Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points
UDC 517.5 The knowledge of the location of zeros and poles Padé and $N$-point Padé approximations to a given function $f$ provides much valuable information about the function being studied.In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious z...
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| author | Jedynak, R. Gilewicz , J. Jedynak, Radoslaw Jedynak, R. Gilewicz , J. |
| author_facet | Jedynak, R. Gilewicz , J. Jedynak, Radoslaw Jedynak, R. Gilewicz , J. |
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| description | UDC 517.5
The knowledge of the location of zeros and poles Padé and $N$-point Padé approximations to a given function $f$ provides much valuable information about the function being studied.In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious zeros and poles appear randomly.Then, it is clear that the control of the position of poles and zeros becomes essential for applications of Padé approximation method.The numerical examples included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and poles.We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because we are interested in the efficiency of numerical application of these approximations. These functions belong to the class of complex-symmetric functions.The PA and NPA to the Stieltjes functions in different regions of the complex plane is also analyzed. It is expected that the appropriate selection of the complex point for the definition of approximant can improve it with respect to the traditional choice of $\zeta = 0.$All considered cases are graphically illustrated.Some unique numerical results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them. |
| doi_str_mv | 10.37863/umzh.v73i8.333 |
| first_indexed | 2026-03-24T02:02:26Z |
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| fulltext |
DOI: 10.37863/umzh.v73i8.333
UDC 517.5
R. Jedynak, J. Gilewicz (K. Pulaski Univ. Technology and Humanities, Radom, Poland)
DISTRIBUTIONS OF ZEROS AND POLES OF \bfitN -POINT PADÉ APPROXIMANTS
TO COMPLEX-SYMMETRIC FUNCTIONS DEFINED AT COMPLEX POINTS
РОЗПОДIЛ НУЛIВ I ПОЛЮСIВ \bfitN -ТОЧКОВИХ НАБЛИЖЕНЬ ПАДЕ
КОМПЛЕКСНО-СИМЕТРИЧНИХ ФУНКЦIЙ,
ВИЗНАЧЕНИХ У КОМПЛЕКСНИХ ТОЧКАХ
The knowledge of the location of zeros and poles Padé and N-point Padé approximations to a given function f provides
much valuable information about the function being studied. In general PAs reproduce the exact zeros and poles of
considered function, but, unfortunately, some spurious zeros and poles appear randomly. Then, it is clear that the control of
the position of poles and zeros becomes essential for applications of Padé approximation method. The numerical examples
included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and
poles. We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because
we are interested in the efficiency of numerical application of these approximations. These functions belong to the class of
complex-symmetric functions. The PA and NPA to the Stieltjes functions in different regions of the complex plane is also
analyzed. It is expected that the appropriate selection of the complex point for the definition of approximant can improve
it with respect to the traditional choice of \zeta = 0. All considered cases are graphically illustrated. Some unique numerical
results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them.
Знання про мiсцезнаходження нулiв i полюсiв наближень Паде та N -точкових наближень Паде для даної функ-
цiї f надає дуже важливу iнформацiю щодо цiєї функцiї. Взагалi, наближення Паде повторюють точнi нулi та
полюси функцiї, але, на жаль, можуть з’являтись iншi нулi та полюси. Зрозумiло, що контроль позицiї нулiв i по-
люсiв є важливим для застосувань методу наближень Паде. Числовi приклади, що наведенi у роботi, демонструють
необхiднiсть знати позицiю нулiв i полюсiв для того, щоб гарантувати збiжнiсть наближення Паде. Нашi дослi-
дження позицiї полюсiв i нулiв наближень Паде та N -точкових наближень Паде виконано для функцiй Стiльтьєса,
оскiльки нас цiкавлять ефективнi числовi застосування таких наближень. Цi функцiї належать до класу комплексно-
симетричних функцiй. Також дослiджено наближення Паде та N -точковi наближення Паде для функцiй Стiльтьєса
у рiзних регiонах комплексної площини. Очiкується, що правильний вибiр комплексної точки для визначення на-
ближення може покращити її вiдносно стандартного вибору \zeta = 0. Для всiх розглянутих випадкiв надано вiдповiднi
iлюстрацiї. Деякi наведенi у статтi унiкальнi числовi результати є достатньо типовими й повиннi спонукати читача
замислитися над ними.
1. Introduction. In the paper published in 1976 by Chisholm et al. [2], the authors claimed that all
poles and zeros of diagonal Padé approximants (PAs) [n/n] to \mathrm{l}\mathrm{n}(1 - z) developed at the complex
point \zeta interlace on the cut \zeta + t(1 - \zeta ), t \in ]1,\infty [. Klarsfeld remarked in 1981 [7] that the zeros
do not follow this rule. The study of this problem was the starting motivation of two Ph.D. theses
directed by the second author [5, 8]. The primary theoretical results obtained from these theses were
published in [6]. However, some interesting numerical observations remain unexplained. Some of
these open problems are presented in this paper.
The paper is organized as follows. In Section 2, we recall the basic terminology related to PAs,
complex-symmetric functions and Stieltjes functions. We also summarize the classical results of
the location of the zeros and poles of PAs to the Stieltjes functions. In Section 3, we discuss the
distribution of the zeros and poles of PAs in a few cases of the development points of PAs to \mathrm{l}\mathrm{n}(1 - z)
and - 1
z
\mathrm{l}\mathrm{n}(1 - z). Section 4 contains a theorem which describes the location of the zeros and poles
c\bigcirc R. JEDYNAK, J. GILEWICZ, 2021
1034 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1035
of PAs at complex and complex conjugate points. Section 5 provides the appropriate theorems to
N-point Padé approximants (NPAs) of the complex-symmetric functions. In Section 6, we describe
numerical observations related to the specific location of zeros and poles of PAs. The problem of
convergence of PA is also considered. Finally, the conclusion contains some concluding remarks. At
the end of the paper, there is an Appendix. It includes the extra material on the problems discussed in
the main party of the manuscript. The paper is illustrated with various numerical examples, mainly
in the form of graphs, to introduce the reader to the problems discussed in the relevant sections.
2. PA, complex-symmetric functions and Stieltjes functions. Let f be an analytic function
having at the points \zeta 1, . . . , \zeta N \in \BbbC the power expansions:
Ci(z) =
pi - 1\sum
k=0
cik(z - \zeta i)
k +O
\Bigl(
(z - \zeta i)
pi
\Bigr)
, i = 1, . . . , N.
Then the NPA [m/n] to f at the points \zeta 1, . . . , \zeta N \in \BbbC noted [11]
[m/n]p1p2...pN\zeta 1\zeta 2...\zeta N
(z) =
Pm(z)
Qn(z)
=
a0 + a1z + . . .+ amzm
1 + b1z + . . .+ bnzn
,
where
m+ n+ 1 = p = p1 + p2 + . . .+ pN
is defined by
f(z) - [m/n](z) = O
\Bigl(
(z - \zeta i)
pi
\Bigr)
, i = 1, 2, . . . , N. (1)
We always suppose that it exists and then (1) leads to
Qn(z)f(z) - Pm(z) = O
\Bigl(
(z - \zeta i)
pi
\Bigr)
, i = 1, 2, . . . , N,
representing m + n + 1 linear equations for unknowns a0, a1, . . . , am, b1, . . . , bn. If N = 1 this
reduces to the classical one-point PA. A compact formula giving this PA at z = 0 is as follows:
Pm(z)
Qn(z)
=
C(m)(z) zC(m - 1)(z) z2C(m - 2)(z) . . . znC(m - n)(z)
cm+1 cm cm - 1 . . . cm - n+1
cm+2 cm+1 cm . . . cm - n+2
...
...
...
...
...
cm+n cm+n - 1 cm+n - 2 . . . cm
1 z z2 . . . zn
cm+1 cm cm - 1 . . . cm - n+1
cm+2 cm+1 cm . . . cm - n+2
...
...
...
...
...
cm+n cm+n - 1 cm+n - 2 . . . cm
,
where
C(k)(z) =
\left\{
\sum k
j=0
cjz
j if k \geq 0,
0 otherwise.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1036 R. JEDYNAK, J. GILEWICZ
The doubly infinite array of PA is called Padé table
n
m
0 1 2 . . .
0 [0/0] [0/1] [0/2] . . .
1 [1/0] [1/1] [1/2] . . .
2 [2/0] [2/1] [2/2] . . .
...
...
...
...
...
The first column in this table contains the truncated power series. In this paper, we are concerned
by PA to the Stieltjes functions
f(z) =
1/R\int
0
d\mu (t)
1 - tz
, d\mu \geq 0, R \geq 0, z \in \BbbC \setminus [R,\infty [, (2)
which belong to the largest class of complex-symmetric functions, i.e., functions satisfying the con-
dition
f(z) = f(z). (3)
In particular, if \zeta and \zeta are two complex conjugate points, then
f(z) =
\infty \sum
n=0
cn(z - \zeta )n and f(z) =
\infty \sum
n=0
cn(z - \zeta )n.
