On the relation between measures defining the Stieltjes and the inverted Stieltjes functions
A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function.
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Інститут математики НАН України
2010
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| Цитувати: | On the relation between measures defining the Stieltjes and the inverted Stieltjes functions / J. Gilewicz, M. Pindor // Український математичний журнал. — 2010. — Т. 62, № 3. — С. 327–331. — Бібліогр.: 3 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1650022025-02-09T12:51:08Z On the relation between measures defining the Stieltjes and the inverted Stieltjes functions Про зв'язок між мірами, що визначають початкову та обернену функції Стільтьєса Gilewicz, J. Pindor, M. Статті A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function. Встановлено формулу для міри оберненої функції Стільтьєса, що виражена через міру початкової функції Стільтьєса. 2010 Article On the relation between measures defining the Stieltjes and the inverted Stieltjes functions / J. Gilewicz, M. Pindor // Український математичний журнал. — 2010. — Т. 62, № 3. — С. 327–331. — Бібліогр.: 3 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165002 517.5 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті |
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Статті Статті Gilewicz, J. Pindor, M. On the relation between measures defining the Stieltjes and the inverted Stieltjes functions Український математичний журнал |
| description |
A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function. |
| format |
Article |
| author |
Gilewicz, J. Pindor, M. |
| author_facet |
Gilewicz, J. Pindor, M. |
| author_sort |
Gilewicz, J. |
| title |
On the relation between measures defining the Stieltjes and the inverted Stieltjes functions |
| title_short |
On the relation between measures defining the Stieltjes and the inverted Stieltjes functions |
| title_full |
On the relation between measures defining the Stieltjes and the inverted Stieltjes functions |
| title_fullStr |
On the relation between measures defining the Stieltjes and the inverted Stieltjes functions |
| title_full_unstemmed |
On the relation between measures defining the Stieltjes and the inverted Stieltjes functions |
| title_sort |
on the relation between measures defining the stieltjes and the inverted stieltjes functions |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
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Статті |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/165002 |
| citation_txt |
On the relation between measures defining the Stieltjes and the inverted Stieltjes functions / J. Gilewicz, M. Pindor // Український математичний журнал. — 2010. — Т. 62, № 3. — С. 327–331. — Бібліогр.: 3 назв. — англ. |
| series |
Український математичний журнал |
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2025-11-26T00:28:41Z |
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| fulltext |
UDC 517.5
J. Gilewicz (Cent. Phys. Théor., France),
M. Pindor* (Inst. Theor. Phys., Warsaw Univ., Poland)
ON THE RELATION BETWEEN MEASURES DEFINING
THE STIELTJES AND THE INVERTED STIELTJES FUNCTIONS
ПРО ЗВ’ЯЗОК МIЖ МIРАМИ, ЩО ВИЗНАЧАЮТЬ
ПОЧАТКОВУ ТА ОБЕРНЕНУ ФУНКЦIЇ СТIЛЬТЬЄСА
A compact formula is found for a measure of the inverted Stieltjes function expressed by the measure of the
original Stieltjes function.
Встановлено формулу для мiри оберненої функцiї Стiльтьєса, що виражена через мiру початкової функ-
цiї Стiльтьєса.
In 1991 Gilewicz [1] posed an open problem of giving an explicit expression for a
measure of the inverted Stieltjes function. Peherstorfer [2] gave an answer to this question
for a certain class of measures for Stieltjes functions. Here we present a completely
different derivation than the one given in [2], actually a quite elementary one, which is
also valid for more general types of measures.
Let us recall that if g is a Stieltjes function
z ∈ C \ (−∞,−R] : g(z) =
1/R∫
0
dµ(t)
1 + tz
(1)
then it is well known that the function h defined by
g(z) =
g(0)
1 + zh(z)
(2)
is also a Stieltjes function (see, e.g., [3]). The function h is called the inverted Stieltjes
function. Formula (1) defines the function which is analytic in the whole complex z-
plane, except for the cut between −R and −∞. Therefore h has the same analytic
properties. However if we want to write h in a similar form, namely
h(z) =
1/R∫
0
dν(t)
1 + tz
,
then it is not quite obvious what is the relation between the measures dµ and dν.
When the relation (2) is reversed, we get
h(z) =
g(0)
zg(z)
− 1
z
and using (1), it can be written as
h(z) =
∫ 1/R
0
tdµ(t)
1+tz∫ 1/R
0
dµ(t)
1+tz
. (3)
*M. Pindor passed away in 2003.
c© J. GILEWICZ, M. PINDOR, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3 327
328 J. GILEWICZ, M. PINDOR
!
