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On the Dirichlet Boundary Value Problem for the Cauchy–Riemann Equations in the Half Disc

Ali Darya1,∗ , Nasir Tagizadeh2

1Faculty of Mathematics Sciences, University of Guilan, Rasht, 19141, Iran
alidarya@phd.guilan.ac.ir

2Faculty of Mathematics Sciences, University of Guilan, Rasht, 19141, Iran
taghizadeh@guilan.ac.ir

∗Correspondence: alidarya@phd.guilan.ac.ir

Abstract. In this article, we investigate the Dirichlet boundary value problem for the Cauchy–Riemann equations in the half disc. First, using the technique of parqueting–reflection and theCauchy–Pompeiu representation formula for a half disc, we obtain an integral representation formulain the half disc. In other words, we construct a unique solution for the Dirichlet boundary valueproblem. Finally, we solve the Dirichlet boundary value problem for both the homogeneous andthe inhomogeneous Cauchy–Riemann equations. In particular, the boundary behaviors at the cornerpoints are considered.

1. Introduction and Preliminaries
Boundary value problems are an essential concept in the field of mathematical analysis andpartial differential equations. They arise when seeking solutions partial differential equationssubject to specific conditions on different parts of the boundary of the domain. The Dirichletboundary value problem is a fundamental concept in mathematical analysis, particularly in thefield of partial differential equations. It deals with finding a solution to a partial differentialequation that satisfies certain prescribed conditions on the boundary of a given domain.One of the most powerful tools for constructing solutions to the Dirichlet problem is the integralrepresentation formula. It provides a way to express the solution in term of an integral over theboundary of domain, which can often simplify the problem and lead to explicit solutions. This formulaallows for the efficient and accurate computation of solutions wide range of partial differentialequations, making it an essential tool in the field of partial differential equations.The parqueting–reflection principle is a technique used in constructing integral representationformulas for the Dirichlet boundary value problem. For specific regions of complex plane whoseboundary consists of sub-arcs of circles or straight lines, the parqueting–reflection method for
Received: 13 Feb 2024.
Key words and phrases. Cauchy–Riemann equation, Dirichlet problem, Integral representation formula, Half disc.1

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https://doi.org/10.28924/ada/ma.4.15
https://orcid.org/0009-0009-7723-9718
https://orcid.org/0000-0002-3865-7943


Eur. J. Math. Anal. 10.28924/ada/ma.4.15 2constructing integral representation formula to solve the Dirichlet boundary value problem for theCauchy–Riemann equation is used. If the boundary of the region has the mentioned characteristics,it is possible to create a new region by reflecting the main region with respect to its boundaryand by reflecting this new region with respect to its boundary, another region is obtained. Bycontinuing this process, we reach pieces of the plane that the union of these pieces provides acover for the complex plane. This cover can be achieved through a single reflection or multipleconsecutive reflections or infinite repetitive reflections. Therefore, the reflection of the main regionat its boundary is repeated to achieve a cover for complex plane.Many results have been obtained for boundary value problems of complex partial differentialequations in some particular domains, see, e.g. [1–16]. In the year 2009, Harmonic boundary valueproblem for the Poisson equation in a half disc was presented by H. Begehr and T. Vaitekhovich [2].In 2012, Y. Wang introduced Schwarz-type boundary value problems for the polyanalytic equationin the half unit disc [16].In this article, in addition to introducing the domain of half disc, we want to express the reflections,covering and points that we obtain at each stage. we also construct the integral representationformula using the parqueting–reflection method and investigate the Dirichlet problem. In particular,we study the explicit solvability of the Dirichlet boundary value problem for both the homogeneousand the inhomogeneous Cauchy–Riemann equations in the half disc.In this article, Let M be the half disc domain in the complex plane C defined by
M = {z ∈ C : |z | < 1, Imz > 0}

where D = {z ∈ C : |z | = 1} and the boundary of M is denoted by ∂M . It is formed by arc of thecircle D and a line segment on the real axis from point −1 to 1. See figure 1.

