Laplace Transform of nested analytic functions via Bell’s polynomials
Keywords:
Laplace transform, Bell, Composite functionsAbstract
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.
References
Bell, E.T. Exponential polynomials. Annals of Mathematics 1934, 35, 258–277.
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.
Caratelli, D. Cesarano, C., Ricci, P.E. Computation of the Bell-Laplace transforms. Dolomites Res. Notes Approx. 2021 14, 74–91.
Cassisa, C., Ricci, P.E. Orthogonal invariants and the Bell polynomials. Rend. Mat. Appl. 2000 (Ser. 7) 20, 293–303.
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.
Fa`a di Bruno, F. Th´eorie des formes binaires. Brero, Turin, 1876.
Ghizzetti, A., Ossicini, A. Trasformate di Laplace e calcolo simbolico. (Italian), UTET, Torino, 1971.
Oberhettinger, F., Badii, L. Tables of Laplace Transforms. Springer-Verlag, Berlin- Heidelberg-New York, 1973.
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
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.
Ricci, P.E. Bell polynomials and generalized Laplace transforms. Integral Transforms Spec. Funct., (2022); doi.org/10.1080/10652469.2022.2059077.
Riordan, J. An Introduction to Combinatorial Analysis. J. Wiley & Sons, Chichester, 1958.
Robert, D. Invariants orthogonaux pour certaines classes d’operateurs. Annales Math´em. pures appl. 1973 52, 81–114.
Roman, S.M., The Fa`a di Bruno Formula. Amer. Math. Monthly 87 (1980), 805–809.
Roman, S.M., Rota, G.C. The umbral calculus. Advanced in Math. 1978 27, 95–188.