Classical results [3]:
1. All poles and zeros of PA [n - 1/n] and [n/n] to the Stieltjes functions (2) at z = 0 interlace
on the cut ]R,\infty [. This property is illustrated in Fig. 1. It is due to a fact that the numerators and
denominators of these PA are related to the orthogonal polynomials P and Q by Pn(x) = xnPn(1/x)
and Qn(x) = xnQn(1/x), respectively. The zeros of the orthogonal polynomials are located inside
of the support of the measure defining those polynomials.
−1
−0.5
0
0.5
1 10 100
Im(z)
Re(z)
zeros [7/8]
poles [7/8 ]
point of development
−1
−0.5
0
0.5
1 10 100
Im(z)
Re(z)
zeros [8/8]
poles [8/8]
point of development
(a) (b)
Fig. 1. Distributions of zeros and poles of (a) PA [n - 1/n] and (b) PA [n/n] for n = 8 to - 1
z
\mathrm{l}\mathrm{n} (1 - z) at z = 0
which are plotted on a semi-logarithmic scale. They interlace on the cut ]R,\infty [.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1037
2. The sequences of these diagonal and subdiagonal PA converge uniformly to the Stieltjes
function in C[R,\infty [.
3. Let us consider the set \{ [n\lambda /n]\} of PA on the sloping diagonal \lambda . Fig. 2 (a) presents the
distributions of zeros [n\lambda /n] to the Stieltjes function
\biggl(
- 1
z
\mathrm{l}\mathrm{n} (1 - z)
\biggr)
for three cases of \lambda =
= \{ 10, 100, 1000\} . We observe that with the increase of this parameter those points tend to the circle
of convergence. We also examine the impact of n when the parameter \lambda is set. Fig. 2 (b) shows the
distributions of zeros for \lambda = 4 and n = \{ 2, 4, 8\} . They look very similar to each other. \lambda = \infty
corresponds to the PA [n/0] and \lambda = 1 to the diagonal PA [n/n]. All zeros of PA [n/0] are located
in the vicinity of the circle of convergence of the Taylor series of the Stieltjes function. This property
is documented in Fig. 2 (c). If \lambda (1 < \lambda < \infty ) decreases to 1 (\lambda \rightarrow 1), the zeros of PA remove from
the circle of convergence increasing the region of convergence D\lambda of PA limited by the position of
zeros (see Fig. 1). The exceeding zeros, i.e., n\lambda - n zeros which are not located on the cut, put
away, surrounding the cut, and (or) go to infinity. This problem was first studied by Baker [1], but
his formula (16.10) is false:
\lambda - 1
\lambda + 1
must be replaced by
\lambda - 1
2
which changes the result radically.
The correct limits of these regions were given in [3, p. 267].
−2.5
−2
−1
0
1
2
−3 −2 −1 0 1 2
Im(z)
Re(z)
zeros [10/1]
zeros [100/1]
zeros [1000/1]
point of development
−5
−4
−2
0
2
4
−6 −4 −2 0 2 4
Im(z)
Re(z)
zeros [8/2]
zeros [16/4]
zeros [32/8]
point of developmen t
−1.5
−1
−0.5
0
0.5
1
−1.5 −1 −0.5 0 0.5 1
Im(z)
Re(z)
zeros [7/0]
zeros [8/0]
point of development
(a) (b) (c)
Fig. 2. Distributions of zeros of (a) PA [\lambda n/n] for \lambda = \{ 10, 100, 1000\} to - 1
z
\mathrm{l}\mathrm{n} (1 - z) at z = 0, (b) PA [4n/n]
for n = \{ 2, 4, 8\} to - 1
z
\mathrm{l}\mathrm{n} (1 - z) at z = 0, and (c) PA [n/0] for n = \{ 7, 8\} to - 1
z
\mathrm{l}\mathrm{n} (1 - z) at z = 0.
These results show how necessary for the convergence of PA is the knowledge of the position
of their zeros and poles. In general PAs reproduce the exact zeros and poles of considered function,
but, unfortunately, some spurious zeros and poles appear randomly. Then it is clear that the control
of the position of poles and zeros becomes essential for applications of PA method. Some beautiful
theorems of convergence of PA outside the set of measure zero or of capacity zero are inapplicable
in practice because these sets are not localizing. Since we are interested in the efficiency of the
numerical application of PA, we relate here our research of localization of poles and zeros of PA
in the case of Stieltjes functions. We also analyze the quality of PA to the Stieltjes functions in
different regions of the complex plane expecting that the appropriate choice of the complex point for
the definition of PA can improve it with respect of the traditional choice of \zeta = 0.
At the end of this section, we shortly present the example of using the method mentioned before
of location zeros of PA to the inverse Langevin function which is a complex-symmetric one, but it
is not related to the Stieltjes function. It is an essential function because it is extensively used in
magnetism and polymer physics. In papers [12 – 14], we derived different approximation formulas to
that function. We evaluated the radius of the convergence of the Taylor series of the inverse Langevin
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1038 R. JEDYNAK, J. GILEWICZ
function by the procedure by Mercer and Roberts and obtained value r = 0.904 for 1500 nonzero
coefficients of the Taylor series expansion. Below we present the truncated Taylor series expansion
of this function
\scrL - 1 (x) = 3x+
9x3
5
+
297x5
175
+
1539x7
875
+O
\bigl(
x9
\bigr)
.
It can be transformed to more convenient form for numerical computations
f(x) =
1 - x2
3x
\scrL - 1 (x) = 1 - 2x2
5
- 6x4
175
+
18x6
875
+O
\bigl(
x9
\bigr)
.
Fig. 3 presents the distributions of zeros of PA [4n/n] to those mentioned before functions for
n = \{ 25, 50, 75\} . Both functions have the same radius of convergence.
−2
−1.5
−1
−0.5
0
0.5
1
1.5
−2 −1.5 −1 −0.5 0 0.5 1 1.5
Im(z)
Re(z)
zeros [100/25]
zeros [200/50]
zeros [300/75]
convergence circle
−2
−1.5
−1
−0.5
0
0.5
1
1.5
−2 −1.5 −1 −0.5 0 0.5 1 1.5
Im(z)
Re(z)
zeros [100/25]
zeros [200/50]
zeros [300/75]
convergence circle
(a) (b)
Fig. 3. Distributions of zeros of PA [4n/n] for n = \{ 25, 50, 75\} to (a) the inverse Langevin function \scrL - 1 (x) and
(b) modified one f(x) =
1 - x2
3x
\scrL - 1 (x) at \zeta = 0.
3. Zeros and poles of PAs to \bfl \bfn (\bfone - \bfitz ) and -
\bfl \bfn (\bfone - \bfitz )
\bfitz
. The first function studied in [2]
is related to the Stieltjes function
f(z) =
1\int
0
dx
1 - xz
= - 1
z
\mathrm{l}\mathrm{n}(1 - z)
defined in the cut-plane \BbbC \setminus [1,\infty [. The zeros and poles of PA defined by a power series of f
expanded at the real points interlace on the cut [1,\infty [. One can question what happens if PA is
defined at the complex point \zeta ? Klarsfeld remarked [7] that the Chisholm result [2] is wrong and
showed that only the poles of PA to \mathrm{l}\mathrm{n}(1 - z) follow the cut \zeta + t(1 - \zeta ), t \geq 1, as mentioned in
Introduction, but not the zeros. This result was generalized in [6] to all m \geq n. For the convenience
let us introduce the following simplified notation for the cut:
\zeta + t(R - \zeta ), t \geq 1, [R,\infty [(\zeta )
directed by the straight line joining the point of development of f : z = \zeta with the branch point
z = R. Let us consider the following power expansions:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1039
if \zeta = 0, then
f(z) = \mathrm{l}\mathrm{n} (1 - z) =
\infty \sum
n=0
cnz
n = -
\infty \sum
n=1
1
n
zn,
if \zeta \not = 0, then
f(z) = \mathrm{l}\mathrm{n} (1 - \zeta ) + \mathrm{l}\mathrm{n}
\biggl(
1 - z - \zeta
1 - \zeta
\biggr)
=
\infty \sum
n=0
c\ast n(z - \zeta )n = \mathrm{l}\mathrm{n} (1 - \zeta ) -
\infty \sum
n=1
1
n
\biggl(
z - \zeta
1 - \zeta
\biggr) n
, (4)
where
c\ast 0 = \mathrm{l}\mathrm{n} (1 - \zeta ), c\ast n =
cn
(1 - \zeta )n
, n \geq 1.
