**BULLETIN***of the* *Bull. Malaysian Math. Sc. Soc. (Second Series) 26 (2003) 13−33*
MALAYSIAN* *

MATHEMATICAL SCIENCES SOCIETY

**Eigenproblem of the Generalized Neumann Kernel **

ALI HASSAN MOHAMED MURID AND MOHAMED M.S.NASSER Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia,

81310 UTM Skudai, Johor, Malaysia

e-mail: ahmm@mel.fs.utm.my and mms_nasser@mel.fs.utm.my

*To the memory of Mohamad Rashidi Md. Razali, *
*a friend, a colleague and source of inspiration *

**Abstract. Recently, the Riemann problem in the interior domain of a smooth Jordan curve was **
solved by transforming its boundary condition to a Fredholm integral equation of the second kind
with the generalized Neumann kernel. The eigenvalues λ = ±1 play an important role in the
solvability of these integral equations. In this paper, the necessary and sufficient conditions for

±1

λ = to be eigenvalues of the generalized Neumann kernel are given and the corresponding eigenfunctions are derived. Some examples are presented.

**1. ** **Introduction **

Suppose that Γ :*t* =*t*(*s*),0≤ *s* ≤ β be a smooth Jordan curve, Ω^{+} and Ω^{−} its interior
and exterior respectively such that the origin of the coordinate system belongs to Ω^{+} and

belongs to . The unit tangent to

∞ Ω^{−} Γ at the point t will be denoted by

) ( / ) ( )

(*t* *t* *s* *t* *s*

*T* = ′ ′ . Let and denote the limiting values of the analytic
function when the point z tends to the point

) (t

*f*^{+} *f*^{−}(t)
)

(z

*f* *t*∈Γ from inside and outside of Γ

respectively. Assume that a, *b and *γ be three real functions of the point *t* ∈Γ all
satisfying the Hölder condition and *a*^{2}(*t*)+*b*^{2}(*t*) ≠ 0 for all *t*∈Γ. The Riemann
Problem in Ω^{+} consists of finding all functions *f* = *u*+*iv* that are analytic in Ω^{+},
continuous on Ω^{+} = Ω^{+} ∪Γ with limiting values of the real and imaginary parts on Γ
satisfying the linear relation

(1.1) .

, ) ( ) ( ) ( ) ( )

(*t* *u*^{+} *t* −*b* *t* *v*^{+} *t* = *t* *t*∈Γ

*a* γ

Let *c*(*t*) ≡ *a*(*t*)+i*b*(*t*),*t* ∈Γ, the boundary condition (1.1) may be rewritten as
(1.2)

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{(}

^{)}

^{,}

^{.}

Re *ct* *f*^{+} *t* =γ *t* *t*∈Γ

We assume that *c*(*t*) =1 on which is no loss of generality as can seen by divided
(1.2) by

Γ )

(t

*c* . When γ(*t*) ≡ 0 we are faced with the homogeneous Riemann problem
(1.3)

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{0}

^{,}

^{.}

Re *ct* *f*^{+} *t* = *t*∈Γ

Similarly, the Riemann problem for exterior domain Ω^{−} consists of finding all
functions *f* = *u*+i*v* that are analytic in Ω^{−} (including at ∞), continuous on

Γ, Ω

=

Ω^{−} ^{−} ∪ and satisfy the boundary condition

(1.4) Γ

∈

=

− ^{−}

− *t* *bt* *v* *t* *t* *t*

*u*
*t*

*a*() ( ) ( ) () γ(),
which is equivalent to

(1.5)

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{(}

^{)}

^{,}

^{.}

Re *ct* *f*^{−} *t* =γ *t* *t* ∈Γ
The homogeneous problem of the exterior domain Ω^{−} is given by

(1.6)

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{0}

^{,}

^{.}

Re *ct* *f*^{−} *t* = *t* ∈Γ

Recently, when and have continuous first order derivatives, Murid et al.

[7] solved the Riemann problem using Fredholm integral equations of the second kind.

The kernel of these integral equations is a generalization to the familiar Neumann kernel so it will be call called generalized Neumann kernel.

) (t

*a* *b*(t)

This kernel is very important in solving Dirichlet problem [3] and Riemann problem
[7] using Fredholm integral equations. The solvability of the Riemann problem (1.2) and
(1.5) depends on the index of the function *c*(t) with respect to the curve Γ, for
definition of the index see [2, pp. 85−89]. However, the solvability of the related integral
equations depend on the eigenvalues λ = ±1. It is found that the possibility of λ = ±1
to be an eigenvalue of the generalized Neumann kernel depends on the index of *c*(t).

The organization of this paper is as follow. In Section 2, we give a brief derivation of the integral equations with generalized Neumann kernel related to the Riemann problem in the interior and exterior domains. Section 3 contains the proof of the continuity of the generalized Neumann kernel and some of its properties. In Section 4, the necessary and sufficient condition for which λ = ±1 is an eigenvalue of the generalized Neumann kernel is given. Section 5 contained two examples. The conclusions are given in Section 6.

**2. ** **Fredholm integral equations related to Riemann problem **

With the help of the Sokhotskyi formula [2, p.25] we are able to extend the results
obtained in [6] and [7] which will play a key role in deriving an integral equation related
to (1.5). Suppose that γ is a real function defined on Γ and satisfies the Hölder
condition. Suppose also that *c*(t) is a complex valued function defined on Γ such that

is continuous on Γand for all )

(t

*c*′ *c*(*t*) ≠ 0 *t* ∈Γ and define the function in
by

)
(z
*L*
Γ

\
**C**

) . )(

( ) ( 2 i 2 ) 1

( *dt*

*z*
*t*
*t*
*c*
*z* *t*

*L* =

### ∫

Γ γ −π (2.1)

Theorem 2.1 derives integral equation related to Riemann problem in the interior
domain Ω^{+} of a smooth Jordan curve Γ. The proof can be found in [7].

**Theorem 2.1. [7] Let **Γ* be a smooth Jordan curve and *Ω^{+}* be its interior. Suppose *
*that * * is an analytic solution of the Riemann problem (1.2) in the interior domain *

. Define * then * * satisfies the integral equation *
)

(z
*f*

Ω+ *g*(*t*) = *c*(*t*)*f*^{+}(*t*),*t* ∈Γ, *g*(t)

. ) ( ) ( )

( ) , ( ) ( PV )

(*t* *N* *c* *t* *w* *g* *w* *dw* *ct* *L* *t*

*g* −

### ∫

_{Γ}= −

^{−}

^{(2.2) }

*where *

. and ,

) , ( ) (

) Im ( ) 1

, )(

( *t* *w* *t* *w*

*t*
*w*

*w*
*T*
*w*
*c*

*t*
*w* *c*

*t*
*c*

*N* ⎥ ∈Γ ≠

⎦

⎢ ⎤

⎣

⎡

= −

π ^{(2.3) }

Similarly we can obtain an integral equation related to Riemann problem (1.5) in the
exterior domain Ω^{−} with an additional condition *f*(∞) = 0.

