McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions
We derive asymptotic expansions of the large zeros of the Coulomb wave functions and for those of their derivatives. The new expansions have the same form as the McMahon expansions of the zeros of the Bessel functions and reduce to them when a parameter is equal to zero. Numerical tests are provided...
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| Цитувати: | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions. Amparo Gil, Javier Segura and Nico M. Temme. SIGMA 20 (2024), 075, 9 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860289341916446720 |
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| author | Gil, Amparo Segura, Javier Temme, Nico M. |
| author_facet | Gil, Amparo Segura, Javier Temme, Nico M. |
| citation_txt | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions. Amparo Gil, Javier Segura and Nico M. Temme. SIGMA 20 (2024), 075, 9 pages |
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| description | We derive asymptotic expansions of the large zeros of the Coulomb wave functions and for those of their derivatives. The new expansions have the same form as the McMahon expansions of the zeros of the Bessel functions and reduce to them when a parameter is equal to zero. Numerical tests are provided to demonstrate the accuracy of the expansions.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 075, 9 pages
McMahon-Type Asymptotic Expansions
of the Zeros of the Coulomb Wave Functions
Amparo GIL a, Javier SEGURA b and Nico M. TEMME c
a) Departamento de Matemática Aplicada y CC, de la Computación, ETSI Caminos,
Universidad de Cantabria, 39005 Santander, Spain
E-mail: amparo.gil@unican.es
URL: http://personales.unican.es/gila/
b) Departamento de Matemáticas, Estadistica y Computación, Universidad de Cantabria,
39005 Santander, Spain
E-mail: javier.segura@unican.es
URL: http://personales.unican.es/segurajj/
c) Valkenierstraat 25, 1825BD Alkmaar, The Netherlands
E-mail: nic@temme.net
Received February 23, 2024, in final form August 07, 2024; Published online August 10, 2024
https://doi.org/10.3842/SIGMA.2024.075
Abstract. We derive asymptotic expansions of the large zeros of the Coulomb wave func-
tions and for those of their derivatives. The new expansions have the same form as the
McMahon expansions of the zeros of the Bessel functions and reduce to them when a pa-
rameter is equal to zero. Numerical tests are provided to demonstrate the accuracy of the
expansions.
Key words: Coulomb wave functions; McMahon-type zeros; asymptotic expansions
2020 Mathematics Subject Classification: 33C47; 33C10; 33C15; 41A60; 65D20; 65H05
Dedicated to Richard B. Paris,
the man who loved asymptotics
1 Introduction
The Coulomb wave functions Fλ(η, ρ) and Gλ(η, ρ) are two linearly independent solutions of the
differential equation
d2w
dρ2
+
(
1− 2η
ρ
− λ(λ+ 1)
ρ2
)
w = 0.
This equation can be transformed into Kummer’s differential equation and the relation with the
Kummer functions is
Fλ(η, ρ) = A1F1(λ+ 1− iη, 2λ+ 2; 2iρ),
Gλ(η, ρ) = iFλ(η, ρ) + iBU(λ+ 1− iη, 2λ+ 2, 2iρ),
A =
|Γ(λ+ 1 + iη)|e−πη/2−iρ(2ρ)λ+1
2Γ(2λ+ 2)
, B = eπη/2+λπi−iσλ(η)−iρ(2ρ)λ+1,
σλ(η) = phΓ(λ+ 1 + iη) (the Coulomb phase shift). (1.1)
This paper is a contribution to the Special Issue on Asymptotics and Applications of Special Functions in
Memory of Richard Paris. The full collection is available at https://www.emis.de/journals/SIGMA/Paris.html
mailto:amparo.gil@unican.es
http://personales.unican.es/gila/
mailto:javier.segura@unican.es
http://personales.unican.es/segurajj/
mailto:nic@temme.net
https://doi.org/10.3842/SIGMA.2024.075
https://www.emis.de/journals/SIGMA/Paris.html
2 A. Gil, J. Segura and N.M. Temme
Coulomb wave functions and their zeros find application in various fields. In particular, they
play an important role in atomic and nuclear physics where they contribute to understanding
phenomena like electron scattering and bound-state properties (see, for example, [4]).
