On the solution of the static Maxwell system
in axially symmetric inhomogeneous media
Kira V. Khmelnytskaya1, Vladislav V. Kravchenko2, Héctor Oviedo3
CINVESTAV-Queretaro, Libramiento Norponiente No. 2000,
Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 MEXICO
Department of Mathematics, CINVESTAV-Queretaro,
Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro,
Qro. C.P. 76230 MEXICO e-mail: vkravchenko@qro.cinvestav.mx
SEPI, ESIME Zacatenco, Instituto Politécnico Nacional, Av. IPN S/N,
C.P. 07738, D.F. MEXICO∗
December 13, 2018
Abstract
We consider the static Maxwell system with an axially symmetric
dielectric permittivity and construct complete systems of its solutions
which can be used for analytic and numerical solution of corresponding
boundary value problems.
1 Introduction
Consider the static Maxwell system
div(εE) = 0, rotE = 0 (1)
where we suppose that ε is a function of the cylindrical radial variable r =√
x21 + x
2: ε = ε(r). Two important situations are usually studied: the
meridional field and the transverse field.
∗Research was supported by CONACYT, Mexico
http://arxiv.org/abs/0704.1492v1
The first case is characterized by the condition that the vector E is inde-
pendent of the angular coordinate θ and the component Eθ of the vector E in
cylindrical coordinates vanishes identically. The vector of such field belongs
to a plane containing the axis x3 and depends only on the distance r to this
axis as well as on the coordinate x3. The field then is completely described
by a two-component vector-function in the plane (r, x3).
The second case is characterized by the condition that the vector E is in-
dependent of x3 and the component E3 is identically zero. The vector of such
field belongs to a plane perpendicular to the axis x3 and the corresponding
model reduces to a two-component vector-function in the plane (x1, x2).
In the present work in both cases we construct a complete system of
solutions of the corresponding model. We use the fact that in both cases the
system (1) reduces to a system describing so-called p-analytic functions [1],
[4], [8], [10], [11], [13], [17], [18]
vy, uy = −
vx. (2)
In the first case the function p is a function of one Cartesian variable x
meanwhile in the second it is a function of r =
x2 + y2. In both cases
we construct an infinite system of so-called formal powers [2], [5]. This is a
complete system of exact solutions of equations (2) generalizing the system
of usual complex powers (z− z0)n, n = 0, 1, 2, . . .. Locally, near the center z0
the formal powers behave asymptotically like powers. Nevertheless in general
their behaviour can be arbitrarily different from that of powers but with a
guarantee of their completeness in the sense that any solution of the consid-
ered equations can be represented as a uniformly convergent series of formal
powers. The general theory of formal powers was developed by L. Bers [2] as
a part of his pseudoanalytic function theory. Its application was restricted
by the fact that only for a quite limited class of pseudoanalytic functions the
explicit construction of formal powers was possible. L. Bers’ results allow
us to construct a complete system of formal powers in the meridional case.
Nevertheless they are not applicable to the model arising from the trans-
verse case. In the recent works [12] and [15] the class of solvable in this sense
systems (2) was substantially extended. In the present work we use these
results for solving the static Maxwell system in the transverse case. This
combination of the relation between the static Maxwell system (1) and the
system (2) together with the classical results of L. Bers on pseudoanalytic
formal powers and new developments in [12] and [15] allow us to obtain a
general solution of the static Maxwell system in the axially symmetric case
in the sense that we construct a complete system of its solutions for both the
meridional and the transverse fields.
2 Reduction of the static Maxwell system to
p-analytic functions
2.1 The meridional case
Introducing the cylindrical coordinates and making the assumptions that
E is independent of the angular variable θ and that the component Eθ is
identically zero we obtain that (1) can be written as follows
∂(rεEr)
∂(εE3)
Denote x = r, y = x3, u = E3 and v = rεEr. Then the system takes the
xε(x)
vy, uy = −
xε(x)
where the subindices denote the derivatives with respect to the corresponding
variables. Thus, in the case of a meridional field the vector E is completely
described by an xε(x)-analytic function ω = u+ iv.
2.2 The transverse case
We assume that E is independent of the longitudinal variable x3 and E3 ≡ 0.
