Integrals Dependent on a Parameter

  • Boris Makarov
  • Anatolii Podkorytov
Part of the Universitext book series (UTX)


In Sect. 7.1, we establish general properties of integrals dependent on a parameter. Section 7.2 is devoted to the Γ function introduced by Euler. This is one of the most important functions representable as an integral dependent on a parameter. Here we collected much information on this function (reflection formulas, the Euler–Gauss and Weierstrass formulas, the Stirling asymptotic formula, etc.). We also prove a theorem axiomatically describing the Γ function in the class of logarithmically convex functions. In Sect. 7.3, we discuss the Laplace method for finding the asymptotic behavior of integrals in the case where the integrand depends exponentially on a large parameter. We give examples illustrating the results of the general method and showing the ways of modification of the method in the cases not covered by the general theory.

Section 7.4 is devoted mainly improper integrals. It also includes the results related to the method of stationary phase, which plays an important role in the study of wave processes.

Systematic use and in-depth exploration of convolution as an important means of smoothing and approximation of functions is characteristic to the book. The last two Sects. 7.5 and 7.6 of this chapter are dedicated to this matter. In particular, convolution is used to prove the Weierstrass classical theorem on approximation of continuous functions by polynomials. Convolution as an approximation means is later used in Chaps.  9 and  10.




  1. [Ar]
    Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964). 7.2.5 MATHGoogle Scholar
  2. [Bou]
    Bourbaki, N.: General Topology. Chapters 5–10. Springer, Berlin (1989). 1.1.3 MATHCrossRefGoogle Scholar
  3. [F]
    Federer, H.: Geometric Measure Theory. Springer, New York (1969). 2.8.1, 8.2.2, 8.4.4, 8.8.1, 10.3 (Ex. 5), 13.2.3 MATHGoogle Scholar
  4. [Fi]
    Fichtenholz, G.M.: Differential and Integral Calculus, vols. I–III. Nauka, Moscow (1970) [in Russian]. 7.4.3 Google Scholar
  5. [Li]
    Littlewood, J.E.: A Mathematician’s Miscellany. Methuen, London (1953). 7.6.4 MATHGoogle Scholar
  6. [LO]
    Leipnik, R., Oberg, R.: Subvex functions and Bohr’s uniqueness theorem. Am. Math. Mon. 74, 1093–1094 (1967). 7.2.8 MathSciNetMATHCrossRefGoogle Scholar
  7. [Z]
    Zorich, V.A.: Mathematical Analysis, vols. I, II. Springer, Berlin (2004). 7.4.3 MATHGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Boris Makarov
    • 1
  • Anatolii Podkorytov
    • 1
  1. 1.Mathematics and Mechanics FacultySt Petersburg State UniversitySt PetersburgRussia

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