In physics applications, λ usually has the values 0, 1, 2, . . . , but here we assume that λ and η
are real. The Kummer function in Fλ(η, ρ) is analytic for all complex values of the parameters,
unless 2λ+2 = −1,−2,−3, . . . . But because of the reciprocal gamma function in the quantity A,
the function Fλ(η, ρ) is analytic at these points. If λ > −1, then the function Fλ(η, ρ) disap-
pears at ρ = 0, which follows from the algebraic factor ρλ+1. Because of this factor, if λ /∈ Z,
then Fλ(η, ρ) becomes multivalued with a branch cut along the negative axis. If we are interested
in negative zeros of Fλ(η, ρ), we can use Kummer’s relation for the Kummer function, which in
this case is
1F1(λ+ 1− iη, 2λ+ 2; 2iρ) = e2iρ1F1(λ+ 1 + iη, 2λ+ 2;−2iρ),
and we can use the method for positive zeros by changing the sign of η.
The function Gλ(η, ρ) becomes unbounded when ρ → 0, unless when λ = η = 0, in which
case
F0(0, ρ) = sin ρ, G0(0, ρ) = cos ρ. (1.2)
The asymptotic expansions of the large zeros of the Coulomb functions to be given in this
paper are related with those of the z-zeros of the Bessel functions Jν(z) and Yν(z) and their
derivatives derived by McMahon [5]. To explain what happens for the Bessel function Jν(z), we
use the well-known representation
Jν(z) =
√
2
πz
(cos θ(ν, z)ϕν(z)− sin θ(ν, z)ψν(z)), (1.3)
where θ(ν, z) = z− 1
2νπ−
1
4π. For large values of z, we have ϕν(z) = 1+O
(
1/z2
)
, ψν(z) = O(1/z),
and the complete asymptotic expansions of these functions are derived in [11, Section 7.4] for
the Hankel functions. For details, we refer to [7, Section 10.17 (i)].
The relation in (1.3) with asymptotic expansions of ϕν(z) and ψν(z) was the starting point
for McMahon [5] to derive the asymptotic expansion of the large zeros of the Bessel func-
tion Jν(z) and similarly for related functions. For details on these expansions we refer to [7,
Section 10.21 (vi)].
For the Coulomb functions, the asymptotic expansions of the large zeros, and the methods
to derive these expansions, have much in common with the expansions of the large zeros of the
Bessel functions. This is not surprising because of the following observations.
(1) Firstly, because the formulas for the Coulomb functions given in the next section and used
for deriving asymptotic expansions of the large zeros have the same analytical form as the
one given in (1.3) for Jν(z).
(2) Secondly, for η = 0 we have Fλ(0, ρ) =
√
πρ/2Jλ+ 1
2
(ρ), and we will verify in a special case
that our expansion of the zeros of Fλ(0, ρ) indeed become the McMahon expansion for the
zeros of this Bessel function.
What does surprise us is that we cannot find much information in the literature about the
ρ-zeros of the Coulomb wave functions. Milton Abramowitz [1] paid some attention to the zeros
of Fλ(η, ρ) in his article on asymptotic expansions of the Coulomb functions. His starting point
is the same equation as ours, but he went further using an iteration method requiring function
evaluations; we give more details in Section 3. Ikebe [3] considered the zeros of Fλ(η, ρ) and
its derivative by computing eigenvalues of matrices following from recurrence relations of the
McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions 3
Coulomb functions. See also [6] for error analysis of this approach. Ball [2] used a similar method
for Bessel functions, Coulomb wave functions, and other special functions. This method is not
based on function evaluations nor on asymptotic expansions, it requires eigenvalue computations
of matrices. The early zeros can be computed efficiently. In fact, our method is efficient for the
large zeros, and we show in examples how it performs for the first zeros.