Theorem 3.1 [6]. Let [m/n] and [m/n]\ast ,m \geq n, be the PAs of the function f(z) = \mathrm{l}\mathrm{n} (1 - z)
developed at the points z = 0 and z= \zeta , respectively. If zk, k = 1, 2, . . . , n, denotes a pole of
[m/n], then
z\ast k = \zeta + zk(1 - \zeta )
denotes the pole of [m/n]\ast . In other words, the poles of [m/n]\ast locate on the cut
[1,\infty [(\zeta ), \zeta + t(1 - \zeta ), t \geq 1. (5)
The zeros of [m/n]\ast locate out of this line.
One can observe that the form (4) is particular for the function \mathrm{l}\mathrm{n}. Other Stieltjes functions do
not follow this rule, i.e., if f(z) =
\sum
anz
n, then f(z) \not = f(\zeta ) +
\sum
an
\biggl(
z - \zeta
1 - \zeta
\biggr) n
for \zeta \not = 0.
(i) Zeros and poles of PA [\bfitn - \bfone /\bfitn ] and [\bfitn /\bfitn ] to \bfl \bfn (\bfone - \bfitz ). This function has an exceptional
property which leads to the Theorem 3.1 giving the location of poles. As shown in Fig. 4 the zeros
are located left of the straight line of poles, approximately in the vicinity of an oblique straight line,
except one zero. In the case m \geq n, Fig. 4 (b) and (c), the poles lie on the line joining the point of
development of f : z = \zeta with the branch point z = 1. When m < n, Fig. 4 (a), the poles deviate
slightly from that line, except one pole (in this example z8 = 101.244 + 309.316i), which is located
quite distant from the mentioned line.
−50
−40
−30
−20
−10
−2 −1. 5 −1 −0.5 0 0.5 1 1.5
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
−50
−40
−30
−20
−10
−2 −1.5 −1 −0.5 0 0.5 1 1.5
Im(z)
Re(z)
zeros [8/8]
poles [8/8]
point of development
−50
−40
−30
−20
−10
−2 −1.5 −1 −0.5 0 0.5 1 1.5
Im(z)
Re(z)
zeros [9/8]
poles [9/8]
point of development
(a) (b) (c)
Fig. 4. Distributions of zeros and poles of (a) PA [n - 1/n], (b) PA [n/n] and (c) PA [n + 1/n] to \mathrm{l}\mathrm{n}(1 - z) at
\zeta = 1 + i for n = 8. Each line connects the point of development of f : z = \zeta with the branch point z = 1.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1040 R. JEDYNAK, J. GILEWICZ
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
zeros [7/8] φ=π/6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
point of development
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
zeros [8/8] φ=π/6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
point of development
(a) (b)
Fig. 5. Distributions of zeros of (a) PA [n - 1/n] and (b) PA [n/n] to \mathrm{l}\mathrm{n}(1 - z) at \zeta = 1 + ei\phi for n = 8 and
\phi = \{ \pi /6, \pi /4, \pi /3, 5\pi /12, \pi /2\} . Each line connects the point of development of f : z = \zeta with the branch
point z = 1.
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
poles [7/8] φ=π/6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
point of development
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
poles [8/8] φ=π/6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
point of development
(a) (b)
Fig. 6. Distributions of poles of (a) PA [n - 1/n] and (b) PA [n/n] to \mathrm{l}\mathrm{n}(1 - z) at \zeta = 1 + ei\phi for n = 8 and
\phi = \{ \pi /6, \pi /4, \pi /3, 5\pi /12, \pi /2\} . Each line connects the point of development of f : z = \zeta with the branch
point z = 1.
This problem was further examined in more detail and included the following cases: a few
equidistant points on the circle of radius 1 around the ramification point, namely \zeta = 1 + ei\phi and
\phi = \{ \pi /6, \pi /4, \pi /3, 5\pi /12, \pi /2\} , and on the other circle \zeta = 1 + i + ei\phi and the same set of \phi
as previous, to analyse the evolution of the location of zeros. The first case is illustrated in Figs. 5
(zeros) and 6 (poles). Figs. 7 (zeros) and 8 (poles) present the second case for the other circle. The
figures mentioned before confirm the same behavior of the location of zeros and poles as in Fig. 4.
In all cases for [n - 1/n] one zero lies near z = 0 and other locate on the straight line which is ro-
tated by a certain angle with reference to a line which connects the point of development of f :
z = \zeta with the branch point z = 1. From Fig. 7 one can conclude that this angle is about
\pi /12 for those points of development presented in this graph. All poles for [n/n] lie on the
straight line which connects the point of development of f : z = \zeta with the branch point z = 1.
Almost all poles for [n - 1/n] lie near the straight line which connects the point of development of
f : z = \zeta with the branch point z = 1, without one pole which locates quite distant from this line.
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1041
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
zeros [7/8] φ=π/6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
point of development
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
zeros [8/8] φ=π/6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
point of development
(a) (b)
Fig. 7. Distributions of zeros of (a) PA [n - 1/n] and (b) PA [n/n] to \mathrm{l}\mathrm{n}(1 - z) at \zeta = 1 + i + ei\phi for n = 8 and
\phi = \{ \pi /6, \pi /4, \pi /3, 5\pi /12, \pi /2\} . Each line connects the point of development of f : z = \zeta with the branch
point z = 1.
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
poles [7/8] φ=π/6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
−10
−8
−6
−4
−2
0
−3 −2 −1 0 1 2
Im(z)
Re(z)
poles [8/8] φ =π /6
φ=π/4
φ=π/3
φ=5π/12
φ=π/2
point of development
(a) (b)
Fig. 8. Distributions of poles of (a) PA [n - 1/n] and (b) PA [n/n] to \mathrm{l}\mathrm{n}(1 - z) at \zeta = 1 + i + ei\phi for n = 8 and
\phi = \{ \pi /6, \pi /4, \pi /3, 5\pi /12, \pi /2\} . Each line connects the point of development of f : z = \zeta with the branch
point z = 1.
(ii) Several progressive deviations from [\bfone ,\infty [(\bfitzeta ) lines of zeros and poles of PA to -
\bfone
\bfitz
\bfl \bfn (\bfone -
- \bfitz ). The zeros of diagonal and subdiagonal PA to f(z) = - 1
z
\mathrm{l}\mathrm{n} (1 - z) computed at \zeta = 0 interlace
on the cut [1,\infty [(0) (Fig. 9).
The zeros of diagonal and subdiagonal PA to f computed at the points 0.2i, . . . , 10i locate on
the under sides of [1,\infty [(\zeta ) consecutive lines (or cuts). The poles of [n/n] and [n - 1/n] (n \leq 8)
locate on the upper sides of these lines. The moving zeros and poles deviate more and more from
[1,\infty [(\zeta ) with | \zeta | forming the shape of the form \wedge . When | \zeta | \rightarrow 0 one receives the distribution of
poles and zeros, which is demonstrated in Fig. 9. From Fig. 10 one concludes that for small | \zeta | such
as two the zeros and poles of diagonal and subdiagonal PA to f locate on [1,\infty [(\zeta ) line.
The location of the zeros of [m/n]\ast remains an open problem. However, the following remarks
can maybe unlock this question. The particular case of a theorem given in [2, p. 217] says that if
[n/n]f is a PA of some function f and \alpha a constant, then
[n/n]f + \alpha = [n/n]f+\alpha .
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1042 R. JEDYNAK, J. GILEWICZ
−1
−0.5
0
0.5
1 10 100
Im(z)
Re(z)
zeros [7/8]
poles [7/8 ]
point of development
−1
−0.5
0
0.5
1 10 100
Im(z)
Re(z)
zeros [8/8]
poles [8/8]
point of development
(a) (b)
Fig. 9. Distributions of zeros and poles of (a) PA [n - 1/n] and (b) PA [n/n] to f(z) = - 1
z
\mathrm{l}\mathrm{n} (1 - z) at \zeta = 0 for
n = 8 which are plotted on a semi logarithmic scale. They interlace on the cut [1,\infty [(0).
−100
−80
−60
−40
−20
0 5 10 15 20 25 30 35
Im(z)
Re(z)
zeros [7/8] k=0.2
poles [7/8] k=0.2
zeros [7/8] k=10
poles [7/8] k=10
point of development
point of development
0
−100
−80
−60
−40
−20
0
0 5 10 15 20 25 30 35
Im(z)
Re(z)
zeros [8/8] k=0.2
poles [8/8] k=0.2
zeros [8/8] k=10
poles [8/8] k=10
point of development
point of development
(a) (b)
Fig. 10. Distributions of zeros and poles of (a) PA [n - 1/n] and (b) PA [n/n] to - 1
z
\mathrm{l}\mathrm{n} (1 - z) at \zeta = ki for
k = \{ 0.2, 10\} and n = 8. Each line connects the point of development of f : z = \zeta with the branch point
z = 1.
It readily leads to the following theorem, where all notations are the same as in Theorem 3.1
except c\ast 0 which is replaced by an arbitrary constant \alpha .