 

Figure 1. Half Disc M

https://doi.org/10.28924/ada/ma.4.15


Eur. J. Math. Anal. 10.28924/ada/ma.4.15 3Now, we introduce some important definitions and properties of complex analytic functions, re-sults which will be required in subsequent sections.
Defining the complex partial differential operators ∂

∂z and ∂
∂z̄ by

∂

∂z
=

1

2
(
∂

∂x
− i

∂

∂y
),

∂

∂z̄
=

1

2
(
∂

∂x
+ i

∂

∂y
).

Let the complex-valued function ω be defined in M and let u and v denote its real and imaginaryparts: ω = u+ iv , where u(x, y) and v(x, y) are real-valued functions. The two partial differentialequations
∂u

∂x
=
∂v

∂y
, (1.1)

∂u

∂y
= −

∂v

∂x
, (1.2)

are called the Cauchy–Riemann equations for the pair of functions u and v . Multiplying the bothsides of the equality (1.2) by i ,
i(
∂u

∂y
+
∂v

∂x
) = 0. (1.3)Adding (1.1) and (1.3) leads to

∂

∂x
(u + iv) + i

∂

∂y
(u −

1

i
v) = 0.

Thus
1

2
(
∂w

∂x
+ i

∂w

∂y
) = 0.

The Cauchy–Riemann equation can be written as ∂ω
∂z̄

= ωz̄ = 0 and this is the condition for ωto be analytic function. Recall the definition of the Pompeiu integral operator
T f (z) = −

1

π

∫
M

f (t)

t − z dξdη,

T f (z) is weakly differentiable with ωz̄ = f when f ∈ Lp(M;C), p > 2 and t = ξ + iη, [14].
2. An Integral Representation Formula for M

In this section, the integral representation formula for the half disc domain is constructed. A con-venient technique for constructing the integral representation formula for domain with boundariesconsisting of arcs and straight lines is given by the parqueting–reflection method. Now using theparqueting–reflection technique for the introduced domain, we obtain a cover for the entire complexplane. This coverage is obtained from three consecutive reflections.
Reflecting any z at circle gives

|z − a| = r ⇒ (z − a) (z̄ − a) = r2 ⇒ zr =
az̄ − aa + r2

z̄ − a .

https://doi.org/10.28924/ada/ma.4.15


Eur. J. Math. Anal. 10.28924/ada/ma.4.15 4Reflecting any z ∈ M at the D, gives
|z | = 1⇒ zz̄ = 1⇒ z∗ =

1

z̄
.

Reflecting any z at the real axis gives z̄ . Therefore, we can obtain the following points
z∗1 =

1

z̄
, z∗2 =

1

z
, z∗3 = z̄ .

Those reflections produce a parqueting of the entire complex plane. To solve the Dirichletboundary value problems for analytic functions the integral representation formula is important.Now, using the Cauchy–Pompeiu representation formula, we construct the integral representationformula for the half disc domain.
Theorem 2.1. Any ω ∈ C1(M;C)

⋂
C(M;C) can be represented as

ω(z) =
1

2πi

∫
∂M

ω(t)

[
1

t − z +
z

tz − 1

]
dt

−
1

π

∫
M

ωt̄(t)

[
1

t − z +
z

tz − 1

]
dξdη, (2.1)

where t = ξ + iη.

Proof. Applying the Cauchy–Pompeiu formula [3]
1

2πi

∫
∂M

ω(t)
dt

t − z −
1

π

∫
M

ωt̄(t)
dξdη

t − z =

{
ω(z) z ∈ M,

0 z /∈ M.
(2.2)

for z ∈ M and z∗2 /∈ M. Substitute the points into the Cauchy–Pompeiu formula (2.2)
ω(z) =

1

2πi

∫
∂M

ω(t)
dt

t − z −
1

π

∫
M

ωt̄(t)
dξdη

t − z . (2.3)
Since point z∗2 is outside the domain M , therefore, the Cauchy–Pompeiu formula is equal to zeroat this point.