**Theorem 2.2. ** *Let *Γ* be a smooth Jordan curve and *Ω^{−}* be its exterior. Suppose that *
* is an analytic solution of the Riemann problem (1.4) in the exterior domain *
)

(*z*

*f* Ω^{−}

*with the condition * *f*(∞) = 0.* Define * * then * * satisfies the integral *
*equation *

, ) ( ) ( )

(*t* *ct* *f* *t*

*g* = ^{−} *g*(t)

. ) ( ) ( )

( ) , )(

( PV )

(*t* *N* *c* *t* *wg* *w* *dw* *ct* *L* *t*

*g* +

### ∫

_{Γ}=

^{+}

^{(2.4) }

*Proof: Let * *f*(z) be a solution of (1.4) in Ω^{−} then *f*(z) is analytic in Ω^{−} and
continuous on Ω^{−} , hence *f*^{−}(*t*) = *f*(*t*), *t*∈Γ. From (1.5) we have

) ( 2 ) ( ) ( ) ( )

(*t* *f* *t* *ct* *f* *t* *t*

*c* + = γ which leads to

. ) ,

( ) ( ) 2 ) ( (

) ) (

( = − + *t*∈Γ

*t*
*c*
*t* *t*
*t* *f*
*c*

*t*
*t* *c*

*f* γ

(2.5)

According to [2, p. 2], *f*(z) satisfies

. ,

) 0 ( i 2

1 +

Γ = ∈Ω

### ∫

*−*

_{w}^{f}

^{w}_{z}

^{z}π ^{(2.6) }

Taking the limit Ω^{+} ∋ *z* →*t*∈Γ and applying Sokhotskyi formula to (2.6), we get

### ∫

Γ =+ (−) 0.

i 2 PV 1 ) 2 (

1 *dw*

*t*
*w*

*w*
*t* *f*

*f* π ^{(2.7) }

Conjugating both side of (2.7) the using (2.5), we get

) 0 (

) ( ) 2 ) ( ( ) (

) ( i

2 PV 1 ) (

) ( ) 2 ) ( (

) ( 2

1 ⎥⎦ =

⎢ ⎤

⎣

⎡ −

+ −

⎥⎦

⎢ ⎤

⎣

⎡ −

−

### ∫

Γ*dw*

*w*
*c*
*w* *w*
*t* *f*
*w*
*w*
*c*

*w*
*c*
*t*

*c*
*t* *t*
*t* *f*
*c*

*t*

*c* γ

π γ

which leads to

*t* *dw*
*w*

*w*
*f*
*w*
*c*

*w*
*t* *c*

*t* *f*
*c*

*t*
*c*

+ −

−

### ∫

Γ) ( ) (

) ( i 2 PV 1 ) ) ( (

) ( 2 1

π

) . )(

( ) ( 2 i 2 PV 1 ) (

) ( 2 2

1 ^{−}

Γ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

+ −

−

= _{c}^{γ}_{t}^{t}_{π}

### ∫

_{c}_{w}^{γ}

_{w}^{w}

_{t}

^{dw}^{ (2.8) }

Using the fact that *T*(*w*) *dw* = *dw* and from the definition of *L*(z), (2.8) becomes

) ( )

) ( ( ) (

) ( i 2 PV 1 ) ) ( (

) ( 2

1 *f* *w* *dw* *L* *t*

*t*
*w*

*w*
*T*
*w*
*c*

*w*
*t* *c*

*t* *f*
*c*

*t*

*c* +

Γ =

− _{π}

### ∫

−^{ (2.9) }

Equation (2.6) may be written in the form

Γ

∈

− =

+

### ∫

Γ*f*

*w*

*dw*

*t*

*t*
*w*

*w*
*t* *T*

*f* ( ) ( ) 0,

i 2 PV 1 ) 2 ( 1

π ^{ (2.10) }

Multiply (2.9) by *c*(t) and add the result to (2.10) multiplied by and using the
definitions of and we get (2.4).

,
)
(t
*c*
)

(t

*g* *N*(c)

**3. ** **Generalized Neumann kernel **

The kernel *N*(*c*) (*t*,*w*) defined by (2.3) is continuous at all points (*t*,*w*)∈Γ×Γ except
for *t* = *w* where it is undefined. In Theorem 3.2, it will be shown that if Γ is
sufficiently smooth then *N*(*c*)(*t*,*w*) is continuous even at *t* = *w* where

Then, the symbol PV appears in (2.2) and (2.4) will be dropped and the equations (2.2) and (2.4) are Fredholm integral equations of the second kind.

. ) , ( ) ( lim ) , ( )

(*c* *w* *w* *N* *c* *t* *w*

*N*

*w*
*t*→

=

**Theorem 3.1. ***Let the smooth Jordan curve *Γ:*t* = *t*(*s*),0≤ *s* ≤ β,* be such that *
* and * *exist and are continuous on *

))
(
(*t* *s*

*c*′ *t*′′(s)

### [

0,β### ]

.*Then the limit of N*(

*c*)(τ,

*w*)

*as*

→ *w*

τ * exists for every w* = *t*(*s*)∈Γ,* and *
)
(
)
,
(
)
(

lim *N* *c* *w* *w* *w*

*w*

*t* =κ

→ (3.1)

*where *

)) . ( (

) ( )) ( ( 2 ) (

) Im ( ) ( 2 ) 1

( ⎥

⎦

⎢ ⎤

⎣

⎡ ′ ′

′ −

′′

= ′

*s*
*t*
*c*

*s*
*t*
*s*
*t*
*c*
*s*
*t*

*s*
*t*
*s*
*w* *t*

κ π (3.2)

*Moreover this limit exists uniformly. *

*Proof: Let *τ = *t*(σ), *w* = *t*(*s*). Then

⎥⎦

⎢ ⎤

⎣

⎡

−

′

= ′

⎥⎦

⎢ ⎤

⎣

⎡

= −

) ( ) (

) ( )) ( (

)) ( Im ( ) ( 1 )

( ) (

) Im ( ) 1

, )(

( σ

σ π

τ τ τ π

*t*
*s*
*t*

*s*
*t*
*s*
*t*
*c*

*t*
*c*
*s*
*t*
*w*

*w*
*T*
*w*
*c*
*w* *c*

*c*

*N* (3.3)

Using

, ) ) ((

) ( ) ( ) )(

( ) ( )

( *t* *s* *t* *s* *s* _{2}^{1}*t* *s* *s* ^{2} *s* ^{2}

*t*σ − = ′ σ − + ′′ σ − +ο σ −

)) ((

) )(

( )) ( ( )) ( ( )) (

(*t* *ct* *s* *c* *t* *s* *t* *s* *s* *s*

*c* σ = + ′ ′ σ − +ο σ −

and

) ( 1 1

1 θ ο θ

θ = − + +

for θ near to 0, we have for s near enough to σ ,

1

) ( ) ) (

( 2

) ( 2 1 1 1 )