Theorem 3.2. Let g(z) = \mathrm{l}\mathrm{n} (1 - z) - \mathrm{l}\mathrm{n} (1 - \zeta ) = \mathrm{l}\mathrm{n}
\biggl(
1 - z
1 - \zeta
\biggr)
, then
[n/n]g(z) =
Pn
\biggl(
z - \zeta
1 - \zeta
\biggr)
Qn
\biggl(
z - \zeta
1 - \zeta
\biggr) =
p0 + p1
\biggl(
z - \zeta
1 - \zeta
\biggr)
+ . . .+ pn
\biggl(
z - \zeta
1 - \zeta
\biggr) n
1 + q1
\biggl(
z - \zeta
1 - \zeta
\biggr)
+ . . .+ qn
\biggl(
z - \zeta
1 - \zeta
\biggr) n
and all zeros and poles of this PA interlace on the cut [1,\infty [(\zeta ) (5). If \alpha \not = 0, then
[n/n]g+\alpha (z) = [n/n]g(z) + \alpha =
\alpha Qn
\biggl(
z - \zeta
1 - \zeta
\biggr)
+ Pn
\biggl(
z - \zeta
1 - \zeta
\biggr)
Qn
\biggl(
z - \zeta
1 - \zeta
\biggr) =
P \ast
n
\biggl(
z - \zeta
1 - \zeta
\biggr)
Q\ast
n
\biggl(
z - \zeta
1 - \zeta
\biggr) .
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1043
If \alpha = 0, then the poles and also the zeros simulate the cut [1,\infty [(\zeta ). The problem consists in
analyzing the behavior of the zeros of P \ast
n as function of complex \alpha , in particular with \alpha = c\ast 0 =
= \mathrm{l}\mathrm{n} (1 - \zeta ) and, if possible, to bound the position of all zeros. One observes that the zeros of PA of
\mathrm{l}\mathrm{n} (1 - z) correspond to the zeros of P \ast
n = \mathrm{l}\mathrm{n} (1 - \zeta )Qn
\biggl(
z - \zeta
1 - \zeta
\biggr)
+ Pn
\biggl(
z - \zeta
1 - \zeta
\biggr)
.
4. Zeros and poles of PAs at complex and complex conjugate points. In this section
[m/n]f (z - \zeta ) and [m/n]\ast f (z - \zeta ) denote the PAs of a complex-symmetric function f at the point
\zeta and its complex conjugate \zeta , respectively.
Theorem 4.1. Let [m/n] =
Pm
Qn
and [m/n]\ast =
P \ast
m
Q\ast
n
be PAs of a complex-symmetric function f
at \zeta and \zeta . Then the zeros and the poles of [m/n] are complex conjugate of the corresponding zeros
and poles of [m/n]\ast .
Proof. Eq. (3) gives
f(z) =
\infty \sum
i=0
ci(z - \zeta )i =
\infty \sum
i=0
ci(z - \zeta )i, (6)
Qn = 1 + q1(z - \zeta ) + . . . + qn(z - \zeta )n and, due to (6), Q\ast
n = 1 + q1(z - \zeta ) + . . . + qn(z - \zeta )n.
Then the zeros of Q\ast
n are the complex conjugate of those of Qn . The same arguments are used for
Pm and P \ast
m which completes the proof.
This fact is documented in Fig. 11.
−15
−10
−5
0
5
10
15
−2 −1.5 −1 −0.5 0 0.5 1 1.5
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
zeros [7/8]
poles [7/8]
point of development
−10
−5
0
5
10
−2 −1.5 −1 −0.5 0 0.5 1 1.5
Im(z)
Re(z)
zeros [8/8]
poles [8/8]
point of development
zeros [8/8]
poles [8/8]
point of development
(a) (b)
Fig. 11. Distributions of zeros and poles of (a) PA [n - 1/n] and (b) PA [n/n] to \mathrm{l}\mathrm{n}(1 - z) at \zeta = 1+ i and \zeta = 1 - i
for n = 8. The zeros and the poles of [m/n] are complex conjugate of the corresponding zeros and poles of
[m/n]\ast .
5. NPAs of the complex-symmetric functions. The following theorem is the consequence of
Theorem 3.1.
Theorem 5.1 [6]. Let f be a complex-symmetric function, then the zeros and poles of 2-point
PA [m/n]ll
\prime
\zeta \zeta
of f are complex conjugate of the zeros and poles of [m/n]l
\prime l
\zeta \zeta
, where l and l\prime are the
arbitrary integers satisfying the condition l + l\prime = m+ n+ 1.
Fig. 12 illustrates the Theorem 5.1. It includes 3 cases of different distribution of pieces of
information. We observe that when the amount of information approaching the symmetrical form
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1044 R. JEDYNAK, J. GILEWICZ
then zeros and poles tend to the real axis. Fig. 12 (c) shows the situation when the symmetric state is
achieved. Then we observe that all zeros and poles interlace on the real cut [1,\infty [(0).
−4
−2
0
2
4
−2 0 2 4 6 8 10 12
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
zeros [7/8]
poles [7/8]
point of development
−4
−2
0
2
4
−2 0 2 4 6 8 10 12
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
zeros [7/8]
poles [7/8]
point of development
−4
−2
0
2
4
−2 0 2 4 6 8 10 12
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
(a) (b) (c)
Fig. 12. Distributions of zeros and poles of (a) [7/8]5 11
1+2i,1 - 2i (rising branch) and [7/8]11 5
1+2i 1 - 2i NPA, (b)
[7/8]7 9
1+2i 1 - 2i and [7/8]9 7
1+2i 1 - 2i, (c) [7/8]8 8
1+2i 1 - 2i to - 1
z
\mathrm{l}\mathrm{n} (1 - z).
Fig. 13 shows distributions of zeros and poles for two different development points (a) \zeta =
= 1 + 0.2i and (b) \zeta = 1 + i. They have the similar distribution as previous ones.
−4
−2
0
2
4
−2 0 2 4 6 8 10 12
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
zeros [7/8]
poles [7/8]
point of development
−4
−2
0
2
4
−2 0 2 4 6 8 10 12
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
zeros [7/8]
poles [7/8]
point of development
(a) (b)
Fig. 13. Distributions of zeros and poles of (a) [7/8]5 11
1+0.2i,1 - 0.2i (rising branch) and [7/8]11 5
1+0.2i 1 - 0.2i NPA,
(b) [7/8]5 11
1+i 1 - i and [7/8]11 5
1+i 1 - i to - 1
z
\mathrm{l}\mathrm{n} (1 - z).
Corollary 5.1 [6]. Let f be a complex-symmetric function. Then the zeros and poles of NPA
[m/n]
l1...lkl
\prime
1...l
\prime
k
\zeta 1...\zeta k\zeta 1...\zeta k
complex conjugate of the zeros and poles of [m/n]
l1...lkl
\prime
1...l
\prime
k
\zeta 1...\zeta k\zeta 1...\zeta k
, where l1 +
+l2 + . . .+ lk + l\prime 1 + l\prime 2 + . . .+ l\prime k = m+ n+ 1.
Theorem 5.2 [6]. Let f be a complex-symmetric function. Then all coefficients of NPA
[m/n]l1...lkl1...lk
\zeta 1...\zeta k\zeta 1...\zeta k
of f, where 2(l1 + l2 + . . .+ lk) = m+ n+ 1, are real.
We performed numerous numerical computations whose results lead to the following theorem.
Theorem 5.3. Let f be a complex-symmetric function. Then all the zeros and poles of NPAs
[n - 1/n]l1...lkl1...lk
\zeta 1...\zeta k\zeta 1...\zeta k
and [n/n - 1]l1...lkl1...lk
\zeta 1...\zeta k\zeta 1...\zeta k
of f, where 2(l1 + l2 + . . . + lk) = m + n + 1, are
real and interlace on the cut of the function.
The general proof of this theorem remains an open problem. We are convinced that the only
candidates between NPAs, whose all the zeros and poles are real, belong to approximants with real
coefficients. It means that they must satisfy Theorem 5.2. We proved it in the simplest form NPAs
[1/2] and [2/1], which is presented in the next example. As shown despite the apparent simplicity of
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1045
the chosen approximants, one gets very complicated mathematical formulas on the coefficients and
then poles and zeros.
Theorem 5.3 is illustrated numerically in Table 1. Only NPAs [3/4] and [4/3] have all the zeros
and poles real. Table 2 confirms that all NPAs from Table 1 have real coefficients.