We derive the asymptotic expansions for the zeros of both Fλ(η, ρ) and Gλ(η, ρ), and of their
derivatives. The approach for these four functions is similar and the results are as simple as
McMahon’s expansion for zeros of the Bessel functions. A minor complication is that we need
to solve a nonlinear equation, which can be done by standard numerical methods, although its
solution can be expressed in terms of the Lambert W -function.
2 Summary of used formulas
We summarise a set of formulas from the literature, see [9, Chapter 31] and [10, Section 33.11].
The following functions are important to describe the large ρ asymptotics:
Pλ(η, ρ) = sin(θλ(η, ρ))Fλ(η, ρ) + cos(θλ(η, ρ))Gλ(η, ρ),
Qλ(η, ρ) = cos(θλ(η, ρ))Fλ(η, ρ)− sin(θλ(η, ρ))Gλ(η, ρ), (2.1)
where
θλ(η, ρ) = ρ− η ln(2ρ)− 1
2
λπ + σλ(η), θ′λ(η, ρ) = 1− η/ρ. (2.2)
Here, σλ(η) is the Coulomb phase shift given in (1.1), and the prime denotes the derivative with
respect to ρ.
By inverting (2.1), we have
Fλ(η, ρ) = sin(θλ(η, ρ))Pλ(η, ρ) + cos(θλ(η, ρ))Qλ(η, ρ),
Gλ(η, ρ) = cos(θλ(η, ρ))Pλ(η, ρ)− sin(θλ(η, ρ))Qλ(η, ρ),
F ′
λ(η, ρ) = cos(θλ(η, ρ))Rλ(η, ρ) + sin(θλ(η, ρ))Sλ(η, ρ),
G′
λ(η, ρ) = − sin(θλ(η, ρ))Rλ(η, ρ) + cos(θλ(η, ρ))Sλ(η, ρ), (2.3)
where
Rλ(η, ρ) = Pλ(η, ρ)θ
′
λ(η, ρ) +Q′
λ(η, ρ), Sλ(η, ρ) = P ′
λ(η, ρ)−Qλ(η, ρ)θ
′
λ(η, ρ).
The functions Pλ(η, ρ) and Qλ(η, ρ) can be written in terms of the Kummer U -functions. For
details, we refer to the cited references.
The following asymptotic expansions follow from those of the U -function. We have for large
values of ρ the expansions
Pλ(η, ρ) ∼
∞∑
k=0
pk
(2ρ)k
, Qλ(η, ρ) ∼
∞∑
k=0
qk
(2ρ)k
,
Rλ(η, ρ) ∼
∞∑
k=0
rk
(2ρ)k
, Sλ(η, ρ) ∼
∞∑
k=0
sk
(2ρ)k
. (2.4)
The coefficients of these expansions follow from simple recurrence relations. Initial values are
p0 = 1, q0 = 0, r0 = 1, s0 = 0,
and for k = 0, 1, 2, 3, . . . , we have
(k + 1)pk+1 = ukpk + vkqk, (k + 1)qk+1 = −vkpk + ukqk, uk = η(2k + 1),
vk = k + k2 − λ2 − λ− η2, rk+1 = pk+1 − 2ηpk − 2kqk,
sk+1 = −qk+1 + 2ηqk − 2kpk.
4 A. Gil, J. Segura and N.M. Temme
3 McMahon-type expansions of the zeros
To start obtaining the expansion of the zeros of Fλ(η, ρ), we look at the first line of (2.3). From
the asymptotic expansions in (2.4), we see that
Pλ(η, ρ) = 1 +O(1/ρ), Qλ(η, ρ) = O(1/ρ), ρ→ ∞.
