Taylor and Maclaurin Series Representations of the Nield-Kuznetsov
Function of the First Kind
T.L. ALDERSON, M.H. HAMDAN
Department of Mathematics and Statistics
University of New Brunswick
100 Tucker Park Road, Saint John, N.B., E2L 4L5
CANADA
Abstract: - Taylor and Maclaurin series and polynomial approximations of the Standard Nield-Kuznetsov
function of the first kind are obtained in this work. Convergence and error criteria are developed. The obtained
series represent alternatives to the existing asymptotic and ascending series approximations of this integral
function, and are expected to provide an efficient method of computation that is valid for all values of the
argument.
Key-Words: - Taylor series, Standard Nield-Kuznetsov function.
Received: July 25, 2021. Revised: February 26, 2022. Accepted: March 19, 2022. Published: April 26, 2022.
1 Introduction
A series of recent articles discussed the importance
of Airy’s ordinary differential equation (ODE) in
fundamental research in the fields of circuits, systems
and signal processing (c.f. [1-3] and the references
therein). Applications of Airy’s ODE to the study of
Schrodinger and Tricomi equations have also been
emphasized, in addition to its importance in the
analysis of Stark equation and the study of Stark
effect, [1-4]. Recent research in the area of Airy’s
ODE reflects the fundamental importance of seeking
solutions to the inhomogeneous version, and the need
for representations and efficient computations of the
integral functions that arise in the processes of
obtaining its general and particular solutions, [5-8].
The above needs give ris to the current work
whose main objective is to develop Taylor series
representation, and Taylor polynomial
approximation, to the Standard Nield-Kuznetsov
function of the first kind, , [5,9]. This function
arises in the general solution to Airy’s, [10],
inhomogeneous ordinary differential equation of the
form, [11]:
where is any constant.
If , general solution of (1) is given by
where are arbitrary constants and and
are the two linearly independent functions
known as Airy’s homogeneous functions of the first
and second kind, respectively. These functions are
defined by the following integrals, [8]:
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The Wronskian of and is non-zero,
as given by, [13]:
(5)
If
or
, general solutions to (1) are given,
respectively, by
where the functions and are the Scorer
functions, [12], given by
If
, obtaining a general solution to (1) in
terms of the Scorer functions requires non-trivial
changes of variables, [8]. A practical need to solve
(1) for values of
arose in the analysis of the
transition layer by Nield and Kuznetsov, [9], who
found it convenient and necessary to introduce an
integral function, , defined by
Hamdan and Kamel, [5], showed that
possesses the integral representation
and provided the following general solution to (1) in
terms of , which they called the Standard
Nield-Kuznetsov Function of the First Kind:
Several properties of were explored by
Hamdan and Kamel, [5], and further features
continue to arise in the literature alongside its non-
trivial computations which invariably rely heavily on
the infinite series representations. In particular, the
following series representations for have been
developed and used in its computations, [5,14,15]:
1.1 Asymptotic Series Approximation:
Based on asymptotic series representations of Airy’s
functions, and , Hamdan and Kamel, [5],
obtained the following asymptotic series for :
where
. If is large, (13) can be
approximated by the following, [9]:
1.2 Ascending Series Representation:
Airy’s particles, and , are defined as, [8, 13]:
wherein is the Gamma function. The Airy’s
particles and the following two series
which employ the Pochhammer symbol, or shifted
factorial
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were employed by Hamdan and Kamel, [5], to obtain
the following expression of :
or, equivalently
Using Cauchy product, (21) can be written as,
[14]:
Series representations (21) and (22) are the
ascending series representations of . They can
be used in the efficient computations of for
small enough values of , [14].
Both asymptotic and ascending series underscore
the importance of investigating properties and
representations of functions associated with the
solutions of Airy’s homogeneous and
inhomogeneous equation. In addition to providing
invaluable insight into the behaviour of solutions to
Airy’s equation and the expanded applications in
mathematical physics, the relationships these arising
functions have with other functions of mathematical
physics serves not only to enrich, but to potentially
expand and deepen mathematical knowledge (cf. [16-
18] and the references therein). This motivates the
current work whose scope is to examine
representations of using Taylor and Maclaurin
series, and its approximations using Taylor and
Maclaurin polynomials.
2 Taylor Series Expansion of
The function is a smooth function with an
derivative expressible in terms of Airy’s
polynomials, [3]. It can therefore be expanded in a
Taylor series, about , of the form:
where
and denotes the derivative of
evaluated at .
If
0
0x
then Taylor series becomes Maclaurin
series, namely:
where
2.1 Derivatives of
In a recent article, Hamdan et.al., [3], obtained
expressions for the higher derivatives of . The
first ten derivatives can be obtained from
equation(10) by direct differentiation, and are
tabulated below.
Table 1. The first ten derivatives of
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The second and higher derivatives of can
be expressed in terms of , and
with coefficients that are
polynomials in . The derivative of , for
, can then be expressed as
For derivative orders , Table 2 lists
:
Table 2. Polynomial Coefficients of ,
and
k
2
x
0
1
3
1
x
0
4
2
x
5
4x
3
6
7
8x
8
9
10
More generally, the degrees of these polynomials
may be determined for arbitrary , and are provided
in the following Table 3 in terms of the floor
function.