( ) (

)

( ^{−}

⎭⎬

⎫

⎩⎨

⎧ − + −

′ + ′′

− −

− =

′ *s* *s*

*s*
*t*

*s*
*t*
*s*

*t*
*s*
*t*

*s*

*t* σ οσ

σ σ

⎭⎬

⎫

⎩⎨

⎧ − + −

′

− ′′

− −

= ( ) ( )

) (

) ( 2 1 1

1 *s* *s*

*s*
*t*

*s*
*t*

*s* σ ο σ

σ

) 1 ) ( (

) ( 2 1

1 ο

σ ′ ^{+}

+ ′′

− −

= *t* *s*

*s*
*t*
*s*

and

) ( ) )) (

( (

) ( )) ( 1 ( )) ( (

)) (

( *s* *s*

*s*
*t*
*c*

*s*
*t*
*s*
*t*
*c*
*s*

*t*
*c*

*t*

*c* ′ ′ − + −

+

= σ ο σ

σ

implies

) 1 )) (

( (

) ( )) ( ( ) (

) ( 2 1 1 ) ( ) (

) ( )) ( (

)) (

( ο

σ σ

σ ′ ′ +

′ − + ′′

= −

−

′

*s*
*t*
*c*

*s*
*t*
*s*
*t*
*c*
*s*
*t*

*s*
*t*
*s*
*s*

*t*
*t*

*s*
*t*
*s*
*t*
*c*

*t*

*c* . (3.4)

Hence

⎥⎦

⎢ ⎤

⎣

⎡ ′ ′

′ −

= ′′

⎥⎦

⎢ ⎤

⎣

⎡

−

′

→ (( ))

) ( )) ( ( ) (

) ( 2 Im 1 ) ( ) (

) ( )) ( (

)) ( Im (

lim *ct* *s*

*s*
*t*
*s*
*t*
*c*
*s*
*t*

*s*
*t*
*s*

*t*
*t*

*s*
*t*
*s*
*t*
*c*

*t*
*c*

*s* σ

σ

σ (3.5)

Thus from (3.3) and (3.5), we have

⎥⎦

⎢ ⎤

⎣

⎡ ′ ′

′ −

′′

= ′

→ (( ))

) ( )) ( ( ) (

) ( 2 Im 1 ) ( )) 1 ( ), ( ( ) (

lim *ct* *s*

*s*
*t*
*s*
*t*
*c*
*s*
*t*

*s*
*t*
*s*

*s* *t*
*t*
*t*
*c*
*N*

*s* σ π

σ

implying (3.1) and (3.2). To show that the limit (3.5), hence also (3.1), exists uniformly,
for any τ,*w*∈Γ, τ = *t*(σ) and *w* =*t*(s), let ε be any given positive real number, we
must find δ(ε) > 0 such that σ −*s* < 0 implies

σ ε

ς ⎥ <

⎦

⎢ ⎤

⎣

⎡ ′ ′

′ −

− ′′

⎥⎦

⎢ ⎤

⎣

⎡

−

′

)) ( (

) ( )) ( ( ) (

) ( 2 Im 1 ) ( ) (

) ( )) ( (

)) ( Im (

*s*
*t*
*c*

*s*
*t*
*s*
*t*
*c*
*s*
*t*

*s*
*t*
*s*

*t*
*t*

*s*
*t*
*s*
*t*
*c*

*t*

*c* . (3.6)

From (3.4) and (3.6), we have

⎥⎦⎤

⎢⎣⎡ +

− −

⎥ =

⎦

⎢ ⎤

⎣

⎡ ′ ′

′ −

− ′′

⎥⎦

⎢ ⎤

⎣

⎡

−

′ 1 (1)

)) Im ( (

) ( )) ( ( ) (

) ( 2 Im 1 ) ( ) (

) ( )) ( (

)) (

Im ( ο

σ σ

σ

*s*
*s*

*t*
*c*

*s*
*t*
*s*
*t*
*c*
*s*
*t*

*s*
*t*
*s*

*t*
*t*

*s*
*t*
*s*
*t*
*c*

*t*
*c*

### [

^{(}

^{1}

^{)}

### ]

^{(}

^{1}

^{)}

Im ο =ο

= (3.7)

Since (3.4) holds for all τ,*w*∈Γ,τ =*t*(σ) and *w* = *t*(s), such that σ near enough to
*s, thus from (3.7) there exists *δ(ε) > 0 such that for all τ,*w*∈Γ, τ = *t*(σ) and

,
)
(s
*t*

*w* = σ −*s* < 0 implies (3.6).

**Theorem 3.2. ***Under the hypotheses of Theorem 3.1 the kernel N*(*c*)(*t*,*w*)* defined by *

⎪⎪

⎩

⎪⎪

⎨

⎧

Γ

∈

⎥ =

⎦

⎢ ⎤

⎣

⎡ ′ ′

′ −

′′

′

Γ

∈

⎥ ≠

⎦

⎢ ⎤

⎣

⎡

= −

, )) ,

( (

) ( )) ( ( 2 ) (

) Im ( ) ( 2

1

, , , ) ,

( ) (

) Im ( 1 ) , ( ) (

*t*
*s* *w*

*t*
*c*

*s*
*t*
*s*
*t*
*c*
*s*
*t*

*s*
*t*
*s*

*t*

*t*
*w*
*t*
*t* *w*

*w*
*w*
*T*
*w*
*c*

*t*
*c*
*w*

*t*
*c*
*N*

π

π (3.8)

*is continuous on *Γ×Γ.* *

*Proof: To prove the continuity of the kernel N*(*c*),(*t*,*w*) at a point (*t*0,*t*0)∈Γ×Γ, it
must be shown that for any given ε > 0, we must find δ > 0 such that

### {

−_{0}, −

_{0}< δ

max *t* *t* *w* *t*

### }

(3.9)implies

. ) , ( ) ( ) , ( )

(*c* *t* *w* −*N* *c* *t*_{0} *t*_{0} < ε

*N* (3.10)

By triangle inequality (3.10) will hold if both

) 2 , ( ) ( ) , ( )

(*c* *t* *w* −*N* *c* *t* *t* < ε

*N* (3.11)

and

2 . ) , ( ) ( ) , ( )

( _{0} _{0} ε

<

−*N* *c* *t* *t*
*t*

*t*
*c*

*N* (3.12)

From Theorem 3.1 the limit lim *N*(*c*)(*t*,*w*) *N*(*c*)(*t*,*t*)

*t*

*w* =

→ exists uniformly for all *t*∈Γ
which implies that there exists δ_{1} > 0 such that *w*−*t* <δ_{1} implies that the inequality
(3.11) is held. The function κ(t) defined by (3.2) is continuous on the compact set ,
hence uniform continuous. Therefore there exists

Γ

2 > 0

δ such that for all *t*∈Γ
satisfies *t*−*t*_{0} < δ_{1} implies κ(*t*)−κ(*t*_{0}) < _{2}^{ε}, then the inequality (3.12) holds. Let

, } , { 2min 1

2 1 δ δ

δ = therefore for all *t*, *w*∈Γ, if max{*t* −*t*_{0} , *w*−*t*_{0} }<δ then
the inequality (3.10) holds, hence *N*(*c*)(*t*,*w*) is continuous at (*t*_{0},*t*_{0})∈Γ×Γ.