Hence, for a large zero of Fλ(η, ρ) the sine function in the first line of (2.3) should be of or-
der O(1/ρ). We write
θλ(η, ρ) = nπ + δ, n ∈ N, (3.1)
and we will see that δ = O(1/ρ). Using this form of θλ(η, ρ), we obtain
sin(θλ(η, ρ)) = cos(nπ) sin(δ), cos(θλ(η, ρ)) = cos(nπ) cos(δ),
and from the first line of (2.3), we obtain
sin(δ)Pλ(η, ρ) + cos(δ)Qλ(η, ρ) = 0, (3.2)
if we assume Fλ(η, ρ) = 0.
Next, we try to find ρ from equation (3.1). We have
ρ− η ln ρ = η ln 2 +
1
2
λπ − σλ(η) + nπ + δ, (3.3)
where δ is the small quantity introduced in (3.1) that must be found together with the zero ρ
for given values of n, η and λ.
Let ρ0 be the solution of the equation
ρ0 − η ln ρ0 = η ln 2 +
1
2
λπ − σλ(η) + nπ. (3.4)
Because the solution should satisfy ρ0 = O(n) for large n, we need to solve this equation
for ρ0 > η. Comparing (3.3) and (3.4), we conclude (ρ− η ln ρ)− (ρ0 − η ln ρ0) = δ. Also, with
a new quantity ε,
ρ = ρ0 + ε =⇒ δ = ε− η ln
(
1 +
ε
ρ0
)
. (3.5)
With this ρ0, δ and ε we try to find a solution ρ of equation (3.2). First, assuming
that ε = O(1/ρ0) for large ρ0, we introduce the expansion
ε ∼
∞∑
k=1
εk
ρk0
, ρ0 → ∞. (3.6)
Using this expansion, we can obtain an expansion of δ in inverse powers of ρ0 as well, and substi-
tute this, with ρ = ρ0 + δ in (3.2). We use the expansions of the functions Pλ(η, ρ) and Qλ(η, ρ)
given in (2.4), with ρ = ρ0 + ε, and collect equal powers of ρ0 to find the coefficients εk.
In this way, we obtain with v0 = −λ2 − λ− η2,
ε1 =
1
2
v0, ε2 =
1
4
η(3v0 + 1), ε3 =
1
24
(
22η2v0 + 17η2 − 7v20 − 6v0
)
. (3.7)
With these coefficients we find the wanted asymptotic expansion of the solution ρ, denoted
by ρn, of equation (3.3)
ρn ∼ ρ0 +
ε1
ρ0
+
ε2
ρ20
+
ε3
ρ30
+ · · · , n→ ∞. (3.8)
McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions 5
From numerical tests, we conclude that this gives indeed the approximation for the n-th
positive zero. Using the expansion in (3.8), with n = 1, λ = 2, η = 3
2 we find ρ0
.
= 9.186 and
Fλ(η, ρ1)
.
= −0.0269. For n = 10 and this expansion with the same λ, η values, we find ρ0
.
= 39.65
and Fλ(η, ρ10)
.
= 0.0000530. More tests are given in Section 4.
Remark 3.1. Abramowitz wrote in his paper [1] on the asymptotics of the Coulomb wave
functions our formula (3.2) in the form δ = − arctan(Qλ(η, ρ)/Pλ(η, ρ) and used (3.1) and (2.2)
to define the iteration
ρn,s = η ln(2ρn,s−1) +
1
2
λπ − σλ(η) + nπ − arctan
Qλ(η, ρn,s−1)
Pλ(η, ρn,s−1)
,
for s = 1, 2, 3, . . . , where a starting value ρn,0 is needed, and the evaluation of the functions in
the arctan-function.
For the analogues of our functions Pλ(η, ρ) and Qλ(η, ρ), Abramowitz derived asymptotic
expansions for large ρ, with λ = 0, which are similar to those in the first line of (2.4). He did
not use expansions of the Kummer U -function, but he derived the expansions using integral
representations of functions related to the Coulomb functions, just as Hankel functions can be
used for the asymptotics of the Bessel functions.