Table 1. Distributions of zeros and poles of NPA [m/n]4 4
1+2i,1 - 2i to - 1
z
\mathrm{l}\mathrm{n} (1 - z)
[1/6] [2/5] [3/4]
zeros poles zeros poles zeros poles
1.92359 – 4.10447 + 4.57707i 1.61757 – 7.07786 – 20.2624i 1.52964 1.22367
– 4.10447 – 4.57707i 3.46555 – 7.07786 + 20.2624i 3.03277 2.11415
1.36082 1.25579 8.96214 4.18872
2.8988 2.27595 16.0223
3.4331 + 5.92542i 5.09943
3.4331 – 5.92542i
[4/3] [5/2] [6/1]
zeros poles zeros poles zeros poles
– 35.5846 1.2437 – 7.9627 1.3174 – 2.4522 1.5287
1.5843 2.2143 0.5564 + 8.9102i 2.6162 – 0.7666 – 4.0899i
3.2911 4.7095 0.5564 – 8.9102i – 0.7666 + 4.0899i
15.1603 1.793 2.5359
4.8891 2.758 – 3.8783i
2.758 + 3.8783i
Table 2. NPA [m/n]4 4
1+2i,1 - 2i to - 1
z
\mathrm{l}\mathrm{n} (1 - z)
[1/6] =
0.9795 - 0.5092z
0.000143z6 - 0.0004171z5 + 0.003797z4 + 0.001442z3 + 0.193z2 - 1.009z + 1
[2/5] =
0.1775z2 - 0.9026z + 0.9954
- 0.0001489z5 - 0.0008228z4 - 0.05352z3 + 0.5503z2 - 1.401z + 1
[3/4] =
- 0.02399z3 + 0.3245z2 - 1.09252z + 0.9976
0.005759z4 - 0.1356z3 + 0.7899z2 - 1.5913z + 1
[4/3] =
- 0.0003542z4 - 0.005507z3 + 0.2245z2 - 0.9694z + 0.9964
- 0.07709z3 + 0.6297z2 - 1.4679z + 1
[5/2] =
0.0001775z5 + 0.00002975z4 + 0.006z3 + 0.03929z2 - 0.6427z + 0.9878
0.2901z2 - 1.1412z + 1
[6/1] =
- 0.000377z6 + 0.001533z5 - 0.009659z4 + 0.01457z3 - 0.07590z2 - 0.1301z + 0.9194
1 - 0.6541z
Example of analytical solution of distribution of zeros and poles of NPAs [\bfone /\bftwo ]\bftwo \bftwo
\bfitzeta ,\bfitzeta
and [\bftwo /\bfone ]\bftwo \bftwo
\bfitzeta ,\bfitzeta
to -
\bfone
\bfitz
\bfl \bfn (\bfone - \bfitz ). Next we find exact solutions for the locations of zeros and poles in the case
of NPA [1/2]22
\zeta \zeta
(z) to the Stieltjes function f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
. The problem can be expressed as
follows:
[1/2]22
\zeta \zeta
(z) =
a0 + a1z
1 + b1z + b2z2
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
1046 R. JEDYNAK, J. GILEWICZ
The Taylor series expansion to the Stieltjes function f developed in the arbitrary point \zeta is given by
the formula
f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
= c0 + c1(z - \zeta ) +O (z - \zeta )2 . (7)
Theorem 5.4. Let f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
and cn(z) be coefficients of the the Taylor series expan-
sion to the Stieltjes function f(z). Then
cn(z) =
n!( - 1)n - 1 \mathrm{l}\mathrm{n}(1 - z)
zn+1
+
n\sum
k=1
( - 1)n - kn!
k(1 - z)kzn - k+1
.
Proof. We use the general Leibniz rule which states that if f and g are n-times differentiable
functions, then the product fg is also n-times differentiable and its nth derivative is given by
(fg)(n)(x) =
n\sum
k=0
\biggl(
n
k
\biggr)
f (n - k)g(k)(x) =
n\sum
k=1
\biggl(
n
k
\biggr)
f (n - k)g(k)(x) + n!f (n)g(0)(x),
where
\biggl(
n
k
\biggr)
=
n!
k!(n - k)!
is the binomial coefficient and f (0)(x) = f(x), g(0)(x) = g(x).
The following formulas for the nth derivatives of both functions f(x) = x - 1 and g(x) =
= - \mathrm{l}\mathrm{n}(1 - x) are valid
f (n)(x) = ( - 1)nn!x - n - 1,
g(n)(x) = (n - 1)!(1 - x) - n, n \geq 1,
g(0)(x) = - \mathrm{l}\mathrm{n}(1 - x).
Using the general Leibniz rule and mentioned before formulas for the nth derivatives of functions f
and g, we get a general formula for the nth derivative of Stieltjes function - \mathrm{l}\mathrm{n}(1 - z)
z
:
\biggl(
- \mathrm{l}\mathrm{n}(1 - x)
x
\biggr) (n)
(x) = (fg)(n)(x) =
= n!( - 1)nx - n - 1( - \mathrm{l}\mathrm{n}(1 - x)) +
n\sum
k=1
\biggl(
n
k
\biggr)
(k - 1)!( - 1)n - k(n - k)!xk - n - 1(1 - x) - k =
=
n!( - 1)n - 1 \mathrm{l}\mathrm{n}(1 - x)
xn+1
+
n\sum
k=1
( - 1)n - kn!
k(1 - x)kxn - k+1
.
Using Theorem 5.4, one can readily obtain c0 = - \mathrm{l}\mathrm{n} (1 - \zeta )
\zeta
and c1 =
1
\zeta
\biggl(
1
1 - \zeta
- \mathrm{l}\mathrm{n} (1 - \zeta )
\zeta
\biggr)
,
which should be inserted in Eq. (7). Then the problem of 2-point Padé to f(z) (Eq. (7)) can be formu-
late by the use of the following matrix representation of the linear system for unknown coefficients
a0, a1, b1, b2 :
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1047\left[
1 \zeta - c0\zeta - c0\zeta
2
1 \zeta - c0\zeta - c0\zeta 2
0 1 - c0 - c1\zeta - 2c0\zeta - c1\zeta
2
0 1 - c0 - c1\zeta - 2c0\zeta - c1\zeta 2
\right]
\left[
a0
a1
b1
b2
\right] =
\left[
c0
c0
c1
c1
\right] . (8)
It can be expressed more conveniently for the further computation using the simple transformation
of rows of the matrix such as adding or subtraction. For simplification one can use the formulas
like \zeta + \zeta = 2\mathrm{R}\mathrm{e} (\zeta ), \zeta - \zeta = 2 \mathrm{I}\mathrm{m} (\zeta ). Finally, we obtain the following matrix equation (Eq. (9))
whose elements are real. One can note that such a transformation can be performed for any matrix
which results from Theorem 5.2. It is evident that unknown coefficients which are found from this
equation must also be real. It leads to the fact that zero is always real (solution of the linear equation
a0 + a1z = 0) but the poles do not have to be real. This fact will be revisited after finding the
coefficients b1 and b2 :
\left[
1 \mathrm{R}\mathrm{e} (\zeta ) - \mathrm{R}\mathrm{e} (c0\zeta ) - \mathrm{R}\mathrm{e} (c0\zeta
2)
0 \mathrm{I}\mathrm{m} (\zeta ) - \mathrm{I}\mathrm{m} (c0\zeta ) - \mathrm{I}\mathrm{m} (c0\zeta
2)
0 1 - \mathrm{R}\mathrm{e} (c0 + c1\zeta ) - \mathrm{R}\mathrm{e} (2c0\zeta + c1\zeta
2)
0 0 - \mathrm{I}\mathrm{m} (c0 + c1\zeta ) - \mathrm{I}\mathrm{m} (2c0\zeta + c1\zeta
2)
\right]
\left[
a0
a1
b1
b2
\right] =
\left[
\mathrm{R}\mathrm{e} (c0)
\mathrm{I}\mathrm{m} (c0)
\mathrm{R}\mathrm{e} (c1)
\mathrm{I}\mathrm{m} (c1)
\right] . (9)
The general solution for unknown coefficients a0, a1, b1, b2 is given in Appendix A. We provide
the exact formulas for zero and poles. Because they are very complicated, we also present the
solution for the particular case of \zeta = 1 + iy, whose behavior can reflect the general situation. It is
easier to analyse the distribution of the zero and poles in the two-dimensional case than spatial (3D)
one. To summarize the obtained results, we show them below in the form of graphs (Fig. 14). They
clearly illustrate the fact that poles are real and interlace the zero on the real cut [1,\infty [(0).
The same procedure was applied to NPA [2/1]2 2
\zeta ,\zeta
. For simplicity, we only provide a solution in
the form of graphs. Fig. 15 confirms that all the zeros and poles are real and interlace on the real
axis.
0
5
10
15
20
25
0 2 4 6 8 y=Im(z)
poles z1
poles z2
zeros z0
(a) (b)
Fig. 14. Distribution of zeros and poles for (a) spatial case [1/2]2 2
x+yi,x - yi NPA to - 1
z
\mathrm{l}\mathrm{n} (1 - z), (b) two-dimensional
case [1/2]2 2
1+yi,1 - yi NPA for continuous values of y = \mathrm{I}\mathrm{m} (z).
