Revised 28 February 2023
Accepted 23 October 2023
Available Online 29 November 2023
- DOI
- https://doi.org/10.55060/s.atmps.231115.006
- Keywords
- Laplace Transform
Bell’s polynomials
Composed functions - Abstract
Bell’s polynomials have been used in many different fields, ranging from number theory to operator 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 composed exponential functions. Furthermore, a code for approximating the LT of general analytic composed functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.
- Copyright
- © 2023 The Authors. Published by Athena International Publishing B.V.
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (https://creativecommons.org/licenses/by-nc/4.0/).
1. INTRODUCTION
The common view that there is no formula for the Laplace Transform (LT) of composed analytic functions is disproved in this article, using Bell’s polynomials [1], as in the case of the derivative of nested functions [2].
Bell’s polynomials appear in very different fields, ranging from number theory [2,3,4,5,6] to operator theory [7], and from differential equations to integral transforms [8].
The importance of the LT is well known and it is not necessary to remind it here.
The second-order Bell polynomials
2. RECALLING THE BELL POLYNOMIALS
The Bell polynomials express the
The first few Bell polynomials are:
The Bell polynomials are given by:
The
The
Therefore, we have the equations:
An explicit expression for the Bell polynomials is given by the Faà di Bruno formula:
A proof of the Faà di Bruno formula can be found in [9]. The proof is based on the umbral calculus (see [10] and the references therein).
Remark 1:
It should be noted that the possibility of constructing the Bell polynomials of index
3. RECALLING THE LAPLACE TRANSFORM
The Laplace Transform, a very useful tool in applied mathematics [12], writes:
The Laplace operator converts a function of a real variable
It can be applied to functions belonging to
Remark 2:
To avoid confusion, we want to stress that the purpose of this article is not to generalize the LT, but only to expand the table of transforms that are often used in applied mathematics problems, and which are reported in the book by Oberhettinger and Badii [13]. Actually, we give an approximation of the LT of composed analytic functions using elementary methods, namely the Taylor expansion and the Bell polynomials.
3.1. Main Properties and an Example
The Laplace transform method gives a rigorous approach to the operational technique introduced by Oliver Heaviside in 1893, in connection with his work in telegraphy.
This transformation is used to solve initial value problems for linear ordinary differential equations:
It can also be used for linear partial differential equations, and in particular in the case of the telegraphists' equation [14], which expresses the voltage
Note that this equation contains, as special cases, the vibrating string equation (when r = g = 0):
The main rules are:
Scaling property:
Action on derivatives:
Convolution theorem:
Using these rules, and others derived from them and reported in suitable tables, the given equation in the time domain
After solving the problem in the frequency domain, the result is transformed back to the time domain, usually by using a table of inverse Laplace transforms or evaluating a Bromwich contour integral in the complex plane.
A simple example is the following.
Consider the harmonic oscillator problem:
Multiplying by
Since:
4. LAPLACE TRANSFORM OF COMPOSED FUNCTIONS
Consider a composed function
We have:
Theorem 3.
Consider a composed function
Proof.
In fact, using the uniform convergence of Taylor's expansion, we can write:
5. THE CASE OF THE EXPONENTIAL FUNCTION
In the particular case when
Further values are reported in Appendix I.
The complete Bell polynomials satisfy the identity:
In this case, the general Eq. (9) reduces to:
In what follows we show the approximation of the LT of nested functions using the computer algebra program Mathematica®.
5.1. First Examples
We start considering the case of the LT of nested exponential functions:
- •
- •
Consider the Bessel function
and the LT of the corresponding exponential function. We find: (15) - •
Let
. We find: (16) - •
Consider the complete elliptic integral of the second kind
and the LT of the corresponding exponential function. We find: (17)
5.2. Graphical Display in Two Known Cases
5.2.1. Test Case #1
Considering the composed function
The distributions of

Magnitude (a) and argument (b) of the Laplace transform of

Magnitude (a) and argument (b) of the Laplace transform of

Distribution of
5.2.2. Test Case #2
Considering the composed function
Using our approximation, we have found:
The distributions of

Magnitude (a) and argument (b) of the Laplace transform of

Magnitude (a) and argument (b) of the Laplace transform of

Distribution of
6. AN EXTENSION OF THE BELL POLYNOMIALS
We limit ourselves to the second-order Bell polynomials,
Consider the differentiable functions
Then the
For example, one has:
The connections with the ordinary Bell polynomials are expressed by the equation:
Consequently, we deduce the theorem:
Theorem 4.
The following recurrence relation for the second-order Bell polynomials holds true:
The first few second-order Bell polynomials are as follows:
Further values are reported in Appendix II.
7. LAPLACE TRANSFORM OF NESTED FUNCTIONS
Let
It results:
Theorem 5.
Consider a nested function
Proof.
It is a straightforward application of the definition of second-order Bell's polynomials.
7.1. Example #1
Assuming
The corresponding inverse LT is approximated by:
7.2. Example #2
Assuming
The corresponding inverse LT is approximated by:
7.3. Example #3
Assuming
The corresponding inverse LT is approximated by:
7.4. Example #4
Assuming
The corresponding inverse LT is approximated by:
Remark 6:
Note also that successive Bell polynomials are represented exclusively by sums, products and powers, avoiding operations that may generate numerical instability. The use of computers allows calculations to be performed stably and quickly, even though the number of products to be added increases rapidly with the number n. In our calculations it was possible to obtain a sufficient approximation by limiting ourselves to order n = 10.
8. CONCLUSION
We have presented a method for approximating the integral of analytic composed functions. Considering the Taylor expansion of the given function and representing their coefficients in terms of Bell’s polynomials, the integral reduces to the computation of an approximating series, which obviously converges if the integral is convergent. This methodology has been applied to the LT of an analytic composed function, starting from the case of analytic nested exponential functions, based on the complete Bell polynomials, computed by using the program Mathematica®, and shown in Appendix I.
In the second part the LT of analytic nested functions is considered, and the second-order Bell’s polynomials used in this approach are reported in Appendix II. We want to stress that, even if we dealt with a basic subject, we have not found in the literature any general method for approximating this type of LTs, a gap which, in our opinion, has been now filled up. A graphical verification of the proposed technique, performed in the case when both the analytical forms of the transform and anti-transform are known, proved the correctness of our results.
The method used in this article has also been applied in other cases such as:
- •
- •
- •
the sine and cosine Fourier transform of particular nested functions [21].
APPENDIX I: TABLE OF COMPLETE BELL POLYNOMIALS

APPENDIX II: TABLE OF SECOND-ORDER BELL POLYNOMIALS
REFERENCES
Cite This Article
TY - CONF AU - Paolo Emilio Ricci AU - Diego Caratelli AU - Sandra Pinelas PY - 2023 DA - 2023/11/29 TI - Laplace Transform Approximation of Nested Functions Using Bell’s Polynomials BT - Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022) PB - Athena Publishing SP - 55 EP - 70 SN - 2949-9429 UR - https://doi.org/10.55060/s.atmps.231115.006 DO - https://doi.org/10.55060/s.atmps.231115.006 ID - Ricci2023 ER -