Laplace Transform of nested analytic functions via Bell’s polynomials

Year 2023, Volume: 7 Issue: 1, 162 - 177, 31.03.2023

Paolo Emilio Ricci Diego Caratelli Sandra Pinelas

https://doi.org/10.31197/atnaa.1187617

Cited By: 1

Abstract

Bell's polynomials have been used in many different fields, ranging from number theory to operators theory. In this article we show a method to compute the Laplace Transform (LT) of nested analytic functions. To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.

Keywords

Laplace transform, Bell, Composite functions

Supporting Institution

No support by our institutions

Thanks

Thanks for invitation.

References

• [1] Bell, E.T. Exponential polynomials. Annals of Mathematics 1934, 35, 258–277.
• [2] Beerends, R.J., Ter Morsche, H.G., Van Den Berg, J.C., Van De Vrie, E.M., Fourier and Laplace Transforms. Cambridge Univ. Press, Cambridge, 2003.
• [3] Caratelli, D. Cesarano, C., Ricci, P.E. Computation of the Bell-Laplace transforms. Dolomites Res. Notes Approx. 2021 14, 74–91.
• [4] Cassisa, C., Ricci, P.E. Orthogonal invariants and the Bell polynomials. Rend. Mat. Appl. 2000 (Ser. 7) 20, 293–303.
• [5] Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Co., 1974; available online at https://doi.org/10.1007/978-94-010-2196-8.
• [6] Fa`a di Bruno, F. Th´eorie des formes binaires. Brero, Turin, 1876.
• [7] Ghizzetti, A., Ossicini, A. Trasformate di Laplace e calcolo simbolico. (Italian), UTET, Torino, 1971.
• [8] Oberhettinger, F., Badii, L. Tables of Laplace Transforms. Springer-Verlag, Berlin- Heidelberg-New York, 1973.
• [9] Orozco L´opez, R. Solution of the Differential Equation y(k) = eay, Special Values of Bell Polynomials, and (k, a)-Autonomous Coefficients. J. Integer Seq., 2021 24, Article 21.8.615
• [10] Qi, F., Niu, D-W., Lim, D., Yao, Y-H., Special values of the Bell polynomials of the second kind for some sequences and functions. J. Math. Anal. Appl., 2020 491 (2), 124382.
• [11] Ricci, P.E. Bell polynomials and generalized Laplace transforms. Integral Transforms Spec. Funct., (2022); doi.org/10.1080/10652469.2022.2059077.
• [12] Riordan, J. An Introduction to Combinatorial Analysis. J. Wiley & Sons, Chichester, 1958.
• [13] Robert, D. Invariants orthogonaux pour certaines classes d’operateurs. Annales Math´em. pures appl. 1973 52, 81–114.
• [14] Roman, S.M., The Fa`a di Bruno Formula. Amer. Math. Monthly 87 (1980), 805–809.
• [15] Roman, S.M., Rota, G.C. The umbral calculus. Advanced in Math. 1978 27, 95–188.