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1048 R. JEDYNAK, J. GILEWICZ
−50
−40
−30
−20
−10
0
10
20
30
0 2 4 6 8 y=Im(z)
zeros
zeros
poles
(a) (b)
Fig. 15. Distribution of zeros and poles for (a) spatial case [2/1]2 2
x+yi,x - yi NPA to - 1
z
\mathrm{l}\mathrm{n} (1 - z), (b) two-dimensional
case [2/1]2 2
1+yi,1 - yi NPA for continuous values of y = \mathrm{I}\mathrm{m}(z).
6. Characteristic location of zeros and poles and convergence of PA: numerical observa-
tions. To appreciate the speed of convergence of diagonal PA [n/n] to Sieltjes functions f in
different directions starting from the point \zeta of development of f one analyze [8] the so-called
equierror curves which border the regions defined by
D\varepsilon = z : | f(z) - [n/n]\zeta (z)| \leq \varepsilon
for different \varepsilon . It was remarked that for the points \zeta equidistant with respect to the ramification point
R (here R = 1) the convergence is a little better for \zeta real. More, the convergence is, in general,
better going from R to \infty , and worse in the opposite direction from \zeta to R. The non-rational Stieltjes
functions f defined by (2) in a complex cut plane have no zeros and no poles. The zeros and poles
of diagonal and subdiagonal PA to f defined at z = 0 localize the cut on [R,\infty [. If the point of
development \zeta is complex, the cut can be z(t) = \zeta + t(R - \zeta ), t \geq 1; in the following we note
this line [R,\infty [(\zeta ). If we observe the concentration of poles and zeros of PA at the vicinity of some
lines as [R,\infty [, we have a natural tendency to interpret this as a localization of the cut. However an
essential problem of approximation concerns the domain of convergence, and the convergence of PA
to Stieltjes functions depends on the position of poles and zeros, the choice of the cut being secondary.
Observing the following numerical results, we remark specific order, the proximity of zeros and (or)
poles to certain lines. Visibly these localizations follow some rules. It is to be discovered. In this
paper, we present the results relative to the function \mathrm{l}\mathrm{n} (1 - z), to the Stieltjes function - \mathrm{l}\mathrm{n} (1 - z)
z
and to its inverted function. In fact, other Stieltjes functions as
\pi \surd
1 - z2
or
\surd
1 + z - 1
z
present
similar properties (Fig. 16).
(i) Exceeding zeros and poles of 6-point PA to \bfitf (\bfitz ) = -
\bfl \bfn (\bfone - \bfitz )
\bfitz
. The zeros of truncated
Mclaurin series are located on the convergence circle (Fig. 17).
The two following figures (Fig. 18 (a), (b)) show that the exceeding zeros of [12/3], [22/3], [30/3]
and the exceeding poles of [3/12], [3/22], [3/30], progress to the convergence circle. The six consid-
ered points of development are: i, - i, 1 + i, 1 - i, - 1 + 2i, - 1 - 2i.
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1049
−3
−2
−1
0
1
2
3
0 5 10 15
Im(z)
Re(z)
zeros [5/6]
poles [5/6]
point of development
−3
−2
−1
0
1
2
3
−40 −30 −20 −10 0
Im(z)
Re(z)
zeros [5/6]
poles [5/6]
point of development
(a) (b)
Fig. 16. Distributions of zeros and poles of NPA [5/6]2 2 2 2 2 2
1+2i1 - 2i - 1+2i - 1 - 2i3+2i3 - 2i to (a) f(z) =
\pi \surd
1 - z2
and
(b) f(z) =
\surd
z + 1 - 1
z
.
−1.5
−1
−0.5
0
0.5
1
1.5
−3 −2 −1 0 1 2
Im(z)
Re(z)
zeros [12/0]
zeros [22/0]
zeros [30/0]
point of development
Fig. 17. Distributions of zeros of [12/0], [22/0], [30/0] PAs to f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
at \zeta = 0.
(ii) Power expansions at the symmetric points on the oblique line. Let us consider the deve-
lopment points on the line D\theta
R = \{ z(t) = R + tei(\theta +
\pi
2
); t \in \BbbR \} , symmetrical with respect to the
ramification point R (Fig. 19).
We observed that the zeros and poles of PA to - \mathrm{l}\mathrm{n} (1 - z)
z
interlace approximatively on D\theta
\bot R.
That fact is documented in Fig. 20 for three different angles \theta = \{ \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1/2), \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1), \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(2)\} .
Figs. 21 and 22 confirm the similar behavior of the zeros and poles of PA for other combinations of
development points and amount of information taken for PAs.
Unfortunately two poles of [1/2]2 2i 2 - i(z) deviate from this line (Fig. 23).
However, a remarkable regularity was observed: the real part of all zeros and poles located in the
vicinity of D\theta
\bot R are almost equal to the zeros and poles located on the real axis (on D0
\bot R) as shown
in Fig. 24.
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1050 R. JEDYNAK, J. GILEWICZ
−6
−4
−2
0
2
4
6
−6 −4 −2 0 2 4
Im(z)
Re(z)
zeros [12/3]
zeros [22/3]
zeros [30/3]
point of development
−8
−6
−4
−2
0
2
4
6
8
−10 −8 −6 −4 −2 0 2 4
Im(z)
Re(z)
poles [3/12]
poles [3/22]
poles [3/30]
point of development
(a) (b)
Fig. 18. Distributions of (a) zeros exceeding of [12/3], [22/3], [30/3] NPAs and (b) exceeding poles of [3/12], [3/22],
[3/30] to f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
.
Fig. 19. D\theta
R and D\theta
\bot R \bot D\theta
R.
−5
0
5
10
15
20
−2 0 2 4 6 8
Im(z)
Re(z)
zeros [4/5]
poles [4/5]
point of development
perpendicular line
−2
0
4
8
12
16
−2 0 4 8 12 16
Im(z)
Re(z)
zeros [4/5]
poles [4/5]
point of development
perpendicular line
−4
0
4
8
12
16
−5 0 5 10 15 20 25 30
Im(z)
Re(z)
zeros [4/5]
poles [4/5]
point of development
perpendicular line
(a) (b) (c)
Fig. 20. Distributions of zeros and poles of [4/5]2 2 2 2
z1z2z3z4 NPA to f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
: (a) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1/2),z1 =
= 1/2i, z2 = 2 - 1/2i, z3 = 3 - i, z4 = - 1 + i, (b) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1) = \pi /4, z1 = i, z2 = 2 - i,
z3 = 3 - 2i, z4 = - 1 + 2i, (c) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(2), z1 = 2i, z2 = 2 - 2i, z3 = 3 - 4i, z4 = - 1 + 4i.
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1051
0
10
20
30
40
−5 0 10 20
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
perpendicular line
0
10
20
30
40
−5 0 10 20 30
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
perpendicular line
0
10
20
30
40
−10 0 10 20 30 40 50 60 70
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
perpendicular line
(a) (b) (c)
Fig. 21. Distributions of zeros and poles of [7/8]4 4 4 4
z1z2z3z4 NPA to f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
: (a) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1/2),z1 =
= 1/2i, z2 = 2 - 1/2i, z3 = 3 - i, z4 = - 1 + i, (b) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1) = \pi /4, z1 = i, z2 = 2 - i,
z3 = 3 - 2i, z4 = - 1 + 2i, (c) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(2), z1 = 2i, z2 = 2 - 2i, z3 = 3 - 4i, z4 = - 1 + 4i.
−10
0
10
20
30
40
50
60
70
−5 0 10 20 30
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
perpendicular line
−10
0
10
20
30
40
50
60
70
−10 0 10 20 30 40 50 60
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
perpendicular line
−10
0
10
20
30
40
50
60
−20 0 20 40 60 80 100
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
perpendicular line
(a) (b) (c)
Fig. 22. Distributions of zeros and poles of [7/8]2 2 2 2 2 2 2 2
z1z2z3z4z5z6z7z8 NPA to f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
: (a) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1/2),
z1 = 1/2i, z2 = 2 - 1/2i, z3 = 3 - i, z4 = - 1 + i, z5 = 4 - 3/2i, z6 = - 2 + 3/2i, z7 = 5 - 2i,
z8 = - 3 + 2i, (b) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1) = \pi /4, z1 = i, z2 = 2 - i, z3 = 3 - 2i, z4 = - 1 + 2i, z5 = 4 - 3i,
z6 = - 2 + 3i, z7 = 5 - 4i, z8 = - 3 + 4i, (c) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(2), z1 = 2i, z2 = 2 - 2i, z3 = 3 - 4i,
z4 = - 1 + 4i, z5 = 4 - 6i, z6 = - 2 + 6i, z7 = 5 - 8i, z8 = - 3 + 8i.
