Jeffreys, B.S.: Methods of Mathematical Physics, 2nd edn. Cambridge University Press, Cambridge (1950) MATHGoogle Scholar
  35. 35.
    Kummer, E.E.: Beitrag zur theorie der function Γ(x). J. Reine Angew. Math. 35, 1–4 (1847) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Leche, V., Meijer, B., Nyström, J.F., Warburg, K., Westrin, T., Meijer, B.: Nordisk familjebok, 2nd edn., Nordisk familjeboks förlags aktiebolag, Stockholm (1904–1926). (in 38 volumes) Google Scholar
  37. 37.
    Lerch, M.: Další studie v oboru Malmsténovských řad. Rozpravy České akademie císare Františka Josefa pro vedy, slovesnost a umení. Trída II, Mathematicko-prírodnická, ročník 3, číslo 28, pp. 1–63 (1894) Google Scholar
  38. 38.
    Liang, J.J.Y., Todd, J.: The Stieltjes constants. J. Res. Natl. Bur. Stand. B, Math. Sci. 76B(3–4), 161–178 (1972) MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Lindelöf, E.: Le calcul des résidus et ses applications à la théorie des fonctions. Gauthier-Villars, Imprimeur Libraire du Bureau des Longitudes, de l’École Polytechnique, Quai des Grands-Augustins, 55, Paris (1905) Google Scholar
  40. 40.
    Malmsten, C.J., Almgren, T.A., Camitz, G., Danelius, D., Moder, D.H., Selander, E., Grenander, J.M.A., Themptander, S., Trozelli, L.M., Föräldrar, Ä., Ossbahr, G.E., Föräldrar, D.H., Ossbahr, C.O., Lindhagen, C.A., Moder, D.H., Syskon, Ä., Lemke, O.V., Fries, C., Laurenius, L., Leijer, E., Gyllenberg, G., Morfader, M.V., Linderoth, A.: Specimen analyticum, theoremata quædam nova de integralibus definitis, summatione serierum earumque in alias series transformatione exhibens (Eng. trans.: “Some new theorems about the definite integral, summation of the series and their transformation into other series”) [Dissertation, in 8 parts]. Upsaliæ, excudebant Regiæ academiæ typographi. Uppsala, Sweden (April–June 1842) Google Scholar
  41. 41.
    Malmstén, C.J.: De integralibus quibusdam definitis seriebusque infinitis (Eng. trans.: “On some definite integrals and series”). J. Reine Angew. Math., 38, 1–39 (1849) [work dated May 1, 1846] MathSciNetCrossRefGoogle Scholar
  42. 42.
    Markushevich, A.I.: Theory of Functions of a Complex Variable, 2nd edn. AMS Chelsea/American Mathematical Society, New York/Providence (2005) [in 3 volumes] Google Scholar
  43. 43.
    McLachlan, N.W.: Complex Variable and Operational Calculus with Technical Applications. Cambridge University Press, London (1942) MATHGoogle Scholar
  44. 44.
    Medina, L.A., Moll, V.H.: A class of logarithmic integrals. Ramanujan J. 20(1), 91–126 (2009) MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Miller, J., Adamchik, V.S.: Derivatives of the Hurwitz Zeta function for rational arguments. J. Comput. Appl. Math. 100, 201–206 (1998) MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Moll, V.H.: The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals. Scentia, Ser. A, Math. Sci. 14, 1–6 (2007) MathSciNetMATHGoogle Scholar
  47. 47.
    Moll, V.H., Amdeberhan, T.: Contemporary Mathematics: Tapas in Experimental Mathematics. American Mathematical Society, Providence (2007) Google Scholar
  48. 48.
    Nan-You, Z., Williams, K.S.: Some results on the generalized Stieltjes constant. Analysis 14, 147–162 (1994) MathSciNetGoogle Scholar
  49. 49.
    Nielson, N.: Handbuch der Theorie der Gammafunktion. Teubner, Leipzig (1906) Google Scholar
  50. 50.