The Neumann kernel arises frequently in the integral equation of potential theory and conformal mapping. If the kernel (3.8) is identical with the Neumann kernel [3, pp. 282−286] so the kernel (3.8) will be called the generalized Neumann kernel and it will be denoted by or only when there is no confusion. When

we write or only N.

1
)
(*t* ≡
*c*

)
,
(
)
(*c* *t* *w*

*N* *N*(c)

,
1
)
(*t* ≡

*c* *N*(*t*,*w*)

Some of the properties of the generalized Neumann kernel are listed in the following remarks.

**Remark 3.1. It is clear that ** is a real kernel thus its adjoint kernel
is given by

)
,
(
)
(*c* *t* *w*
*N*
)

, ( )

*(*c* *t* *w*

*N* *N*^{*}(*c*)(*t*,*w*) = *N*(*c*)(*w*,*t*) = *N*(*c*)(*w*,*t*). Moreover, for all
and we have

*w*

*t* ≠ *t*,*w*∈Γ,

. ) , ( ˆ) ) (

( ) ( / ) (

) ( / ) Im ( 1 )

( ) (

) Im ( ) 1 , ( )

( *N* *c* *t* *w*

*t*
*w*

*w*
*T*
*w*
*c*
*w*
*T*

*t*
*c*
*t*
*T*
*w*

*t*
*t*
*T*
*t*
*c*

*w*
*t* *c*

*w*
*c*

*N* ⎥ = −

⎦

⎢ ⎤

⎣

⎡

− −

⎥ =

⎦

⎢ ⎤

⎣

⎡

= −

π

π (3.13)

Therefore *N*^{*}(*c*) = −*N*(*c*ˆ) where *c*ˆ(*t*) = *T*(*t*)/(*t*) =*T*(*t*)*c*(*t*),*t*∈Γ. Since Γ is a
smooth Jordan curve, *T*(t) is the unit tangent vector at *t*∈Γ, therefore

where

)

) (

(*t* *e*^{i}^{t}

*T* = ^{θ}

)

θ(t is the angle between the tangent to the contour Γ and the real axis. In going
round the contour Γ in anticlockwise direction θ(t) acquires the increment 2π.
Therefore ind_{Γ}(*T*) =1 and hence

. ) ( ind 1 ) ( ind ) ( ind ) (

ind_{Γ} *c* = _{Γ} *T* − _{Γ} *c* = − _{Γ} *c* (3.14)

**Remark 3.2. ** If Γ is the unit circle then *T*(*w*) = *iw* and

⎥⎦⎤

⎢⎣⎡ + −

⎥⎦

⎢ ⎤

⎣

⎡

−

− −

⎥ =

⎦

⎢ ⎤

⎣

⎡

− +

= −

*t*
*w*

*w*
*w*

*c*
*w*
*t*
*w*

*t*
*c*
*w*
*c*
*t*

*w*
*w*
*T*
*w*

*c*

*w*
*c*
*w*
*c*
*t*
*w* *c*

*t*
*c*

*N* i

1Im ) ( i ) ( ) Im ( 1 ) ( )

(

) ( ) ( ) Im ( ) 1 , ( )

( π π π

⎥⎦

⎢ ⎤

⎣

⎡

−

− − + −

⎥⎦

⎢ ⎤

⎣

⎡ −

−

− −

−

− −

= *w* *t*

*w*
*t*

*w*
*w*
*w*

*c*
*w*
*t*

*w*
*t*
*c*
*w*
*c*
*w*
*c*

*w*
*t*
*w*

*t*
*c*
*w*

*c* i i

i 2

1 ) (

i ) ( ) ( ) ( i ) ( ) ( i 2

1

π π

⎥⎦

⎢ ⎤

⎣

⎡ +

−

− −

= ( ) ()

) ( ) ( 2

1 2

1

*t*
*c*

*t*
*w*
*c*

*w*
*t*

*w*
*t*
*c*
*w*
*c*
π
π

which implies that *N*(*c*)(*t*,*w*) is symmetric when Γ is the unit circle.

**4. ** **Eigenvalue of the generalized Neumann kernel ****N****(****c****)(****t,****w****)**

In this section we give the main results of this paper. First we need the following
theorems from [2, pp. 221 & 226] with slight modifications. The first theorem discusses
the solvability of the Riemann problem in the interior domain Ω^{+} and the second
theorem discusses the solvability in the exterior domain Ω^{−}.

**Theorem 4.1. [2, p. 222] ** *Let * *x* = ind_{Γ}(*c*),*in the case x* ≤ 0* the homogeneous *

*Riemann problem (1.3) * *has * * linearly independent solutions and the *
*non-homogeneous problem (1.2) is absolutely soluble and its solution depends linearly *

*on * * arbitrary real constants. In the case * * the homogeneous Riemann *
*problem (1.3) has only the trivial solution and the non-homogeneous problem (1.2) is *

*soluble only if * * conditions are satisfied. If the latter conditions are satisfied the *
*non-homogeneous problem has a unique solution. *

1

2 +

− *x*

1

2 +

− *x* *x* > 0

1
2*x*−

**Theorem 4.2. [2, p. 226] ** *Let * *in the case * * the homogeneous *
*Riemann problem (1.6) in the exterior domain *

,
)
(
ind *c*

*x* = _{Γ} *x* ≥ 0

Ω−* has *2*x*+1* linearly independent *
*solutions and the non-homogeneous problem (1.4) is absolutely soluble and its solution *
*depends linearly on *2*x*+1*arbitrary real constants. In the case x* < 0* the homogeneous *
*Riemann problem (1.6) in the exterior domain *Ω^{−}* has only the trivial solution and the *
*non-homogeneous problem is soluble only if *−2*x*−1* conditions are satisfied. If the *
*latter conditions are satisfied the non-homogeneous problem has a unique solution. *

**Remark 4.1. If ** *f*(z) is any analytic function in Ω^{−} and vanishes at infinity then its
Taylor series in the vicinity of *z* = ∞ is given by ( ) = ^{1} + _{2}^{2} + _{3}^{3} + .

