On some general integral formulae

Year 2024, Volume: 7 Issue: 1, 1 - 11, 15.03.2024

Norbert Ortner , Peter Wagner

https://doi.org/10.33205/cma.1406998

Abstract

We repeat and reformulate some more or less known general integral formulae and deduce from them some applications in a concise way. We then present some general double integral formulae which play an essential role in the calculation of fundamental solutions to homogeneous elliptic operators. In particular, this yields generalizations of definite integrals found in standard integral tables. In the final section, the area of an ellipsoidal hypersurface in $\textbf{R}^n$ is represented by a hyperelliptic integral.

Keywords

Leray’s formula, elliptic integrals, definite double integrals

References

• L. A. A˘ızenberg, A. P. Yuzhakov: Integral representations and residues in multidimensional complex analysis, AMS, Providence, RI (1983).
• G. Boros, V. H. Moll: Irresistible integrals, Cambridge University Press, Cambridge (2004).
• Yu. A. Brychkov, O. I. Marichev and A. P. Prudnikov: Integrals and series. Vol. 1 (Elementary functions), Gordon &Breach, New York (1986).
• J. Dieudonné: Éléments d’analyse, Tome 3, Gauthiers-Villars, Paris (1970).
• A. Eagle: The elliptic functions as they should be, Galloway and Porter, Cambridge (1958).
• A. Erdélyi, Ed.: Tables of integral transforms, Vol. I, McGraw-Hill, New York (1954).
• J. Faraut, K. Harzallah: Deux cours d’analyse harmonique, Birkhäuser, Boston (1987).
• I. Fredholm: Sur l’intégrale fondamentale d’une équation différentielle elliptique à coefficients constants, Rend. Circ. Mat. Palermo, 25 (1908), 346–351; OEuvres complètes: pp. 117–122.
• H. G. Garnir, J. Gobert: Fonctions d’une variable complexe, Dunod, Paris (1965).
• I. M. Gel’fand, G. E. Shilov: Generalized functions. Vol. I (Properties and operations), Academic Press, New York (1964).
• I. S. Gradshteyn, I. M. Ryzhik: Table of integrals, series and products, Academic Press, New York (1980).
• W. Gröbner, N. Hofreiter: Integraltafel, 2. Teil: Bestimmte Integrale, 5th ed., Springer, Wien (1973).
• A. Gonzalez–Dominguez, S. E. Trione: On the Laplace transforms of retarded Lorentz-invariant functions, Adv. Math., 31 (1979), 51–62.
• J. Leray: Hyperbolic differential equations, Institute of Advanced Study, Princeton (1952).
• D. S. Mitrinovi´c, J. D. Keˇcki´c: The complex method of residues, D. Reidel, Dordrecht (1984).
• N. Ortner, P. Wagner: Fundamental solutions of linear partial differential operators, Springer, New York (2015).
• N. Ortner, P. Wagner: A distributional version of Frullani’s integral, Bull. Sci. Math., 186 (2023), 103272.
• L. Schwartz: Théorie des distributions, 2nd ed., Hermann, Paris (1966).
• S. E. Trione: Sur une formule de J. Leray, Trabajos de Matematica 31, Buenos Aires (1981).
• P. Wagner: On the fundamental solutions of a class of elliptic quartic operators in dimension 3, J. Math. Pures Appl. 81 (2002), 1191–1206.