−2
−1
0
1
2
3
4
0 1 2
Im(z)
Re(z)
zeros [1/2]
poles [1/2]
point of development
perpendicular line
−1
0
1
2
3
0 1 2 3
Im(z)
Re(z)
zeros [1/2]
poles [1/2]
point of development
perpendicular line
−2
−1
0
1
2
3
0 1 2 3 4 5 6
Im(z)
Re(z)
zeros [1/2]
poles [1/2]
point of development
perpendicular line
(a) (b) (c)
Fig. 23. Distributions of zeros and poles of [1/2]2 2
z1z2 NPA to f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
: (a) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1/2), z1 = 1/2i,
z2 = 2 - 1/2i, (b) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(1) = \pi /4, z1 = i, z2 = 2 - i, (c) \theta = \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(2), z1 = 2i, z2 = 2 - 2i.
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1052 R. JEDYNAK, J. GILEWICZ
0
10
20
30
40
50
−10 0 10 20 30 40
Im(z)
Re(z)
zeros [7/8]
poles [7/8]
point of development
zeros [7/8]
poles [7/8]
point of development
Fig. 24. Distributions of zeros and poles of [7/8]4 4 4 4
z1z2z3z4 NPA and [7/8]4 4 4 4
z\ast 1z\ast 2z\ast 3z\ast 4
to f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
for z1 = i,
z2 = 2 - i, z3 = 4 - 3i, z4 = - 2 + 3i and z\ast 1 = 1 + i, z\ast 2 = 1 - i, z\ast 3 = 1 + 3i, z\ast 4 = 1 - 3i.
(iii) PAs of the inverted Stieltjes function. If f is a Stieltjes function, then the function g defined
by
f(z) =
f(0)
1 - zg(z)
is also a Stieltjes function and is called inverted function [1, 3, 4, 9, 10]. The inverted function to
f(z) = - \mathrm{l}\mathrm{n} (1 - z)
z
is
g(z) = - 1
z
- 1
\mathrm{l}\mathrm{n} (1 - z)
.
The zeros and poles of PA [n - 1/n] and [n/n] of g computed at \zeta = i interlace in the vicinity of
the straight line [1,\infty [(i). It is a natural behavior indicating the cut chosen by PA (Fig. 25).
−20
−15
−10
−5
0
−5 0 5 10 15 20
Im(z)
Re(z)
poles [7/8]
zeros [7/8]
point of development
−20
−15
−10
−5
0
5
−5 0 5 10 15 20
Im(z)
Re(z)
poles [8/8]
zeros [8/8]
point of development
(a) (b)
Fig. 25. Distributions of zeros and poles of (a) PA [7/8] and (b) PA [8/8] computed at \zeta = i to g(z) = - 1
z
-
- 1
\mathrm{l}\mathrm{n} (1 - z)
. Each line connects the point of development of f : z = \zeta with the branch point z = 1.
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1053
7. Conclusions. Numerical experiments carried out with PA and NPA developed at the complex
points revealed certain regularity related to their distribution of poles and zeros. It can be seen that
positions of the poles and zeros for PA to the Stieltjes function expanded at \zeta = ki (for numerically
tested k = \{ 0.2, . . . , 10\} ) follow some well-defined corridors. In each case, they are included in a
certain fan (i.e., regions having the shape of a wide “V”) pointed on the ramification point R and
directed by [R,\infty [(R) line. When k \rightarrow 0 they lie on the real axis. An open question consists
of determining the angles of this fan concerning the line [R,\infty [(R) in each case. The paper also
provides some new results relating to the distribution of zeros and poles of PA and NPA using the
Taylor series of functions developed at the complex points and their conjugate points. The theorem
for the case, when all the zeros and poles of NPA are real, is also formulated. It is derived on the base
of the numerous numerical studies. We evidenced it in two cases [1/2] and [2/1] NPAs. At present
many general problems in this field remain still open which can encourage the readers to solve them.
Appendix A. Exact solution for [\bfone /\bftwo ]\bftwo \bftwo
\bfitzeta ,\bfitzeta
NPA to -
\bfone
\bfitz
\bfl \bfn (\bfone - \bfitz ). This section provides the
general solution for unknown coefficients a0, a1, b1, b2 which satisfy the matrix equation (8):
a0 =
=
\mathrm{I}\mathrm{m} (\zeta )(c20c1\zeta \mathrm{I}\mathrm{m} (\zeta ) + c1\zeta (c0)
2 \mathrm{I}\mathrm{m} (\zeta ) - 2c0c0\mathrm{I}\mathrm{m} (c0)\mathrm{R}\mathrm{e} (\zeta ))
\mathrm{I}\mathrm{m}(\zeta (c1\zeta +2c0))\mathrm{I}\mathrm{m}(\zeta (\mathrm{R}\mathrm{e}(c1\zeta )+\mathrm{R}\mathrm{e} (c0) - c0))+\mathrm{I}\mathrm{m}(c1\zeta +c0)\mathrm{I}\mathrm{m}(\zeta (c0\zeta - \mathrm{R}\mathrm{e}(\zeta (c1\zeta +2c0))))
,
a1 =
=
4c0 | c0| 2 (\mathrm{R}\mathrm{e} (\zeta ) - \zeta ) - 4c20c1\mathrm{I}\mathrm{m} (\zeta )2 + (c0)
2 \bigl( \zeta (c1\zeta + 4c0) + c1\zeta
\bigl(
\zeta - 2\zeta
\bigr)
- 4c0\mathrm{R}\mathrm{e} (\zeta )
\bigr)
4(\mathrm{I}\mathrm{m}(\zeta (c1\zeta +2c0))\mathrm{I}\mathrm{m}(\zeta (\mathrm{R}\mathrm{e}(c1\zeta )+\mathrm{R}\mathrm{e}(c0) - c0))+\mathrm{I}\mathrm{m}(c1\zeta +c0)\mathrm{I}\mathrm{m}(\zeta (c0\zeta - \mathrm{R}\mathrm{e}(\zeta (c1\zeta +2c0)))))
,
b1 =
=
\mathrm{I}\mathrm{m} (\zeta (c1\zeta + 2c0))\mathrm{I}\mathrm{m} (c0 - \zeta \mathrm{R}\mathrm{e} (c1)) + \mathrm{I}\mathrm{m} (c1)\mathrm{I}\mathrm{m} (\zeta (\mathrm{R}\mathrm{e} (\zeta (c1\zeta + 2c0)) - c0\zeta ))
\mathrm{I}\mathrm{m} (\zeta (c1\zeta + 2c0))\mathrm{I}\mathrm{m}(\zeta (\mathrm{R}\mathrm{e}(c1\zeta )+\mathrm{R}\mathrm{e}(c0) - c0))+\mathrm{I}\mathrm{m}(c1\zeta +c0)\mathrm{I}\mathrm{m}(\zeta (c0\zeta - \mathrm{R}\mathrm{e}(\zeta (c1\zeta +2c0))))
,
b2 =
=
\mathrm{I}\mathrm{m} ((c1\zeta + c0)\mathrm{I}\mathrm{m} (\zeta \mathrm{R}\mathrm{e} (c1) - c0) + c1\mathrm{I}\mathrm{m} (\zeta (c0 - \mathrm{R}\mathrm{e} (c1\zeta + c0))))
\mathrm{I}\mathrm{m} (\zeta (c1\zeta + 2c0))\mathrm{I}\mathrm{m}(\zeta (\mathrm{R}\mathrm{e}(c1\zeta )+\mathrm{R}\mathrm{e}(c0) - c0))+\mathrm{I}\mathrm{m}(c1\zeta +c0)\mathrm{I}\mathrm{m}(\zeta (c0\zeta - \mathrm{R}\mathrm{e}(\zeta (c1\zeta +2c0))))
.
We present below the exact formulas for zero (\zeta 0) and poles (\zeta 1, \zeta 2) of approximant.