*z*
*a*
*z*
*a*
*z*

*z* *a*

*f* Suppose

that the function *f*_{1}(*z*) is defined by *f*_{1}(*z*) = *zf*(*z*) then its Taylor series in the vicinity

of *z* = ∞ is given = + + 2 +

3 2

1 1( )

*z*
*a*
*z*

*a* *a*

*z*

*f* and hence *f*_{1}(*z*) is analytic in Ω^{−} and
bounded at infinity.

**Corollary 4.1. ** *Let * *, if * * then the homogeneous Riemann problem *
(1.6) in the exterior domain * with the condition *

)
(
ind *c*

*x* = _{Γ} *x* > 0

Ω− *f*(∞) = 0* has *2*x*−1* linearly *
*independent solutions. If * * then the homogeneous Riemann problem (1.6) in the *
*exterior domain * * with the condition *

≤ 0
*x*

Ω− *f*(∞) = 0* has only the trivial solution. *

*Proof: To prove this Corollary, we shall prove that the homogeneous Riemann problem *
(1.6) with the condition *f*(∞) = 0 is equivalent to the homogeneous Riemann problem

(4.1)

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{0}

Re *c*_{1} *t* *f*_{1}^{−} *t* =

where *c*_{1}(*t*) = *c*(*t*)/*t*,*t* ∈Γ, and is merely analytic at infinity. Suppose that
is a solution to (1.6) with and define According to
the Remark 3.1 is analytic in

*f*1 *f*(z)

0 ) (∞ =

*f* *f*_{1}(*z*) = *zf*(*z*), *z*∈Ω^{−}.
)

1(*z*

*f* Ω^{−} and bounded at infinity. Since

we get is a solution to (4.1). Similarly let is a solution to (4.1) and define Then is analytic in

Γ

∈

= ^{−}

− *t* *tf* *t* *t*

*f*_{1} () (), *f*_{1}(*z*) *f*_{1}(*z*)

. ,

/ ) ( )

(*z* = *f*_{1} *z* *z* *z*∈Ω^{−}

*f* *f*(z) Ω^{−},

vanishes at infinity and Thus is a solution to (1.6) with . Therefore the homogeneous Riemann problem (1.6) with the condition is equivalent to the homogeneous Riemann problem (4.1) in Let

then

. , / ) ( )

( = _{1}^{−} ∈Γ

− *t* *f* *t* *t* *t*

*f* *f*(z)

0
)
(∞ =
*f*

0 ) (∞ =

*f* Ω^{−}.

)
(
ind _{1}

1 *c*

*x* = _{Γ} *x*_{1} = ind_{Γ}(*c*)−ind_{Γ}(*t*) = *x*−1. If then from
Theorem 4.2 the homogeneous Riemann problem (4.1) and hence (1.6) with the condition

has exactly real linearly independent solution. If

> 0

*x* *x*_{1} ≥ 0,

0 ) (∞ =

*f* 2*x*−1 *x* ≤ 0 then

according to Theorem 4.2 the homogeneous Riemann problem (4.1) and hence (1.6) with the condition

,

1 < 0
*x*

0 ) (∞ =

*f* has only the trivial solution.

The following theorems discuss the eigen problem of the generalized Neumann kernel. We shall discuss only the cases λ = ±1 which are very important in the discussion of the solvability of the Fredholm integral related to Riemann problem.

**Theorem 4.3. ** *If x* = ind_{Γ}(*c*) > 0,* then *λ = −1* is an eigenvalue of the generalized *
*Neumann kernel N*(*c*)(*t*,*w*).* *

*Proof: Since * according to Corollary 4.1, the homogeneous Riemann problem
(1.6) with the condition has

,

> 0
*x*

0 ) (∞ =

*f* 2*x*−1 linearly independent solutions in

. From Theorem 2.2 satisfy the homogeneous Fredholm integral equation

1 2 , , 2 , 1 , ) (

, = −

Ω^{−} *f*_{j}*z* *j* *x* *c*(*t*) *f*_{j}^{−}(*t*)

. 0 )

( ) ( ) , ( ) ( )

( )

(

### ∫

Γ−

− *t* + *N* *c* *t* *w* *cw* *f* *w* *dw* =
*f*

*t*

*c* _{j}* _{j}* (4.2)

Hence λ = −1 is an eigenvalue of *N*(c).

**Theorem 4.4. ** If *x* = ind_{Γ}(*c*) > 0* and *φ(t)* is a real eigenfunction of the generalized *
*Neumann kernel N*(*c*)(*t*,*w*)* corresponding to the eigenvalue *λ = −1* then the function *

* defined by *
)

Φ(z

*z* *dw*
*w*
*w*
*c*

*z* =

### ∫

Γ*w*−

Φ ( )( )

) ( i i 2 ) 1

( φ

π (4.3)

*is a solution to the homogeneous Riemann problem (1.6) in *Ω^{−}.* *

*Proof: The function *Φ(z) defined by (4.3) is analytic in (**C**\Γ) ∪{∞}. Taking the
limit Ω^{+} *z* →*t*∈Γ and using the Sokhotskyi formula to (4.3), we get

*dw*
*t* *w*

*w*
*w*
*c*

*w*
*T*
*t*
*t* *c*

*t*
*t*

*c* ( )

) ( ) (

) ( ) 1 (

) ( i ) ( ) (

2 φ

φ π

### ∫

Γ ++ −

=

Φ (4.4)

Taking the imaginary part of both sides of (4.4) we have

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{(}

^{)}

^{(}

^{,}

^{)}

^{(}

^{)}

^{.}

Im

2

### ∫

Γ+ = +

Φ *t* *t* *N* *t* *w* *w* *dw*

*t*

*c* φ φ (4.5)

Since φ is an eigenfunction of *N*(*c*)(*t*,*w*) corresponding to the eigenvalue λ = −1,
(4.5) reduces to the following homogeneous Riemann problem in Ω^{+}

(4.6)

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{0}

^{.}

Im *ct* Φ^{+} *t* =

Since according to Theorem 4.1 the problem (4.6) has only the trivial solution.

Therefore for all . Consequently, according to [2, p. 25] the function ,

> 0
*x*

0 ) ( =

Φ *z* *z*∈Ω^{+}

### ( ) *z*

### Φ

defined by (4.3) satisfies(4.7) .

) ( i ) ( )

(*t* *t* *t*

*c* Φ^{−} = −φ

Taking the real part of both sides of (4.7), we find that Φ(z) is a solution of homogeneous Riemann problem (1.6).