Abramowitz computed the first three zeros of F0(η, ρ), for a few η-values. See Table 5 in
Section 4 for a selection of these values. He did not give starting values but one may try
ρn,0 = ρ0 defined in (3.4).
Remark 3.2. When η = 0, the F -Coulomb functions become J-Bessel functions. We have
Fλ(0, ρ) =
√
πρ/2Jλ+ 1
2
(ρ) and (3.4) gives ρ0 =
(
1
2λ+ n
)
π. It is not difficult to verify that the
first coefficients given in (3.7) become the first coefficients in McMahon’s expansion of the zeros
of the J-Bessel function, see [7, Section 10.21 (vi)].
For the function Gλ(η, ρ), we see in the second line of (2.1) that for a first approximation
we have to use the zeros of the cosine function, and we change (3.1) and (3.4) by replacing n
with n− 1
2 . This defines δ for this case and with this new θλ(η, ρ) the second line of (2.3) becomes
the same as in (3.2). It follows that the coefficients εk in the expansion of ε in (3.6) are the same
as those for the zeros of Fλ(η, ρ), with different ρ0 in the expansion. This property corresponds
with McMahon’s expansion for the zeros of the Y -Bessel function: the asymptotic expansions
of the zeros of Jν(x) and Yν(x) have the same coefficients, but the series have a different large
parameter.
For the derivative F ′
λ(η, ρ), we change (3.1) and (3.4) similarly, replacing n with n− 1
2 , and
the equation for the zeros corresponding with (3.2) becomes
− sin(δ)Rλ(η, ρ) + cos(δ)Sλ(η, ρ) = 0. (3.9)
This gives new coefficients for the expansion of ε, and we write (3.5) as
ρ̂ = ρ0 + ε̂ =⇒ δ = ε̂− η ln
(
1 +
ε̂
ρ0
)
,
and the expansions corresponding with (3.6) and (3.8) in the form
ε̂ ∼
∞∑
k=1
ε̂k
ρk0
, ρ̂n ∼ ρ0 +
ε̂1
ρ0
+
ε̂2
ρ20
+
ε̂3
ρ30
+ · · · , n→ ∞. (3.10)
The first coefficients are, again with v0 = −λ2 − λ− η2,
ε̂1 =
1
2
v0, ε̂2 =
1
4
η(3v0 − 1), ε̂3 =
1
24
(
22η2v0 − 19η2 − 7v20 + 6v0
)
. (3.11)
6 A. Gil, J. Segura and N.M. Temme
Finally, we consider G′
λ(η, ρ). We take, as for Fλ(η, ρ), (3.1) and (3.4). Hence, the expansions
of the zeros of G′
λ(η, ρ) follow from (3.9) and have the same coefficients as in (3.11).
We summarise for all four cases the choices of θλ(η, ρ), ρ0 and the equation needed for
obtaining the expansions.
(1) Fλ(η, ρ)
θλ(η, ρ) = nπ + δ, n ∈ N,
ρ0 − η ln ρ0 = η ln 2 +
1
2
λπ − σλ(η) + nπ, sin(δ)Pλ(η, ρ) + cos(δ)Qλ(η, ρ) = 0.
(2) Gλ(η, ρ)
θλ(η, ρ) =
(
n− 1
2
)
π + δ, n ∈ N,
ρ0 − η ln ρ0 = η ln 2 +
1
2
λπ − σλ(η) +
(
n− 1
2
)
π,
sin(δ)Pλ(η, ρ) + cos(δ)Qλ(η, ρ) = 0.
(3) F ′
λ(η, ρ)
θλ(η, ρ) =
(
n− 1
2
)
π + δ, n ∈ N,
ρ0 − η ln ρ0 = η ln 2 +
1
2
λπ − σλ(η) +
(
n− 1
2
)
π,
− sin(δ)Rλ(η, ρ) + cos(δ)Sλ(η, ρ) = 0.