1/R
C
C
Cc
"
Fig. 1. The contour C for the Cauchy integral and contours Cc and Cρ into which C is deformed.
In order to find an integral representation of h we use the standard Cauchy formula.
However we will work in the ζ-plane where ζ = −1/z, because then both branch points
of h lie inside a bounded domain. Therefore, if we set h̃(ζ) = h(−1/ζ), then formula (3)
becomes
h̃(ζ) =
∫ 1/R
0
tdµ(t)
ζ−t∫ 1/R
0
dµ(t)
ζ−t
. (4)
On the other hand
h̃(ζ) =
1
2πi
∫
C
h̃(ξ)dξ
ξ − ζ
,
where the contour C encircles counterclockwise the point ζ and does not contain the
points 0 and 1/R (see Fig. 1).
Now (see Fig. 2) we can deform C in such a way that it becomes the contours Cc
(encircling both branch points and the cut joining them, clockwise) and Cρ (being a
circle of radius ρ > 1/R where we move counterclockwise). Thus
h̃(ζ) =
1
2πi
∫
Cc
h̃(ξ)dξ
ξ − ζ
+
∫
Cρ
h̃(ξ)dξ
ξ − ζ
.
Obviously ∫
Cρ
h̃(ξ)dξ
ξ − ζ
=
2π∫
0
ρieiϕh̃(ρeiϕ)dϕ
ρeiϕ − ζ
.
By (4), if ξ →∞, then
h̃(ξ)→
∫ 1/R
0
tdµ(t)∫ 1/R
0
dµ(t)
.
Therefore when ρ→∞
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
ON THE RELATION BETWEEN MEASURES DEFINING THE STIELTJES . . . 329
C1r C+
C– C2r
1/R
Fig. 2. The final form of the contour Cc.
∫
Cρ
h̃(ξ)dξ
ξ − ζ
→
∫ 1/R
0
tdµ(t)∫ 1/R
0
dµ(t)
2π∫
0
idϕ.
On the other hand the contour Cc can be split into four paths: C+, C1r, C− and C2r as
indicated on Fig. 2. The contours C1r and C2r are circles of radius r arround the points
0 and 1/R, respectively. When r → 0, integrals over C+ and C− become integrals
between 0 and 1/R over the upper and lower lips of the cut joining the branch points,
while integrals over C1r and C2r converge to 0. As a consequence, we deduce
h̃(ζ) =
∫ 1/R
0
tdµ(t)∫ 1/R
0
dµ(t)
+
1
2πi
∫
C+
h̃(z)dz
z − ζ
+
∫
C−
h̃(z)dz
z − ζ
.
Our task now is to express the difference of integrals over the upper and lower lips of the
cut by an integral containing the measure dµ defining the function g. If the bounded and
nondecreasing function µ is differentiable, then dµ(t) may be written in the form µ′(t)dt
and we may directly use (4). In the following we shall use dµ(t) = µ′(t)dt for more
general situations meaning that we consider such dµ(t) that there exists a distribution
µ′(t) with the necessary properties for the existence of the integrals considered. The
same for dν(t) and ν′(t).
In particular we consider a case where µ contains a contribution from a Heaviside
function, i.e., we take
µ(t) = GH(t) + σ(t), H(t) =
1, for t > 0,
0, for t ≤ 0,
where now σ has no jump at t = 0. In this case (4) becomes
h̃(ζ) =
∫ 1/R
0
tdσ(t)
ζ−t
G
ζ +
∫ 1/R
0
dσ(t)
ζ−t
. (5)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
330 J. GILEWICZ, M. PINDOR
Using now the well known Sokhotskij – Plemelj formulae for (5) with ζ = x± iε for the
sum of integrals over C+ and C− we get∫
C+
h̃(z)dz
z − ζ
+
∫
C−
h̃(z)dz
z − ζ
=
= −
1/R∫
0
σ′(x)
[
G+
∫ 1/R
0
σ′(t)dt
]
[
G
x −
∫ 1/R
0
σ′(t)−σ′(x)
t−x dt− σ′(x) log
(
1
Rx − 1
)]2
+ π2(σ′(x))2
dx
x− ζ
.
