Schwarz problem in lens and lune

Begehr, Heinrich; Vaitekhovich, Tatyana

doi:10.1080/17476933.2013.799152

Cited by 33 publications

(11 citation statements)

References 5 publications

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“…z = m 2 z−1 z−m 2 , is reflected at ∂ D −m 1 (r 1 ) onto z re = −αz+β βz−α . Observing now |z| 2 − 2Re[m 2 z] + 1 = 0 shows |αz re + β| 2 − 2Re[m 2 (αz re + β)(βz re + α)] + |βz re + α)| 2 = 0, see [5], compare also [10], [11]. Lemma 1.1 For z ∈ D its reflection 1/z at the unit circle ∂D is reflected at ∂ D −m 1 (r 1 ) onto…”

mentioning

confidence: 73%

“…Namely,

z \in \partial D_{m_{2}} false(r_{2} false)

, i.e.

\bar{z} = \frac{m_{2} z - 1}{z - m_{2}}

, is reflected at

\partial D_{- m_{1}} false(r_{1} false)

onto

z_{r e} = \frac{- α z + β}{β z - α} .

Observing now

{| z |}^{2} - 2 Re false[m_{2} z false] + 1 = 0

shows

false| α z_{r e} {+ β |}^{2} - 2 Re false[m_{2} (α z_{r e} + β) (β \bar{z_{r e}} + α) false] + false| β \bar{z_{r e}} + α {) |}^{2} = 0

, see , compare also , .…”

Section: Parqueting Of the Complex Plane Through Reflections Of A Stripmentioning

confidence: 99%

“…A hyperbolic strip in the unit disc

D = {| z | < 1}

is the intersection of

boldD

with the complements of two non‐intersecting closed discs with non‐void intersection with

boldD

the boundaries of which are orthogonal to the unit circle

\partial D = {| z | = 1}

. (Hyperbolic half planes are treated in with the parqueting‐reflection principle , .) For simplicity the two discs are chosen as

D_{- m_{1}} false(r_{1} false)

and

D_{m_{2}} false(r_{2} false)

where

0 < m_{1}, m_{2}, m_{1}^{2} = r_{1}^{2} + 1, m_{2}^{2} = r_{2}^{2} + 1,

satisfy

0 < r_{1}, r_{2}, r_{1} + r_{2} < m_{1} + m_{2}

so that

- 1 < r_{1} - m_{1} < 0 < m_{2} - r_{2} < 1 .

Here

D_{m} false(r false) = {false| z - m false| < r} .

Then the strip is

D = D ∖ false(\bar{D_{- m_{1}} (r_{1})} \cup \bar{D_{m_{2}} (r_{2})} false) = {false| z false| < 1, r_{1} < false| z + m_{1} false|, r_{2} < false|

…”

Section: Parqueting Of the Complex Plane Through Reflections Of A Stripmentioning

confidence: 99%

“…A hyperbolic strip in the unit disc D = {|z| < 1} is the intersection of D with the complements of two nonintersecting closed discs with non-void intersection with D the boundaries of which are orthogonal to the unit circle ∂D = {|z| = 1}. (Hyperbolic half planes are treated in [11] with the parqueting-reflection principle [1], [10].) For simplicity the two discs are chosen as D −m 1 (r 1 ) and D m 2 (r 2 ) where 0 < m 1 , m 2 , m 2 1 = r 2 1 + 1, m 2 2 = r 2 2 + 1, satisfy 0 < r 1 , r 2 , r 1 + r 2 < m 1 + m 2 so that −1 < r 1 − m 1 < 0 < m 2 − r 2 < 1.…”

Section: Parqueting Of the Complex Plane Through Reflections Of A Stripmentioning

confidence: 99%

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Neumann function for a hyperbolic strip and a class of related plane domains

Akel

Begehr

2016

Mathematische Nachrichten

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The harmonic Neumann function is constructed for a class of hyperbolic strips inside the unit disc via the parqueting‐reflection principle. As it turns out this Neumann function is related to the respective Green function, see [the second author, Green function for a hyperbolic strip and a class of related plane domains, Appl. Anal. 93(2014), 2370–2385], in the same way as for the case of e.g. the unit disc or half planes, etc. On this basis the Neumann problem for the Poisson equation is solved in an explicit way. Such explicit solutions serve for applications as in engineering or mathematical physics.

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mentioning

confidence: 73%

“…Namely,

z \in \partial D_{m_{2}} false(r_{2} false)

, i.e.

\bar{z} = \frac{m_{2} z - 1}{z - m_{2}}

, is reflected at

\partial D_{- m_{1}} false(r_{1} false)

onto

z_{r e} = \frac{- α z + β}{β z - α} .

Observing now

{| z |}^{2} - 2 Re false[m_{2} z false] + 1 = 0

shows

false| α z_{r e} {+ β |}^{2} - 2 Re false[m_{2} (α z_{r e} + β) (β \bar{z_{r e}} + α) false] + false| β \bar{z_{r e}} + α {) |}^{2} = 0

, see , compare also , .…”

Section: Parqueting Of the Complex Plane Through Reflections Of A Stripmentioning

confidence: 99%

“…A hyperbolic strip in the unit disc

D = {| z | < 1}

is the intersection of

boldD

with the complements of two non‐intersecting closed discs with non‐void intersection with

boldD

the boundaries of which are orthogonal to the unit circle

\partial D = {| z | = 1}

. (Hyperbolic half planes are treated in with the parqueting‐reflection principle , .) For simplicity the two discs are chosen as

D_{- m_{1}} false(r_{1} false)

and

D_{m_{2}} false(r_{2} false)

where

0 < m_{1}, m_{2}, m_{1}^{2} = r_{1}^{2} + 1, m_{2}^{2} = r_{2}^{2} + 1,

satisfy

0 < r_{1}, r_{2}, r_{1} + r_{2} < m_{1} + m_{2}

so that

- 1 < r_{1} - m_{1} < 0 < m_{2} - r_{2} < 1 .

Here

D_{m} false(r false) = {false| z - m false| < r} .

Then the strip is

D = D ∖ false(\bar{D_{- m_{1}} (r_{1})} \cup \bar{D_{m_{2}} (r_{2})} false) = {false| z false| < 1, r_{1} < false| z + m_{1} false|, r_{2} < false|

…”

Section: Parqueting Of the Complex Plane Through Reflections Of A Stripmentioning

confidence: 99%

Section: Parqueting Of the Complex Plane Through Reflections Of A Stripmentioning

confidence: 99%

See 2 more Smart Citations

Neumann function for a hyperbolic strip and a class of related plane domains

Akel

Begehr

2016

Mathematische Nachrichten

Self Cite

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“…As analogue to this formula, another method can be applied which gives the covering of the entire complex plane C by reflection of the given domain at its boundary. The method is fully described in numerous papers of Begehr and other authors; see, for example [1][2][3][4][5][6][7][8][9][10][11][12]. Our aim is to find the solution of the Dirichlet boundary value problem for the Poisson equation through the Poisson integral formula.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet Problem for Complex Poisson Equation in a Half Hexagon Domain

Shupeyeva

2016

Journal of Complex Analysis

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Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Aksoy

Begehr

Çelebi

2019

Mathematische Nachrichten

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Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.

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Schwarz problem in lens and lune

Cited by 33 publications

References 5 publications

Neumann function for a hyperbolic strip and a class of related plane domains

Neumann function for a hyperbolic strip and a class of related plane domains

Dirichlet Problem for Complex Poisson Equation in a Half Hexagon Domain

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

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