(4) G′
λ(η, ρ)
θλ(η, ρ) = nπ + δ, n ∈ N,
ρ0 − η ln ρ0 = η ln 2 +
1
2
λπ − σλ(η) + nπ, − sin(δ)Rλ(η, ρ) + cos(δ)Sλ(η, ρ) = 0.
For all four cases, we have verified by using numerical calculations and graphs that the n
used in these relations approximate the nth positive zero of the Coulomb wave function or its
derivative. Some of the tests performed are shown in the next section. For λ = η = 0, the first
positive zeros have the proper n-value, which trivially follows from (1.2).
4 Numerical tests
For testing the approximations obtained with the McMahon-type expansions
(
ρMc
n
)
for the zeros
of Coulomb functions, we use the numerical method described in [8] implemented in Maple
with a large number of digits. Tables 1, 2, 3 and 4 show tests for the first 10 zeros of the
functions F1.3(2.1, ρ), G1.3(2.1, ρ), F
′
1.3(2.1, ρ) and G′
1.3(2.1, ρ), respectively. We use 6 terms in
the expansions (3.8) and (3.10). High-accuracy numerical values of the zeros of the functions
obtained with the method described in [8], are given in the second column of the tables. The
third column in all tables shows the relative errors obtained in the comparisons. As can be seen,
an accuracy close to 10−8 can be obtained for the largest considered zeros. As expected, the
accuracy of the approximations improves as n increases.
Additionally, a test of the influence of the parameters λ, η on the accuracy of the approxi-
mations ρMc
n to the zeros of Fλ(η, ρ), is shown in Figure 1. In this figure, we show the minimum
McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions 7
n ρMc
n ρn, using the method in [8] Rel. error
1 9.28 . . . 9.276226087098264 6.8× 10−4
2 13.321 . . . 13.32061436693835 5.3× 10−5
3 17.0494 . . . 17.04925305758087 9.0× 10−6
4 20.6332 . . . 20.63316305105047 2.3× 10−6
5 24.13198 . . . 24.13196399208639 7.5× 10−7
6 27.57415 . . . 27.57414717920683 2.9× 10−7
7 30.975729 . . . 30.97572598757761 1.3× 10−7
8 34.346662 . . . 34.34666006955555 6.1× 10−8
9 37.693593 . . . 37.69359261174668 3.2× 10−8
10 41.021189 . . . 41.02118854245900 1.7× 10−8
Table 1. Test for the McMahon-type approximations ρMc
n to the first 10 zeros of the Coulomb func-
tion F1.3(2.1, ρ).
n ρMc
n ρn, using the method in [8] Rel. error
1 6.95 . . . 6.925107084382577 4.9× 10−3
2 11.36 . . . 11.35971565567721 1.6× 10−4
3 15.2094 . . . 15.20913702648054 2.0× 10−5
4 18.8545 . . . 18.85445602183751 4.3× 10−6
5 22.39103 . . . 22.39100849194709 1.2× 10−6
6 25.85895 . . . 25.85894221100473 4.6× 10−7
7 29.27929 . . . 29.27928968958546 1.9× 10−7
8 32.664553 . . . 32.66455053595783 8.7× 10−8
9 36.0228 . . . 36.02279903910762 4.3× 10−8
10 39.359572 . . . 39.35957112638164 2.3× 10−8
Table 2. Test for the McMahon-type approximations ρMc
n to the first 10 zeros of the Coulomb func-
tion G1.3(2.1, ρ).
n ρMc
n ρn, using the method in [8] Rel. error
1 6.8 . . . 6.740012285516214 2.0× 10−2
2 11.34 . . . 11.33586159146655 4.5× 10−4
3 15.2 . . . 15.19947063325694 5.3× 10−5
4 18.8493 . . . 18.84912765706333 1.1× 10−5
5 22.38767 . . . 22.38760195810186 3.2× 10−6
6 25.85659 . . . 25.85656409550572 1.1× 10−6
7 29.27754 . . . 29.27752955366132 4.6× 10−7
8 32.663199 . . . 32.66319220425298 2.1× 10−7
9 36.02172 . . . 36.02171734983164 1.0× 10−7
10 39.35869 . . . 39.35868838281058 5.5× 10−8
Table 3. Test for the McMahon-type approximations ρMc
n to the first 10 zeros of the function F ′
1.3(2.1, ρ).