0 =
1

2πi

∫
∂M

ω(t)
zdt

tz − 1
−

1

π

∫
M

ωt̄(t)
zdξdη

tz − 1
, (2.4)

By combining the obtained equations, the integral representation formula is obtained. �

The integral representation formula (2.1), serves to solve the related the Dirichlet problem forthe Cauchy–Riemann equations in M .
3. Dirichlet Problem for the Cauchy–Riemann Equation in M

In this section, we study the Dirichlet boundary value problem for the homogeneous and the in-homogeneous Cauchy–Riemann equations. In the following, we solve the Dirichlet boundary valueproblem for the homogeneous Cauchy–Riemann equation.

https://doi.org/10.28924/ada/ma.4.15


Eur. J. Math. Anal. 10.28924/ada/ma.4.15 5

Theorem 3.1. The Dirichlet problem for the homogeneous Cauchy–Riemann equation ωz̄ = 0, in M,

ω = γ, on ∂M, γ ∈ C(∂M;C)
(3.1)

with given γ ∈ C(∂M;C), γ(±1) = 0, is solvable, if and only if
1

2πi

∫
∂M

γ(t)

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt = 0, (3.2)

and the unique solution can be presented as

ω(z) =
1

2πi

∫
∂M

γ(t)

[
1

t − z +
z

tz − 1

]
dt, z ∈ M. (3.3)

Proof. Let ω defined by (3.3) be a solution to the Dirichlet problem. Then the equality
ω(z) = γ(t), t ∈ ∂M, (3.4)

holds. We consider the following function
h(z) =

1

2πi

∫
∂M

γ(t)

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt, (3.5)

and take the difference
ω(z)− h(z) =

1

2πi

∫
∂M

γ(t)

[
1

ζ − z +
z

tz − 1

]
dt −

1

2πi

∫
∂M

γ(t)

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt

=
1

2πi

∫
∂M

γ(t)

[
1

t − z −
1

t − z̄ +
z

tz − 1
−

z̄

tz̄ − 1

]
dt

=
1

2πi

∫
∂M∩D

γ(t)

[
t

t − z +
t̄

t̄ − z̄ −
t̄

t̄ − z −
t

t − z̄

]
dt

t

+
1

2πi

∫ 1

−1

γ(s)

[
1

s − z −
1

s − z̄ +
z

sz − 1
−

z̄

s z̄ − 1

]
ds

=
1

2πi

∫
∂M∩D

γ(t)

[
1− |z |2

|t − z |2
−

1− |z |2

|t̄ − z |2

]
dt

t

+
1

2πi

∫ 1

−1

γ(s)

[
z − z̄
|s − z |2

−
z − z̄
|1− zs|2

]
ds.

Studying the boundary behavior of the boundary integral implies computations on the differentparts of the boundary ∂M . For |t0| = 1, Imt0 > 0. As |t̄ − t0|2 6= 0, 1− |t0|2 = 0 and |s − t0|2 =

|1− t0s|2 . Thus,
lim
z→t

(ω(z)− h(z)) = γ(t).For |t0| < 1, Imt0 = 0. Since |t − t0| = |t̄ − t0| , |1− t0s|2 6= 0, t − t̄0 = 0. Thus,
lim
z→t

(ω(z)− h(z)) = γ(t).

Now, we consider the boundary behavior at the corner points. Let
ω(z)− h(z) =

1

2πi

∫
∂M∩D

γ(t)

[
t + z

t − z −
t̄ + z

t̄ − z

]
dt

t

https://doi.org/10.28924/ada/ma.4.15


Eur. J. Math. Anal. 10.28924/ada/ma.4.15 6we can write
ω(z)− h(z) =

1

2πi

∫
∂M∩D

γ(t)
t + z

t − z
dt

t
+

1

2πi

∫
∂M∩D

γ(t)
t̄ + z

t̄ − z
dt

t̄

=
1

2πi

∫
∂M∩D

γ(t)
t + z

t − z
dt

t
−

1

2πi

∫
∂M∩D

γ(t̄)
t + z

t − z
dt

t

=
1

2πi

∫
∂M∩D

Υ(t)
t + z

t − z
dt

twhere
Υ(t) =

{
γ(t), Imz ≥ 0,

−γ(t̄), Imz < 0.From the properties of the Poisson kernel for unit disc [2,3], we have
lim
z→t

(ω(z)− h(z)) = Υ(t).