Then from (1) we have that the vector (E1, E2)
T is the gradient of a function
u = u(x1, x2) which satisfies the two-dimensional equation
div(ε∇u) = 0. (3)
Denote x = x1, y = x2, z = x+ iy and consider the system
vy, uy = −
vx. (4)
It is easy to see that if the function ω = u + iv is its solution then u is
a solution of (3), and vice versa [14], if u is a solution of (3) in a simply
connected domain Ω then choosing
v = A(iεuz), (5)
where
A [Φ] (x, y) = 2
ReΦdx+ ImΦdy
+ c, (6)
c is an arbitrary real constant, Γ is an arbitrary rectifiable curve in Ω leading
from (x0, y0) to (x, y) we obtain that ω = u + iv is a solution of (4). Here
the subindex z means the application of the operator ∂z =
(∂x + i∂y). Note
that due to the fact that u is a solution of (3) the function Φ = iεuz satisfies
the condition ∂y ReΦ − ∂x ImΦ = 0 and hence the integral in (6) is path-
independent. For a convex domain then expression (6) can be written as
follows
A [Φ] (x, y) = 2
ReΦ(η, y)dη +
ImΦ(x0, ξ)dξ
Note that v is a solution of the equation
∇v) = 0.
Thus, equation (3) (and hence the system (1) in the case under consideration)
is equivalent to the system (4) in the sense that if ω = u+ iv is a solution of
(4) then its real part u is a solution of (3) and vice versa, if u is a solution
of (3) then ω = u+ iv, where v is constructed according to (5) is a solution
of (4).
We reduced both considered cases the meridional and the transverse to
the system describing p-analytic functions. In the first case p = xε(x) is a
function of one Cartesian variable and in the second p = ε(r), r =
x2 + y2.
As we show below in both cases we are able to construct explicitly a complete
system of formal powers and hence a complete system of exact solutions of
the corresponding Maxwell system. Let us notice that equation (3) with ε
being a function of the variable r was considered in the recent work [6] whith
applications to electrical impedance tomography. The algorithm proposed
in that work implies numerical solution of a number of ordinary differential
equations arising after a standard separation of variables. Our construction
of a complete system of solutions of (3) is based on essentially different ideas
and does not require solving numerically any differential equation.
3 p-analytic functions and formal powers
3.1 The main Vekua equation
Consider the system describing p-analytic functions
vy, uy = −
vx, (7)
where we suppose that p is a positive and continuously differentiable function
of x and y. Together with tis system we consider the following Vekua equation
which due to its importance in relation to second-order elliptic equations of
mathematical physics is called [14], [15] the main Vekua equation
W, (8)
where f =
p. The function ω = u + iv is a solution of (7) iff [15] W =
uf + iv/f is a solution of (8).
In [15] there was proposed a method for explicit construction of the system
of formal powers corresponding to the main Vekua equation under a quite
general condition on f . Here we briefly describe the method for which we
need first to recall some basic definitions from L. Bers’ theory of formal
powers [2].
Let F and G be a couple of solutions of a Vekua equation
Wz = a(F,G)W + b(F,G)W in Ω (9)
such that Im(FG) > 0. Then (F,G) is said to be a generating pair corre-
sponding to (9). The complex functions a(F,G) and b(F,G) are called charac-
teristic coefficients of the pair (F,G) and it can be seen that
a(F,G) = −
FGz − FzG
FG− FG
, b(F,G) =
FGz − FzG
FG− FG
Together with these characteristic coefficients another pair of characteristic
coefficients is introduced in relation to the notion of the (F,G)-derivative:
A(F,G) = −
FGz − FzG
FG− FG
, B(F,G) =
FGz − FzG
FG− FG
where the z means the application of the operator ∂z =
− i ∂
. As in
the present work we do not use explicitly the notion of the (F,G)-derivative,
we refer the interested reader to [2] for its definition and properties. However
we do need the concept of characteristic coefficients for defining the following
important object.
Definition 1 Let (F,G) and (F1, G1) - be two generating pairs in Ω corre-
sponding to the Vekua equations with coefficients a(F,G), b(F,G) and a(F1,G1)
and b(F1,G1) respectively. Then (F1, G1) is called successor of (F,G) and
(F,G) is called predecessor of (F1, G1) if
a(F1,G1) = a(F,G) and b(F1,G1) = −B(F,G).