8 A. Gil, J. Segura and N.M. Temme
n ρMc
n ρn, using the method in [8] Rel. error
1 9.24 . . . 9.226939712774167 2.0× 10−3
2 13.308 . . . 13.30627800305222 1.4× 10−4
3 17.0426 . . . 17.04225058479286 2.3× 10−5
4 20.629 . . . 20.62896049608348 5.7× 10−6
5 24.12918 . . . 24.12914248690917 1.8× 10−6
6 27.57213 . . . 27.57211363372210 7.0× 10−7
7 30.97419 . . . 30.97418664616960 3.1× 10−7
8 34.345457 . . . 34.34545207910902 1.4× 10−7
9 37.69262 . . . 37.69261810059473 7.5× 10−8
10 41.020385 . . . 41.02038500317911 4.1× 10−8
Table 4. Test for the McMahon-type approximations ρMc
n to the first 10 zeros of the function G′
1.3(2.1, ρ).
η n = 2 n = 3
(1)10.974 (1)14.567
η = 1.5 (2)10.97336 (2)14.566337
(3)10.97335 (3)14.566335
(1)12.403 (1)16.110
η = 2 (2)12.4053 (2)16.11047
(3)12.4052 (3)16.11044
(1)13.786 (1)17.596
η = 2.5 (2)13.7885 (2)17.5954
(3)13.7879 (3)17.5953
(1)15.130 (1)19.033
η = 3 (2)15.1349 (2)19.0356
(3)15.1335 (3)19.0352
Table 5. Comparison of values for the second and third zeros of F0(η, ρ) for a few η-values. (1) Values
given in [1]; (2) McMahon-type approximations; (3) Values obtained with the numerical method given
in [8].
value of n for which the relative accuracy obtained with the McMahon-type approximations is
better than 10−6. The calculations have been made by fixing the value of λ and varying the
values of η. The accuracy is checked, as before, using the numerical method given in [8]. Results
obtained for two different values of λ are shown for comparison. Similar results are obtained for
the approximations to the zeros of the other functions.
As a final test, in Table 5, we compare the McMahon-type approximations with some of
the values for the zeros of F0(η, ρ) appearing in the table given in Abramowitz’s paper [1] for
a few η-values. The values obtained with the numerical method [8] are also shown. As can be
seen, few discrepancies in the last digit for some of the values given in [1] are found.
Acknowledgements
We thank the referees for helpful and constructive remarks on an earlier version of the paper. We
acknowledge financial support from Ministerio de Ciencia e Innovación, Spain, project PID2021-
127252NB-I00 (MCIN/AEI/10.13039/501100011033/ FEDER, UE).
McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions 9
Figure 1. Minimum value of n for which the relative accuracy obtained with the McMahon-type ap-
proximations ρMc
n to the zeros of Fλ(η, ρ) is better than 10−6.
References
[1] Abramowitz M., Asymptotic expansions of Coulomb wave functions, Quart. Appl. Math. 7 (1949), 75–84.
[2] Ball J.S., Automatic computation of zeros of Bessel functions and other special functions, SIAM J. Sci.
Comput. 21 (1999), 1458–1464.
[3] Ikebe Y., The zeros of regular Coulomb wave functions and of their derivatives, Math. Comp. 29 (1975),
878–887.