In particular
lim
z→±1

(ω(z)− h(z)) = γ(±1) = 0,

is seen because of the continuity of Υ at ±1.

Similar to what was done above, from the properties of the Poisson kernel for half plane [2], wehave
lim
z→t

(ω(z)− h(z)) = γ(t). (3.6)
By (3.4) and (3.6), we have

lim
z→t

h(z) = 0, t ∈ ∂M.

Then, from the maximum principle for analytic functions h(z) = 0 for z ∈ M, which is given ascondition (3.2).
Conversely, If the condition (3.2) is is satisfied, then, the analytic function ω can be expressedas

ω(z) = ω(z)− h(z)

=
1

2πi

∫
∂M

γ(t)

[
1

t − z +
z

tz − 1

]
dt −

1

2πi

∫
∂M

γ(t)

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt

=
1

2πi

∫
∂M

γ(t)

[
1

t − z −
1

t − z̄ +
z

tz − 1
−

z̄

tz̄ − 1

]
dt.

Hence,
lim
z→t

ω(z) = γ(t), t ∈ ∂M. (3.7)
Follows again from the properties of the Poisson kernel. �

https://doi.org/10.28924/ada/ma.4.15


Eur. J. Math. Anal. 10.28924/ada/ma.4.15 7In the next stage, we investigate the Dirichlet problem for the inhomogeneous Cauchy–Riemannequation. To solve this problem, we reduce the inhomogeneous problem into a homogeneous one,using definition and properties of the Pompeiu operator, and then find a solution for it using theprevious theorem.
Theorem 3.2. The Dirichlet boundary value problem for the inhomogeneous Cauchy–Riemann equa-
tion

ωz̄ = f (z), z ∈ M, f ∈ Lp(M;C), p > 2,

ω = γ, on ∂M, γ ∈ C(∂M;C), (3.8)
is solvable if and only if for z ∈ M,

1

2πi

∫
∂M

γ(t)

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt =

1

π

∫
M

f (t)

[
1

ζ − z̄ +
z̄

tz̄ − 1

]
dξdη, (3.9)

and its solution can be uniquely expressed as

ω(z) =
1

2πi

∫
∂M

γ(t)

[
1

t − z +
z

tz − 1

]
dt −

1

π

∫
M

f (t)

[
1

t − z +
z

tz − 1

]
dξdη. (3.10)

where t = ξ + iη.

Proof. By the Theorem 2.1, If the Dirichlet problem (3.8) is solvable, its can be expressed in theform of (3.10). Let ϕ(z) = ω(z)− T f (z), by applying the ∂z̄ operator to the function ’ we have,
∂z̄ϕ = ∂z̄ω − ∂z̄T f ⇒ ∂z̄ϕ = f − f = 0,

ϕ = ω − T f ⇒ ϕ = γ − T f .

