Abstract
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.
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Notes
- 1.
In particular, if \(\widetilde{Y}=[-\infty,+\infty]\), then the cases a=±∞ are possible.
- 2.
Carl Friedrich Gauss (1777–1855)—German mathematician.
- 3.
Karl Theodor Wilhelm Weierstrass (1815–1897)—German mathematician.
- 4.
Adrien-Marie Legendre (1752–1833)—French mathematician.
- 5.
James Stirling (1692–1770)— Scotish mathematician.
- 6.
To the best of our knowledge, this was first published by H. Bohr and J. Mollerup in “Laerebog i Matematisk Analyse” in 1922 (see e.g. [LO]).
- 7.
Pierre-Simon Laplace (1749–1827)—French mathematician.
- 8.
George Neville Watson (1886–1965)—British mathematician.
- 9.
Ludwig Otto Hesse (1811–1874)—German mathematician.
- 10.
We recall that by λ 1 we denote the one-dimensional Lebesgue measure.
- 11.
Paul Adrien Maurice Dirac (1902–1984)—British physicist.
- 12.
Vladimir Andreevich Steklov (1863–1926)—Russian mathematician.
- 13.
A brief and clear account of the idea of the method (as well as a clever parody of a formal and pseudo-scientific style of exposition) can be found in the remarkable book [Li], Sect. 11.
References
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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
Fichtenholz, G.M.: Differential and Integral Calculus, vols. I–III. Nauka, Moscow (1970) [in Russian]. 7.4.3
Littlewood, J.E.: A Mathematician’s Miscellany. Methuen, London (1953). 7.6.4
Leipnik, R., Oberg, R.: Subvex functions and Bohr’s uniqueness theorem. Am. Math. Mon. 74, 1093–1094 (1967). 7.2.8
Zorich, V.A.: Mathematical Analysis, vols. I, II. Springer, Berlin (2004). 7.4.3
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Makarov, B., Podkorytov, A. (2013). Integrals Dependent on a Parameter. In: Real Analysis: Measures, Integrals and Applications. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5122-7_7
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