**Corollary 4.2. ** *If * * then the eigenfunctions of * * corresponding *
*to the eigenvalue *

0 ) ( ind >

= _{Γ} *c*

*x* *N*(c)

−1
λ = * are *

(4.8) 1

2 , , 2 , 1 , , ) ( ) ( i )

(*t* = *ct* *f*_{j}^{−} *t* *t*∈Γ *j* = *x*−

φ*j*

*where * * are the linearly independent solution of the homogeneous Riemann problem *
(1.6) in the exterior domain * with the condition *

*f**j*

Ω− *f*(∞) = 0.* *

*Proof: From Theorem 4.3, *λ = −1 is an eigenvalue of and the functions
are linearly independent solutions of the
homogeneous Fredholm integral equation

)
(c
*N*
1

2 , , 2 , 1 , ) ( ) ( i )

(*t* ≡ *ct* *f*_{j}^{−} *t* *j* = *x*+

φ*j*

. 0 )

( ) , ( ) ( )

(*t* +

### ∫

Γ*N*

*c*

*t*

*w*φ

*w*

*dw*=

φ (4.9)

Since Re[*c*(*t*) *f*_{j}^{−}(*t*)] = 0, hence φ* _{j}*(t) are real eigenfunctions of
corresponding to the eigenvalue

)
,
(
)
(*c* *t* *w*
*N*
.

−1

λ = We next show that these are the only
independent eigenfunctions of *N*(*c*)(*t*,*w*) corresponding to the eigenvalue λ = −1.
Let φ be any real eigenfunction of *N*(*c*)(*t*,*w*)corresponding to the eigenvalue λ = −1.
From Theorem 4.4 the function *G*(z) defined by

) . ( ) (

) ( i i 2 ) 1

( *dt*

*z*
*t*
*t*
*c*
*z* *t*

*G* =

### ∫

Γ φ −π (4.10)

is a solution to the homogeneous Riemann problem (1.6) with *G*(∞) = 0 and
Since

. ) ( i ) ( )

(*t* *G* *t* *t*

*c* ^{−} = −φ *f** _{j}*,

*j*=1,2, , 2

*x*−1 are the linearly independent solutions of the homogeneous Riemann problem (1.6) with

*f*(∞) = 0, hence

Consequently,

1

### .

2

1 ( )

)

(*z* ^{=}

### ∑

_{k}

^{x}_{=}

^{−}

*b*

*k*

*f*

*k*

*z*

*G*

### ∑

=^{−}

### ∑

−

=

−

− = =

= i () () ^{2} _{1}^{1}i () () ^{2} _{1}^{1} ().
)

(*t* *ct* *G* *t* _{k}^{x}*b*_{k}*ct* *f*_{k}*t* _{k}^{x}*b** _{k}*φ

_{k}*t*φ

Therefore φ_{1},φ_{2}, ,φ_{2}_{x}_{−}_{1} are the only linearly independent real eigenfunctions of
corresponding to the eigenvalue

)
,
(
)
(*c* *t* *w*

*N* λ = −1.

**Theorem 4.5. ** If *x* =ind_{Γ}(*c*) ≤ 0,* then *λ =1* is an eigenvalue of the generalized *
*Neumann kernel N*(c).

*Proof: Since * , according to Theorem 4.1 the homogeneous Riemann problem
(1.3) has linearly independent solution

From theorem 2.1 satisfy the homogeneous Fredholm integral equation 0

*x*≤

1

2 +

− *x* *f** _{j}*(

*z*),

*j*=1,2, ,−2

*x*+1,

*z*∈Ω

^{+}. )

(
)
(*t* *f* *t*
*c* _{j}^{+}

, 0 )

( ) ( ) , ( ) ( )

( )

( −

### ∫

Γ =+

+ *t* *N* *c* *t* *w* *c* *w* *f* *w* *dw*

*f*
*t*

*c* _{j}* _{j}* (4.11)

and hence λ =1 is an eigenvalue of *N*(c).

**Theorem 4.6. ** *If x* =ind_{Γ}(*c*) ≤ 0* and *φ(t)* is a real eigenfunction of the generalized *
*Neumann kernel N*(*c*)(*t*,*w*)* corresponding to the eigenvalue *λ =1* then the function *

* defined by *
)

Φ(z

*z* *dw*
*w*
*w*
*c*

*z* =

### ∫

Γ*w*−

Φ ( )( )

) ( i i 2 ) 1

( φ

π (4.12)

*is a solution to the homogeneous Riemann problem (1.3) in *Ω^{+}.

*Proof: The function * Φ(z) defined by (4.12) is analytic in (**C**\Γ)∪{∞} with
Taking the limit

. 0 ) (∞ =

Φ Ω^{−} *z* →*t*∈Γ and using the Sokhotskyi formula to
(4.12), we get

### ∫

Γ−

+ −

−

=

Φ *w* *dw*

*t*
*w*
*w*
*c*

*w*
*T*
*t*
*t* *c*

*t*
*t*

*c* ( )

) )(

( ) ( ) ( ) 1

( i ) ( ) (

2 φ

φ π (4.13)

Taking the imaginary part of both sides of (4.13) we have

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{(}

^{)}

^{(}

^{,}

^{)}

^{(}

^{)}

^{.}

Im

2

### ∫

Γ− = − +

Φ *t* *t* *N* *t* *w* *w* *dw*

*t*

*c* φ φ (4.14)

Since φ is an eigenfunction of *N*(*c*)(*t*,*w*)corresponding to the eigenvalue λ =1,
(4.14) reduces to the following homogeneous Riemann problem in Ω^{−},

(4.15)

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{0}

^{.}

Im *ct* Φ^{−} *t* =

with the condition Φ(∞) = 0. Since according to Corollary 4.1 the problem (4.15) has only the trivial solution. Therefore

,

≤ 0
*x*

0 ) ( =

Φ *z* for all . Consequently,
according to [2, p. 25] the function defined by (4.12) satisfies

Ω−

*z*∈
)

Φ(z

(4.16) .

) ( i ) ( )

(*t* *t* *t*

*c* Φ^{+} = φ

Taking the real part of both sides of (4.16), we find that Φ(z) is a solution of (1.3).

**Corollary 4.3. ** *If * * then the eigenfunctions of * * corresponding to *
*the eigenvalue *

0 ) (

ind ≤

= _{Γ} *c*

*x* *N*(c)

=1
λ * are *

(4.17) 1

2 , , 2 , 1 , , ) ( ) ( i )

(*t* = *ct* *f*_{j}^{+} *t* *t*∈Γ *j* = − *x*+

φ*j*

*where * * are the linearly independent solution of the homogeneous Riemann problem *
(1.3) in the interior domain

*f**j*

+. Ω

*Proof: From Theorem 4.5, *λ =1 is an eigenvalue of and the functions
are linearly independent solutions of the
homogeneous Fredholm integral equation

)
(c
*N*
1

2 , , 2 , 1 , ) ( ) ( i )

(*t* ≡ *ct* *f*_{j}^{+} *t* *j* = − *x*+

φ*j*

. 0 )

( ) , ( ) ( )

(*t* −

### ∫

Γ*N*

*c*

*t*

*w*ϕ

*w*

*dw*=

ϕ (4.18)

Since Re[*c*(*t*)*f*_{j}^{+}(*t*)]= 0, hence φ* _{j}*(t) are real eigenfunctions of the
corresponding to the eigenvalue

)
(c
*N*
.