[4] Luna B.K., Papenbrock T., Low-energy bound states, resonances, and scattering of light ions, Phys. Rev. C
100 (2019), 054307, 17 pages, arXiv:1907.11345.
[5] Mcmahon J., On the roots of the Bessel and certain related functions, Ann. of Math. 9 (1894), 23–30.
[6] Miyazaki Y., Kikuchi Y., Cai D., Ikebe Y., Error analysis for the computation of zeros of regular Coulomb
wave function and its first derivative, Math. Comp. 70 (2001), 1195–1204.
[7] Olver F.W.J., Maximon L.C., Bessel functions, in NIST Handbook of Mathematical Functions, Cambridge
University Press, Cambridge, 2010, 215–286.
[8] Segura J., Reliable computation of the zeros of solutions of second order linear ODEs using a fourth order
method, SIAM J. Numer. Anal. 48 (2010), 452–469.
[9] Temme N.M., Asymptotic methods for integrals, Ser. Anal., Vol. 6, World Scientific Publishing, Hackensack,
NJ, 2014.
[10] Thompson I.J., Coulomb wave functions, in NIST Handbook of Mathematical Functions, Cambridge Uni-
versity Press, Cambridge, 2010, 741–756.
[11] Watson G.N., A treatise on the theory of Bessel functions, 2nd ed., Cambridge Math. Lib., Cambridge
University Press, Cambridge, 1944.
https://doi.org/10.1090/qam/28479
https://doi.org/10.1137/S1064827598339074
https://doi.org/10.1137/S1064827598339074
https://doi.org/10.2307/2005300
https://doi.org/10.1103/PhysRevC.100.054307
https://arxiv.org/abs/1907.11345
https://doi.org/10.2307/1967501
https://doi.org/10.1090/S0025-5718-00-01241-2
https://doi.org/10.1137/090747762
https://doi.org/10.1142/9195
1 Introduction
2 Summary of used formulas
3 McMahon-type expansions of the zeros
4 Numerical tests
References
|
| id | nasplib_isofts_kiev_ua-123456789-212345 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T19:24:48Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gil, Amparo Segura, Javier Temme, Nico M. 2026-02-05T09:54:24Z 2024 McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions. Amparo Gil, Javier Segura and Nico M. Temme. SIGMA 20 (2024), 075, 9 pages 1815-0659 2020 Mathematics Subject Classification: 33C47; 33C10; 33C15; 41A60; 65D20; 65H05 arXiv:2402.14537 https://nasplib.isofts.kiev.ua/handle/123456789/212345 https://doi.org/10.3842/SIGMA.2024.075 We derive asymptotic expansions of the large zeros of the Coulomb wave functions and for those of their derivatives. The new expansions have the same form as the McMahon expansions of the zeros of the Bessel functions and reduce to them when a parameter is equal to zero. Numerical tests are provided to demonstrate the accuracy of the expansions. We thank the referees for helpful and constructive remarks on an earlier version of the paper. We acknowledge financial support from Ministerio de Ciencia e Innovación, Spain, project PID2021-127252NB-I00 (MCIN/AEI/10.13039/501100011033/FEDER, UE). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions Article published earlier |
| spellingShingle | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions Gil, Amparo Segura, Javier Temme, Nico M. |
| title | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions |
| title_full | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions |
| title_fullStr | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions |
| title_full_unstemmed | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions |
| title_short | McMahon-Type Asymptotic Expansions of the Zeros of the Coulomb Wave Functions |
| title_sort | mcmahon-type asymptotic expansions of the zeros of the coulomb wave functions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212345 |
| work_keys_str_mv | AT gilamparo mcmahontypeasymptoticexpansionsofthezerosofthecoulombwavefunctions AT segurajavier mcmahontypeasymptoticexpansionsofthezerosofthecoulombwavefunctions AT temmenicom mcmahontypeasymptoticexpansionsofthezerosofthecoulombwavefunctions |