Then consider the homogeneous Dirichlet problem
ϕz̄ = 0, in z ∈ M, ϕ = γ − T f , on ∂M. (3.11)

which is equivalent to equation (3.11) By the Theorem 3.2 the solvability condition for equation(3.14) is
1

2πi

∫
∂M

(γ(t)− T f (t))

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt = 0,

Using the properties of the integral operator
1

2πi

∫
∂M

T f (t)

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt =

1

π

∫
M

f (t̃)
1

2πi

∫
∂M

[
1

t − z̄ +
z̄

tz̄ − 1

]
dt

t − t̃
d ξ̃dη̃

=
1

π

∫
M

f (t̃)

[
1

t̃ − z̄
+

z̄

t̃ z̄ − 1

]
dξ̃dη̃.

which is just condition (3.9).

https://doi.org/10.28924/ada/ma.4.15


Eur. J. Math. Anal. 10.28924/ada/ma.4.15 8On the other hand, if the condition of solvability (3.9) is satisfied, then (3.10) can be expressedas follows.
ω(z) =

1

2πi

∫
∂M

γ(z)

[
1

t − z +
z

tz − 1
−

1

t − z̄ −
z̄

tz̄ − 1

]
dt.

−
1

π

∫
M

f (z)

[
1

t − z +
z

tz − 1
−

1

t − z̄ −
z̄

tz̄ − 1

]
dξdη. (3.12)

Since the area integral tends to 0 as z → t ∈ ∂M, by the proof of Theorem (3.2), (3.12) impliesthat
lim
z→t

ω(z) = γ(t), t ∈ ∂M.Now, we are going to investigate the uniqueness of the Dirichlet problem solution. Assume that ωand w are two solutions to the Dirichlet problem, therefore we have
ωz̄ = f , in z ∈ M, ω = γ on ∂M,

wz̄ = f , in z ∈ M, w = γ, on ∂M.By subtracting the above two relations, we conclude that
(ω − w)z̄ = 0, in M

ω − w = 0. in M,this completes the proof. �

Acknowledgments
The authors would like to express their sincere gratitude to the Editor in Chief, Associate Editorand Referees for their valuable comments that led to considerable improvement of the article.

Declarations
There is no conflict of interest related to the present research. The work has not been publishedbefore and is not under consideration elsewhere. This research received no particular grant fromany funding agency.

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[6] H. Emkanpour, N. Taghizadeh, Three boundary value problems of the Cauchy–Riemann equation in eclipse domain,Complex Var. Epllitic Equ. 67 (2022) 510–529.[7] Y. Gao, Y. Zhao, B. Zhao, Boundary value problems of holomorphic vector functions in 1D QCs, Physica B. 394(2007) 56–61.[8] V.V. Karachik, Class of Neumann-type problems for the polyharmonic equation in a ball, Comput. Math. Math. Phys.60 (2020) 144–162.[9] V.G. Nikolaev, Schwarz problem for j-analytic functions in an ellipse, Comput. Math. Math. Phys. 62 (2022) 1089–1111.[10] A.V. Petukhov, A.O. Savchenko, Solution of the exterior boundary value problem for the Helmholtz equation usingoverlapping domain decomposition, Comput. Math. Math. Phys. 62 (2020) 784–796.[11] K. Ravikumar, K. Ramkumar, D. Chalishajar, Existence and stability results for second-order neutral Stochasticdifferential equations with random impulses and Poisson jumps, Eur. J. Math. Anal. 1 (2021) 1.[12] N. Taghizadeh, M. Mirzazadeh, F. Farahrooz, Exact solutions of the nonlinear Schrödinger equation by the firstintegral method, J. Math. Anal. Appl. 374 (2011) 549–553.[13] N. Taghizadeh, V.S. Mohammadi, Some boundary value problems for the Cauchy–Riemann equation in half lens,Eurasian Math J. 9 (2018) 73–84.[14] I.N. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford, 1962.[15] Y. Wang, X. Zhao, Schwarz boundary value problem for the Cauchy–Riemann equation in a rectangle, Bound. ValueProbl. 2016 (2016) 7.[16] Y. Wang, Schwarz-type boundary value problems for the polyanalytic equation in the half unit disc, Complex Var.Epllitic Equ. 57 (2012) 983–993.

https://doi.org/10.28924/ada/ma.4.15

	1. Introduction and Preliminaries
	2. An Integral Representation Formula for M
	3. Dirichlet Problem for the Cauchy–Riemann Equation in M
	Acknowledgments
	Declarations
	References