=1

λ We next show that these are the only
independent eigenfunctions of *N*(c) corresponding to the eigenvalue λ =1. Let φ be
any real eigenfunction of *N*(c) corresponding to the eigenvalue λ =1. From Theorem
4.6 the function *G*(z) defined by

) . ( ) (

) ( i i 2 ) 1

( *dt*

*z*
*t*
*t*
*c*
*z* *t*

*G* =

### ∫

Γ φ −π (4.19)

is a solution to the homogeneous Riemann problem (1.3) and

Therefore and hence

. ) ( i ) ( )

(*t* *G* *t* *t*

*c* ^{+} = φ

### ∑

^{−}=

^{+}

= ^{2}_{1} ^{1} ( )
)

(*z* _{k}^{x}*b*_{k}*f*_{k}*z*
*G*

(4.20)

### ∑

^{−}=

^{+}

### ∑

+

−

=

− = − = −

−

= i () () ^{2}_{1} ^{1} i () () ^{2}_{1} ^{1} ().
)

(*t* *ct* *G* *t* _{k}^{x}*b*_{k}*ct* *f*_{k}*t* _{k}^{x}*b** _{k}*φ

_{k}*t*ϕ

Therefore φ1,φ2* _{q}*, ,φ

_{−}2

_{x}_{+}1 are the only linearly independent real eigenfunctions of corresponding to the eigenvalue

) (c

*N* λ =1.

**Corollary 4.4. ** Suppose that *x* = ind_{Γ}(*c*) ≤ 0.* If *φ* _{j}*(

*t*),

*t*∈Γ,

*j*=1,2, ,

*are linearly independent real eigenfunctions of*

*corresponding to the*

*eigenvalue*

1

2 +

− *x* *N*(c)

=1

λ * then the general solution of the homogeneous Riemann problem (1.3) *
*is given by *

*z* *dt*
*t*
*t*
*c*
*a* *t*

*z*

*f** _{h}* =

### ∑

^{−}

*=*

_{j}

^{x}^{+}

_{j}### ∫

_{Γ}

*− ) ( ) (*

^{j}) ( i i 2 ) 1

( ^{2} ^{1}

1

φ

π (4.21)

*where a** _{j}*,

*j*=1,2, , −2

*x*+1

*are arbitrary real constants.*

*Proof. From Theorem 4.6 the function * *f** _{h}*(z) is a solution to (1.3). Since φ

*are linearly independent, then so are the integrals in (4.21). According to Theorem 4.1 and since*

_{j}*f*

*(z) contains −2*

_{h}*x*+1 arbitrary real constants, the functions is the general solution to(1.3).

)
(z
*f*_{h}

The following two Theorems are proved only for the unit disk where the kernel in this case is symmetric.

)
,
(
)
(*c* *t* *w*
*N*

**Theorem 4.7. ***Suppose that *Γ* is the unit disk. If x*= ind_{Γ}(*c*) ≤ 0* then *λ = −1* is not *
*an eigenvalue of the generalized Neumann kernel N*(c).

*Proof: In accordance with the Fredholm’s Alternative to prove that *λ = −1 is not an
eigenvalue to *N*(c) it is sufficient to prove that the homogeneous equation

0 )

( ) , ( ) ( )

(*t* +

### ∫

Γ*N*

*c*

*t*

*w*ρ

*w*

*dw*=

ρ (4.22)

has only the trivial solution. Let ρ(t) be any solution of (4.22) and let us set ) .

( ) (

) ( i

2 ) 1

( *dw*

*z*
*w*
*w*
*c*
*z* *w*

*H* =

### ∫

Γ ρ −π (4.23)

Taking the limit Ω^{+} *z* →*t* ∈Γ and using the Sokhotskyi formula to (4.23), we get
*tdw*

*w*
*w*
*w*
*c*

*t*
*t* *c*

*t*
*H*
*t*

*c* = +

### ∫

Γ −+ ( )

) (

) ( i PV 1 ) ( ) ( ) (

2 ρ

ρ π (4.24)

Taking the real parts of both sides of (4.24) and using (4.22) we conclude that is a solution to the homogeneous Riemann problem (1.3). Since

)
(z
*H*
,

≤ 0

*x* according to
Corollary 4.3 the kernel *N*(c) has an eigenvalue λ =1 with the corresponding linearly
independent real eigenfunctions φ_{1},φ_{2}, ,φ_{−}_{2}_{x}_{+}_{1}. Since Γ is the unit circle then

is symmetric and the eigenfunctions )

,
(
)
(*c* *t* *w*

*N* φ* _{j}* may be assumed to be orthonormal

[9, p. 129], i.e.

, 1 2 , , 2 , 1 , , )

( )

( = = − +

### ∫

Γφ

_{i}*t*φ

_{j}*t*

*dt*δ

_{ij}*i*

*j*

*x*(4.25)

where δ* _{ij}* =1 if

*i*=

*j*and δ

*= 0 if From Corollary 4.4 the general solution of the homogeneous Riemann problem (1.3) is given by*

_{ij}.
*j*
*i* ≠

, ) ,

( ) (

) ( i i 2 ) 1

( ^{−}_{=}^{2}_{1}^{+}^{1} _{Γ} ∈Ω^{+}

=

### ∑

^{a}### ∫

_{c}_{t}*−*

_{t}

^{t}

_{z}

^{dt}

^{z}*z*

*f*_{h}_{i}^{x}* _{i}* φ

^{i}π (4.26)

where *a** _{i}*,

*i*=1,2, ,−2

*x*+1 are arbitrary real constants. Thus there exists −2

*x*+1 certain real constants

*a*

_{1},

*a*

_{2}, ,

*a*

_{−}

_{2}

_{x}_{+}

_{1}such that

. ) ,

( ) (

) ( i

i 2

1 )

( ) (

) ( i

2 ) 1 (

1 2

1 +

Γ

+

−

=

Γ ∈Ω

= −

= −

### ∫ ∑

### ∫

_{c}

_{w}

_{w}^{w}

_{z}

^{dw}

_{c}_{w}

_{w}^{a}

_{z}^{w}

^{dw}

^{z}*z*
*H*

*x*

*i* *i*φ*i*

π ρ

π (4.27)

Letting

### ∑

^{−}=

^{+}∈Γ

= ^{2}_{1} ^{1} (), ,
)

(*t* _{i}^{x}*a** _{i}*φ

_{i}*t*

*t*

φ (4.28)

then we get from (4.27)

+

Γ = ∈Ω

−

### ∫

_{c}_{(}

^{(}

_{w}^{w}_{)}

^{)}

_{(}−

_{w}^{i}

^{(}

_{z}^{w}_{)}

^{)}

^{dw}^{0}

^{,}

*i*

^{z}2

1 ρ φ

π (4.29)

According to [2, p. 25] the function *G*(z) defined in Ω^{−} by

−

Γ = ∈Ω

−

− −

=

### ∫

_{c}

_{w}^{w}

_{w}

_{z}^{w}

^{dw}

^{z}*z*

*G* 0,

) ( ) (

) ( i ) ( i 2 ) 1

( ρ φ

π (4.30)

is analytic in

### Ω

^{−}, vanishes at infinity and satisfies

(4.31) .