Table 3. Degrees of Coefficient Polynomials
Polynomial
Degree
3
3
3
The k+1st derivative of takes the form:
In order to be able to compute the k+1st derivative
with the knowledge of the derivative, Hamdan
et.al., [3], established the following relationships
between the polynomial coefficients in (27) and (28):
Once the derivative is known, the polynomial
coefficients from (27) can be used in (29)-(31) to
compute the polynomial coefficients in (28).
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Now, using (27) in (24), the following
coefficients are obtained in the Taylor series
expansion of :
(32)
and (23) can be written as:
(33)
In order to find , and in terms
of previously calculated polynomials, (29)-(31)
are used in the form:
(34)
(35)
(36)
Using (34)-(36) in (32), the following
coefficients are obtained:
(37)
Equation (45)(33) can then by replaced by the
following final form of Taylor series expansion
of about :
(38)
If then and
(39)
(40)
Equation (38) becomes the following Maclaurin
series expansion of :
(41)
3 Convergence of Taylor Series of
This series converges for values of satisfying
, where is the radius of convergence
defined by
where
is finite since the maximum
degrees of the polynomials involved in and
are comparable. Hence, the radius of
convergence is infinite and series (38) converges
for all . The same is true for Maclaurin series (41).
This furnishes the following Theorem on
convergence.
Theorem 1.
The Taylor series expansion (38) of about
converges for all values of .
4 Values of Polynomials and
Derivatives of at
Values of polynomials and
derivatives of at are shown in the
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following Table 4 for At the outset, it is
noted that for any given derivative, only one of the
polynomials is non-zero at . Furthermore,
is non-zero whenever is non-zero,
and the following recursive relations can easily be
established:
Table 4. Values of Coefficient Polynomials and
Derivatives of at
k
2
0
0
1
3
1
0
0
4
2
0
0
5
6
0
7
0
8
9
0
10
0
11
12
0
13
0
14
15
0
It may also be convenient to note the closed
formulae for the respective values as represented in
the following table. As indicated, the values depend
on the congruence of modulo 3, and employ the
triple factorial notation,
Table 5. Values of Coefficient Polynomials and
at as related to congruence of
modulo 3.
k
0
0
0
0
0
0
0
2
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5 Taylor Polynomial Approximation
to
If the Taylor series of is terminated after n+1
terms, then a Taylor polynomial, , of degree n
results. This polynomial approximates the function
near
0
xx
, namely
Equation (48) takes the following form in terms of
Airy’s polynomials:
If
0
0x
then the Taylor polynomial becomes
Maclaurin polynomial, :
where can be generated using (43).
As an example, the 14th degree Maclaurin polynomial
approximation of takes the form
6 Remainder and Error Terms
When approximating by an degree Taylor
polynomial, , an error term,
, is introduced. Explicitly, is given by:
On an arbitrary interval around ,
continuity of and each of it’s derivatives
deems that is bounded, say
. As such, Taylor’s inequality
provides
For all . Consequently,
Taking limits in (55) shows
In other words, is equal to it’s Taylor Series
(everywhere).
7 Tangent Line Approximation
If n=1 then Taylor polynomial approximation to
becomes:
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In the first derivative of , Airy’s
polynomials are not involved. Therefore, using (10),
the following expressions for and are
obtained, respectively
Using (59) and (60) in (58) yields
Equation (61) is the tangent line approximation to
near . It is written here in terms of
Airy’s functions and integrals.
Equation (58) also gives an approximation to the
slope of the tangent line, , in terms of the
slope of the secant line, namely
If , then the right-hand-side of (59) is zero
and
8 Sample Results
In using 10 terms of series ascending series (21),
Alzahrani et.al. obtained the following values for
when 10 decimal places are retained:
In using asymptotic series (13), which is valid for
, the following approximation is obtained:
0.2071421427
It is believed that the computed value of is
more accurate when the ascending series is used.
By comparison, in using Maclaurin polynomial,
(52), of various degrees, the following
approximations are obtained for (Table 6) and
(Table 7) while retaining 10 decimal places.
All Maclaurin polynomials used give an excellent
agreement with the values computed using ascending
series (21).
Table 6. Computed Values of Using
Maclaurin Polynomials
)
-0.1591549430
-0.1671126902
-0.1672547928
-0.1672560818
-0.1672568251
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Table 7. Computed Values of Using
Maclaurin Polynomials
)
-0.0015915494
-0.001591628977
-0.0015916289784
-0.0015916289784
-0.0015916289784
9 Conclusion
In this work, Taylor and Maclaurin series expansions
of the Standard Nield-Kuznetsov function of the first
kind, , were obtained in order to provide further
insight into the behavior of this integral function.
Convergence criteria were also investigated in order
to show that Taylor series representation of
converges for all . Errors incurred in representing
this function by Taylor and Maclaurin polynomials
were quantified and tangent line approximation was
obtained. Results obtained in computing using
Maclaurin polynomial agree well with results
obtained using ascending series representation for
small values of x.
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Contribution of individual authors
Both authors reviewed the literature, formulated the
problem, provided independent analysis, and jointly
wrote and revised the manuscript.
Sources of funding
No financial support was received for this work.
Creative Commons Attribution
License 4.0 (Attribution 4.0
International , CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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