    Ostrowski, A.M.: On some generalisations of the Cauchy–Frullani integral. Proc. Natl. Acad. Sci. USA 35, 612–616 (1949) MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Poisson, S.D.: Second mémoire sur la distribution de l’électricité à la surface des corps conducteurs (lu le 6 septembre 1813). Mémoires de la classe des sciences mathématiques et physiques de l’Institut Impérial de France, partie II Chez Fermin Didot, Imprimeur de l’Institut Impérial de France et Libraire pour les mathématiques, pp. 163–274, rue Jacob, no. 24, Paris, France (1814) Google Scholar
  52. 52.
    Poisson, S.D.: Suite du mémoire sur les intégrales définies, inséré dans les deux précédents volumes de ce journal. J. Éc. Polytech. XI(18), 295–341 (1820) Google Scholar
  53. 53.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vols. I–IV. Gordon and Breach, New York (1992) MATHGoogle Scholar
  54. 54.
    Riemann, B.: Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse. Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, pp. 136–144 (1859) Google Scholar
  55. 55.
    Schlömilch, O.: Uebungsaufgaben für Schüler. Lehrsatz. Grunert Arch. Math. Phys. XII(IV), 415 (1849), part XXXV Google Scholar
  56. 56.
    Schlömilch, O.: Ueber eine Eigenschaft gewisser Reihen. Z. Math. Phys. III, 130–132 (1858) Google Scholar
  57. 57.
    Sloane, N.J.A.: Sequence A115252. The On-Line Encyclopedia of Integer Sequences (2006).
  58. 58.
    Smirnov, V.I.: A Course of Higher Mathematics. vol. I–V. Pergamon, London (1964) MATHGoogle Scholar
  59. 59.
    Spiegel, M.R.: Theory and Problems of Complex Variables with an Introduction to Conformal Mapping and Its Application. McGraw-Hill, New York (1968) Google Scholar
  60. 60.
    Sveshnikov, A.G., Tikhonov, A.N.: Theory of Functions of a Complex Variable. Nauka, Moscow (1967). [in Russian] MATHGoogle Scholar
  61. 61.
    Bierens de Haan, D.: Tables d’intégrales définies. Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Deel IV, Amsterdam (1858) CrossRefMATHGoogle Scholar
  62. 62.
    Bierens de Haan, D.: Nouvelles tables d’intégrales définies. P. Engels, Libraire Éditeur, Leide (1867) MATHGoogle Scholar
  63. 63.
    Gårding, L.: Mathematics and Mathematicians: Mathematics in Sweden Before 1950. American Mathematical Society/London Mathematical Society, Providence (1994) Google Scholar
  64. 64.
    Le Gendre, A.M.: Exercices de calcul intégral sur divers ordres de transcendantes et sur les quadratures Tomes I–III. Mme Ve Courcier, Imprimeur–Libraire pour les Mathématiques, rue du Jardinet, no. 12, quartier Saint-André-des-Arc, Paris (1811–1817) Google Scholar
  65. 65.
    Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, Oxford (1939) MATHGoogle Scholar
  66. 66.
    Titchmarsh, E.C.: The Theory of the Riemann Zeta-Function, 2nd edn. Clarendon, Oxford (1986) MATHGoogle Scholar
  67. 67.
    Vardi, I.: Integrals, an introduction to analytic number theory. Am. Math. Mon. 95, 308–315 (1988) MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Vilceanu, R.O.: An application of Dirichlet L-series to the computation of certain integrals. Bull. Math. Soc. Sci. Math. Roum. 51(99)(2), 159–173 (2008) MathSciNetMATHGoogle Scholar
  69. 69.
    Volkovyskii, L.I., Lunts, G.L., Aramanovich, I.G.: A Collection of Problems on Complex Analysis. Pergamon, London (1965) MATHGoogle Scholar
  70. 70.
    Weisstein, E.W.: Vardi’s integral. From MathWorld—a Wolfram web resource (2006).
  71. 71.
    Whittaker, E., Watson, G.N.: A Course of Modern Analysis. an Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions, 3rd edn. Cambridge University Press, Cambridge (1920) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of ToulonLa Valette du Var (Toulon)France

Personalised recommendations