, ) ( i ) ( ) ( )

(*t* *G*^{−} *t* = *t* − *t* *t*∈Γ

*c* ρ φ

Letting *G*(*z*) ≡ i*G*(*z*), then from (4.31) include that *G*(z) is a solution to the Riemann
problem

### [

^{(}

^{)}

^{(}

^{)}

### ]

^{(}

^{)}

Re *ct* *G*^{−} *t* =φ *t* (4.32)

in Ω^{−} with the condition *G*(∞) = 0. Since *x* ≤ 0, according to [2, p. 300], there exists
a real function η(*t*), *t* ∈Γ depends on −2*x*+1 real arbitrary constants such that

) . ( ) (

) ( i

2 ) 1

( *dt*

*z*
*t*
*t*
*c*
*z* *t*

*G* =

### ∫

Γ η −π (4.33)

Taking the limit Ω^{−} *z* →*t* ∈Γ and using the Sokhotskyi formula to (4.33), we get
*t* *dw*

*w*
*w*
*w*
*c*

*t*
*t* *c*

*t*
*G*
*t*

*c* = − +

### ∫

Γ −− ( )

) (

) ( i PV 1 ) ( ) ( ) (

2 η

η π (4.34)

Taking the real parts of both sides of (4.24) and using (4.32) we conclude that η is a solution to Fredholm integral equation

. ) ( 2 )

( ) , ( ) ( )

(*t* *N* *c* *t* *w* η *w* *dw* φ *t*

η −

### ∫

Γ = − (4.35)Since λ =1 is an eigenvalue, in accordance with Fredholm alternative [4, p. 45] φ(t) must be orthogonal with all eigenfunctions of the adjoint kernel Since

is symmetric, hence

.
)
,
(
)
(*c* *w* *t*
*N*
)

,
(
)
(*c* *t* *w*

*N* φ(t) must be orthogonal with φ_{1},φ_{2}, ,φ_{−}_{2}_{x}_{+}_{1}.

Therefore for all we must have Using

(4.28) we get

1 2 , , 2 ,

1 − +

= *x*

*j*

### ∫

Γφ(*t*)φ

*(*

_{j}*t*)|

*dt*|= 0.

. 1 2 , , 2 , 1 , 0 )

( ) i (

2

1 2 1

1

### ∫

= = − +### ∑

^{−}

*=*

_{i}

^{x}^{+}

*a*

_{i}_{Γ}φ

_{i}*t*φ

_{j}*t*

*dt*

*j*

*x*

π (4.36)

Using (4.25) in (4.36) we conclude that *a** _{j}* = 0 for all

*j*=1,2, ,−2

*x*+1. Consequently φ(

*t*) = 0. Substituting φ(

*t*) = 0 in (4.32) we have

Γ

∈

− *t* = *t*

*G*
*t*

*c*() ()] 0,
[

Re with *G*(∞) =0. Since *x* ≤ 0, then from Corollary 4.1 we
get *G*(*z*) = 0 for all *z*∈Ω^{−} and hence *G*^{−}(*t*) = 0,*t* ∈Γ which implies that

Substituting and .

, 0 )

( = ∈Γ

− *t* *t*

*G* *G*^{−}(*t*) = 0 φ(*t*) = 0 in (4.31) we get ρ(*t*) = 0.
Our assertion is thereby proved.

**Theorem 4.8. ** Suppose that Γ*is the unit circle. If x*= ind_{Γ}(*c*)> 0* then *λ =1* is not *
*an eigenvalue of the generalized Neumann kernel N*(*c*)(*t*,*w*).

*Proof. Since * is the unit circle, from Remark 3.2 the kernel is symmetric.

In accordance with the Fredholm’s Alternative to prove

Γ *N*(*c*)(*t*,*w*)

=1

λ is not an eigenvalue to it is sufficient to prove that the homogeneous equation

)
(c
*N*

0 )

( ) , ( ) ( )

(*t* −

### ∫

Γ*N*

*c*

*t*

*w*μ

*w*

*dw*=

μ (4.37)

or, since *N*(*c*)(*t*,*w*) is symmetric, the associated homogenous equation
0

) ( ) , ( ) ( )

(*t* −

### ∫

Γ*N*

*c*

*w*

*t*μ

*w*

*dw*=

μ (4.38)

has only the trivial solutions. Let μ(t) be any solution of (4.38)
and *c*_{1}(*t*) = *T*(*t*)/*c*(*t*), *t* ∈Γ. From Remark 3.1, *N*(*c*)(*w*,*t*) = −*N*(*c*_{1})(*t*,*w*),

and (4.38) becomes 0

1 ) (

ind _{1}

1 = _{Γ} *c* = −*x* ≤

*x*

. 0 )

( ) , ( ) ( )

(*t* +

### ∫

Γ*N*

*c*

_{1}

*t*

*w*μ

*w*

*dw*=

μ (4.39)

In view of Theorem 4.7 λ = −1 is not an eigenvalue of *N*(*c*_{1}) and hence μ(*t*) = 0.
Consequently λ =1 is not an eigenvalue of *N*(c).

**5. ** **Examples **

**Example 1. Suppose that ** *c*(*t*)≡1 and Γ is any smooth Jordan curve. Since
according to Corollary 4.3 has a simple eigenvalue

, 0

Ind =

= _{Γ}*c*

*x* *N*(c) λ =1.

Since the homogeneous Riemann problem Re[*f*(*t*)] = 0 has only one independent
solution *f*(*z*)= iω,ω is arbitrary real number. Therefore the eigenfunction of
corresponding to the eigenvalue

)
(c
*N*

=1

λ is θ(*t*) =1.

**Example 2. ** Consider the case when *c*(*t*)= *t** ^{n}*,

*n*is an integer and Γ is the unit circle. First we consider the case

*n*≤ 0. From Corollary 4.3 λ =1 is an eigenvalue to

with linearly independent eigenfunctions. According to [2, p. 221] the homogeneous Riemann problem

) (c

*N* −2*n*+1

0 )]

( ) ( [

Re *ct* *f* *t* = has the general solution

(4.40)

### ∑

^{−}=

^{+}

= ^{2}_{1} ^{1} ( )
)

( _{k}^{n}_{k}_{k}

*H* *z* *c* *f* *z*

*f*