Zeros of an infinite series
$begingroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
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$begingroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
New contributor
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
New contributor
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Let $sum_{j=1}^{infty}a_{j}$ be a convergent series of positive numbers and ${z_{j}}_{j=1}^infty$ a closed discrete subset of the open unit disc $mathbb{D}$. Then $h(z):=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ is a meromorphic function on $mathbb{D}$.
If we only consider the case of infinite sum, does $h(z)=sum_{j=1}^{infty}frac{a_{j}}{z-z_{j}}$ always have infinitely many zeros on $mathbb{D}$? Note that $h$ never vanishes outside $mathbb{D}$.
This question comes from the paper Bounded Projective Functions and Hyperbolic Metrics with Isolated Singularities (Example 1.1 and Question 3.3).
cv.complex-variables
cv.complex-variables
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
New contributor
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 20 at 4:04
Peter Mortensen
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
New contributor
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 20 at 4:04
Peter Mortensen
1635
edited Jan 20 at 4:04
Peter Mortensen
1635
edited Jan 20 at 4:04
Peter Mortensen
1635
1635
asked Jan 19 at 14:03
Yu FengYu Feng
634
New contributor
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 19 at 14:03
Yu FengYu Feng
634
asked Jan 19 at 14:03
Yu FengYu Feng
634
634
New contributor
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Yu Feng is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
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$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
$begingroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
$endgroup$
This was conjectured by J. Borcea, and a counterexample was constructed by J. Langley:
MR2317957
Langley, J. K.
Equilibrium points of logarithmic potentials on convex domains,
Proc. Amer. Math. Soc. 135 (2007), no. 9, 2821–2826.
His counterexample has an additional property that $z_k$ tend to a limit on the unit circle.
However, your question has positive answer under some additional conditions imposed on $z_k$; this is proved in the same paper of Langley.
Notice that a similar question in the plane (under the assumptions $z_ktoinfty$, and
$$sum_k a_k/|z_k|<infty,$$
$f$ is meromorphic in the plane) is unsolved, despite a lot of research on this question.
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
edited Jan 20 at 3:59
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
answered Jan 19 at 14:29
Alexandre EremenkoAlexandre Eremenko
49.9k6138256
49.9k6138256
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
1
1
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
What do you mean by "the is true"?
$endgroup$
– Peter Mortensen
Jan 20 at 2:49
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
$begingroup$
@Peter Mortensen: Thanks. I corrected the misprint.
$endgroup$
– Alexandre Eremenko
Jan 20 at 3:53
add a comment |
Yu Feng is a new contributor. Be nice, and check out our Code of Conduct.
Yu Feng is a new contributor. Be nice, and check out our Code of Conduct.
Yu Feng is a new contributor. Be nice, and check out our Code of Conduct.
Yu Feng is a new contributor. Be nice, and check out our Code of Conduct.
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