Going back to z variable we get
h(z) =
∫ 1/R
0
tσ′(t)dt
G+
∫ 1/R
0
σ′(t)dt
+
G+
1/R∫
0
σ′(t)dt
×
×
−
1/R∫
0
σ′(x)dx
x
[(
G
x −
∫ 1/R
0
σ′(t)−σ′(x)
t−x dt− σ′(x) log
(
1
Rx − 1
))2
+ π2(σ′(x))2
]+
+
1/R∫
0
σ′(x)dx
x(1 + zx)
[(
G
x −
∫ 1/R
0
σ′(t)−σ′(x)
t−x dt− σ′(x) log
(
1
Rx − 1
))2
+ π2(σ′(x))2
]
.
This formula can be stated in a more compact form using the fact that
G
x
−
1/R∫
0
σ′(t)− σ′(x)
t− x
dt− σ′(x) log
(
1
Rx
− 1
)
± πiσ′(x) =
1/R∫
0
dµ(t)
x± iε− t
, (6)
where, by (1), we immediately see that the right-hand side is just g(−1/z)/z for z =
= x± iε. Another simplification comes from the following observation:
1/R∫
0
σ′(x)dx
x
[(
G
x −
∫ 1/R
0
σ′(t)−σ′(x)
t−x dt− σ′(x) log
(
1
Rx − 1
))2
+ π2(σ′(x))2
] =
− 1
2πi
1/R∫
0
dx
x
(
G
x − PV
∫ 1/R
0
σ′(t)dt
t−x + πiσ′(x)
) −
−
1/R∫
0
dx
x
(
G
x − PV
∫ 1/R
0
σ′(t)dt
t−x − πiσ′(x)
)
. (7)
The denominators of the two integrals above are
x
G
x
−
1/R∫
0
σ′(t)dt
t− x
= x
1/R∫
0
dµ(t)
x− t
= g
(
− 1
x
)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
ON THE RELATION BETWEEN MEASURES DEFINING THE STIELTJES . . . 331
for x just below and just above the interval [0, 1/R] (i.e., over C− and C+ (see (6))).
Therefore the integrals on the right-hand side of (7) may be written as
1
2πi
∫
Cρ
dz
z
(
G
z −
∫ 1/R
0
σ′(t)dt
t−z
) − 1
2πi
lim
r→0
∫
C1r
dz
z
(
G
z −
∫ 1/R
0
σ′(t)dt
t−z
) (8)
(see Fig. 2) because an integral over the small circle around 1/R converges to 0 with
the radius r of that circle. Now, the integral over Cρ can be deformed to a large circle
of radius ρ going to∞. In this limit the integral is
2πi
∫ 1/R
0
tσ′(t)dt(
G+
∫ 1/R
0
σ′(t)dt
)2 .
To understand what is the limit of the integral over C1r when r → 0, we recall that
z = reiϕ on C1r. Looking now at the explicit form of the denominator of the integrand
in (8) and using
lim
r→0
reiϕ
1/R∫
0
σ′(t)dt
t− reiϕ
= 0,
it follows that if G 6= 0 then the limit of the integral over C1r is 0. The same is true if
G = 0, and
lim
r→0
1/R∫
0
σ′(t)dt
t− reiϕ
=∞.
Finally, if the above limit is finite, then
lim
r→0
∫
C1r
dz
g (−1/z)
=
2πi∫ 1/R
0
σ′(t)dt
t
.
Therefore, since
G+
1/R∫
0
σ′(t)dt = g(0),
then our final formulae are either
h(z) = g(0)
1
2πi
lim
r→0
∫
C1r
dξ
G+ ξ
∫ 1/R
0
dσ(t)
ξ−t
+
1/R∫
0
tdσ(t)
(1 + tz) |g (−1/t)|2
or
ν′(t) = g(0)
δ(t) lim
r→0
1
2πi
∫
C1r
dξ
G+ ξ
∫ 1/R
0
dσ(u)
ξ−u
+
tσ′(t)
|g (−1/t)|2
,
where δ(t) = H ′(t) is the Dirac distribution. In particular this result shows that if the
measure dµ defining g contains a δ (Dirac) at the origin (that is, if G 6= 0), then the
measure dν defining h does not contain δ at the origin, and vice-versa.
1. Gilewicz J. The open problems // J. Comput. Appl. Math. – 1993. – 48. – P. 230.
2. Peherstorfer F. The open problems // Ibid. – 1993. – 48. – P. 230 – 233.
3. Gilewicz J. Approximants de Padé // Lect. Notes Math. – 1978. – 667. – 207 p.
Received 01.12.09
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 3
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