The zero is obtained from the linear equation a0 + a1\zeta 0 = 0:
\zeta 0 = -
4\mathrm{I}\mathrm{m} (\zeta )
\Bigl(
c20c1\zeta \mathrm{I}\mathrm{m} (\zeta ) + c1\zeta (c0)
2 \mathrm{I}\mathrm{m} (\zeta ) - 2c0c0\mathrm{I}\mathrm{m} (c0)\mathrm{R}\mathrm{e} (\zeta )
\Bigr)
4c0 | c0| 2 (\mathrm{R}\mathrm{e} (\zeta ) - \zeta ) - 4c20c1\mathrm{I}\mathrm{m} (\zeta )2 + (c0)
2 \bigl( \zeta (c1\zeta + 4c0) + c1\zeta
\bigl(
\zeta - 2\zeta
\bigr)
- 4c0\mathrm{R}\mathrm{e} (\zeta )
\bigr)
while the poles from the quadratic equation 1 + b1\zeta + b2\zeta
2 = 0:
\zeta 1 =
- b1 -
\surd
\Delta
2b2
, \zeta 2 =
- b1 +
\surd
\Delta
2b2
,
where the discriminant of the quadratic equation is equal
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1054 R. JEDYNAK, J. GILEWICZ
\Delta =
P (\zeta )
Q(\zeta )
,
P (\zeta ) = (\mathrm{I}\mathrm{m} (\zeta (c1\zeta + 2c0))\mathrm{I}\mathrm{m} (c0 - \zeta \mathrm{R}\mathrm{e} (c1)) + \mathrm{I}\mathrm{m} (c1)\mathrm{I}\mathrm{m} (\zeta (\mathrm{R}\mathrm{e} (\zeta (c1\zeta + 2c0)) - c0\zeta )))
2 -
- 4 \mathrm{I}\mathrm{m} ((c1\zeta + c0)\mathrm{I}\mathrm{m} (\zeta \mathrm{R}\mathrm{e} (c1) - c0) + c1\mathrm{I}\mathrm{m} (\zeta (c0 - \mathrm{R}\mathrm{e} (c1\zeta + c0))))(\mathrm{I}\mathrm{m} (\zeta (c1\zeta +
+2c0))\mathrm{I}\mathrm{m} (\zeta (\mathrm{R}\mathrm{e} (c1\zeta ) + \mathrm{R}\mathrm{e} (c0) - c0)) + \mathrm{I}\mathrm{m} (c1\zeta + c0)\mathrm{I}\mathrm{m} (\zeta (c0\zeta - \mathrm{R}\mathrm{e} (\zeta (c1\zeta + 2c0))))),
Q(\zeta ) = (\mathrm{I}\mathrm{m} (\zeta (c1\zeta + 2c0))\mathrm{I}\mathrm{m} (\zeta (\mathrm{R}\mathrm{e} (c1\zeta ) + \mathrm{R}\mathrm{e} (c0) - c0))+
+\mathrm{I}\mathrm{m} (c1\zeta + c0)\mathrm{I}\mathrm{m} (\zeta (c0\zeta - \mathrm{R}\mathrm{e} (\zeta (c1\zeta + 2c0)))))
2,
c0(\zeta ) = - \mathrm{l}\mathrm{n}(1 - \zeta )
\zeta
,
c1(\zeta ) =
1
(1 - \zeta )\zeta
+
\mathrm{l}\mathrm{n}(1 - \zeta )
\zeta 2
.
Those solutions are presented in Fig. 14 (a). To analyse the behavior of distributions zero and poles,
we also chose the particular case of \zeta = 1 + iy and y > 0. In this case, we obtain the relatively
simple results.
a0 =
y(\pi 2(\pi y2 - 4y + \pi ) + 4(\pi y2 + 4y + \pi ) \mathrm{l}\mathrm{n}2(y) - 8\pi (y2 - 1) \mathrm{l}\mathrm{n}(y))
(\pi 2 - 4)(y2 + 1)2
,
a1 = - y(\pi 2(\pi - 2y) + 4(2y + \pi ) \mathrm{l}\mathrm{n}2(y) + 8\pi \mathrm{l}\mathrm{n}(y))
(\pi 2 - 4)(y2 + 1)2
,
b1 = - 2( - 2\pi y3 + (\pi 2 - 4)y2 + 2(\pi y2 + 4y + \pi )y \mathrm{l}\mathrm{n}(y) + 2\pi y + \pi 2 - 4)
(\pi 2 - 4)(y2 + 1)2
,
b2 =
- (4 + \pi 2)y2 + 4\pi y + 4(2y + \pi )y \mathrm{l}\mathrm{n}(y) + \pi 2 - 4
(\pi 2 - 4)(y2 + 1)2
.
The zero is given by the equation
\zeta 0 =
\pi 2(\pi y2 - 4y + \pi ) + 4(\pi y2 + 4y + \pi ) \mathrm{l}\mathrm{n}2(y) - 8\pi (y2 - 1) \mathrm{l}\mathrm{n}(y)
\pi 2(\pi - 2y) + 4(2y + \pi ) \mathrm{l}\mathrm{n}2(y) + 8\pi \mathrm{l}\mathrm{n}(y)
while the discriminant of the quadratic equation
\Delta =
P1(\zeta )
(\pi 2 - 4)2(y2 + 1)4
,
P1(\zeta ) = 4(( - 2\pi y3 + (\pi 2 - 4)y2 + 2(\pi y2 + 4y + \pi )y \mathrm{l}\mathrm{n}(y) + 2\pi y + \pi 2 - 4)2 -
- (\pi 2 - 4)(y2 + 1)2( - (4 + \pi 2)y2 + 4\pi y + 4(2y + \pi )y \mathrm{l}\mathrm{n}(y) + \pi 2 - 4)).
Those results are shown in Fig. 14 (b).
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DISTRIBUTIONS OF ZEROS AND POLES OF N -POINT PADÉ APPROXIMANTS. . . 1055
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2. J. S. R. Chisholm, A. C. Genz, M. Pusterla, A method for computing Feynman amplitudes with branch cuts, J.
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Received 17.09.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 8
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| id | umjimathkievua-article-333 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:26Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/23/10aad4c6f0f87c0827e1d0122e3ebe23.pdf |
| spelling | umjimathkievua-article-3332025-03-31T08:47:35Z Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points Distributions of zeros and poles of N-point Pade approximants to complex-symmetric functions defined at complex points Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points Jedynak, R. Gilewicz , J. Jedynak, Radoslaw Jedynak, R. Gilewicz , J. . UDC 517.5 The knowledge of the location of zeros and poles Padé and $N$-point Padé approximations to a given function $f$ provides much valuable information about the function being studied.In general PAs reproduce the exact zeros and poles of considered function, but, unfortunately, some spurious zeros and poles appear randomly.Then, it is clear that the control of the position of poles and zeros becomes essential for applications of Padé approximation method.The numerical examples included in the paper show how necessary for the convergence of PA is the knowledge of the position of their zeros and poles.We relate our research of localization of poles and zeros of PA and NPA in the case of Stieltjes functions because we are interested in the efficiency of numerical application of these approximations. These functions belong to the class of complex-symmetric functions.The PA and NPA to the Stieltjes functions in different regions of the complex plane is also analyzed. It is expected that the appropriate selection of the complex point for the definition of approximant can improve it with respect to the traditional choice of $\zeta = 0.$All considered cases are graphically illustrated.Some unique numerical results presented in the paper, which are sufficiently regular should motivate the reader to reflect on them. УДК 517.5 Pозподіл нулів і полюсів $n$-точкових наближень паде комплексно-симетричних функцій, визначених у комплексних точках Знання про місцезнаходження нулів і полюсів наближень Паде та $N$-точкових наближень Паде для даної функції $f$ надає дуже важливу інформацію щодо цієї функції.Взагалі, наближення Паде повторюють точні нулі та полюси функції, але, на жаль, можуть з'являтись інші нулі та полюси.Зрозуміло, що контроль позиції нулів і полюсів є важливим для застосувань методу наближень Паде.Числові приклади, що наведені у роботі, демонструють необхідність знати позицію нулів і полюсів для того, щоб гарантувати збіжність наближення Паде.Наші дослідження позиції полюсів і нулів наближень Паде та $N$-точкових наближень Паде виконано для функцій Стільтьєса, оскільки нас цікавлять ефективні числові застосування таких наближень. Ці функції належать до класу комплексно-симетричних функцій.Також досліджено наближення Паде та $N$-точкові наближення Паде для функцій Стільтьєса у різних регіонах комплексної площини. Очікується, що правильний вибір комплексної точки для визначення наближення може покращити її відносно стандартного вибору $\zeta = 0.$Для всіх розглянутих випадків надано відповідні ілюстрації.Деякі наведені у статті унікальні числові результати є достатньо типовими й повинні спонукати читача замислитися над ними. Institute of Mathematics, NAS of Ukraine 2021-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/333 10.37863/umzh.v73i8.333 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 8 (2021); 1034 - 1055 Український математичний журнал; Том 73 № 8 (2021); 1034 - 1055 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/333/9091 Copyright (c) 2021 Radoslaw Jedynak |
| spellingShingle | Jedynak, R. Gilewicz , J. Jedynak, Radoslaw Jedynak, R. Gilewicz , J. Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points |
| title | Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points |
| title_alt | Distributions of zeros and poles of N-point Pade approximants to complex-symmetric functions defined at complex points Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points |
| title_full | Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points |
| title_fullStr | Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points |
| title_full_unstemmed | Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points |
| title_short | Distributions of zeros and poles of $N$-point Padé approximants to complex-symmetric functions defined at complex points |
| title_sort | distributions of zeros and poles of $n$-point padé approximants to complex-symmetric functions defined at complex points |
| topic_facet | . |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/333 |
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