Definition 2 A sequence of generating pairs {(Fm, Gm)}, m = 0,±1,±2, . . .
, is called a generating sequence if (Fm+1, Gm+1) is a successor of (Fm, Gm).
If (F0, G0) = (F,G), we say that (F,G) is embedded in {(Fm, Gm)}.
For any generating pair (F,G) the corresponding (F,G)-integral is defined
as follows
wd(F,G)ζ = F (z) Re
G∗wdζ +G(z) Re
F ∗wdζ
where Γ is a rectifiable curve leading from z0 to z and (F
∗, G∗) is an adjoint
generating pair defined by the equations
F ∗ = −
FG− FG
, G∗ =
FG− FG
If w is an (F1, G1) - pseudoanalytic function (i.e., it is a solution of the Vekua
equation with the coefficients a(F1,G1) and b(F1,G1)) then its (F,G)-integral is
path-independent.
Now we are ready to introduce the definition of formal powers.
Definition 3 The formal power Z
m (a, z0; z) with center at z0 ∈ Ω, coeffi-
cient a and exponent 0 is defined as the linear combination of the genera-
tors Fm, Gm with real constant coefficients λ, µ chosen so that λFm(z0) +
µGm(z0) = a. The formal powers with exponents n = 1, 2, . . . are defined by
the recursion formula
Z(n)m (a, z0; z) = n
(n−1)
m+1 (a, z0; ζ)d(Fm,Gm)ζ. (10)
This definition implies the following properties.
m (a, z0; z) is an (Fm, Gm)-pseudoanalytic function of z.
2. If a′ and a′′ are real constants, then Z
′+ia′′, z0; z) = a
m (1, z0; z)+
m (i, z0; z).
3. The asymptotic formulas
Z(n)m (a, z0; z) ∼ a(z − z0)n, z → z0 (11)
hold.
Writing Z(n)(a, z0; z) we indicate that the formal power corresponds to
the generating pair (F,G).
The definition of formal powers shows us that in order to obtain Z(n)(a, z0; z)
we need to have first the formal power Z
(n−1)
1 (a, z0; z) for which it is necessary
to calculate Z
(n−2)
2 (a, z0; z) and so on. Thus, the problem of construction of
formal powers of any order for a given generating pair (F,G) reduces to the
construction of a corresponding generating sequence. Then definition 3 gives
us a simple algorithm for constructing the formal powers. In other words,
one needs a pair of exact solutions for each of the infinite number of Vekua
equations corresponding to a generating sequence.
In the next subsection we show how this seemingly difficult task can be
accomplished in a quite general situation. Meanwhile here we recall some well
known results in order to explain that the system of formal powers in fact
represents a complete system of solutions of a corresponding Vekua equation.
First of all, let us notice that due to the property 2 of formal powers for every
n (and for a fixed z0) it is sufficient to construct only two formal powers:
Z(n)(1, z0; z) and Z
(n)(i, z0; z), then for any coefficient a the corresponding
formal power Z(n)(a, z0; z) is a linear combination of the former two.
An expression of the form
n=0 Z
(n)(an, z0; z) is called a formal polyno-
mial. Under the conditions imposed in this work on the function ε and on
the domain of interest Ω (see section 4 and for more details [15]) the fol-
lowing Runge-type theorem is valid where following [2] we say that a series
converges normally in a domain Ω if it converges uniformly on every bounded
closed subdomain of Ω.
Theorem 4 [3] A pseudoanalytic function defined in a simply connected do-
main can be expanded into a normally convergent series of formal polynomi-
In other words a pseudoanalytic function can be represented as an infinite
linear combination of the functions
Z(n)(1, z0; z), Z
(n)(i, z0; z)
Moreover, if we know that a pseudoanalytic function W satisfies the Hölder
condition on the boundary of a domain of interest Ω (a common require-
ment when a boundary value problem is considered) then, e.g., the following
estimate in the C(Ω)-norm is available.
Theorem 5 [16] Let W be a pseudoanalytic function in a domain Ω bounded
by a Jordan curve and satisfy the Hölder condition on ∂Ω with the exponent
α (0 < α ≤ 1). Then for any ǫ > 0 and any natural n there exists a
pseudopolynomial of order n satisfying the inequality
|W (z)− Pn(z)| ≤
Const
for any z ∈ Ω
where the constant does not depend on n, but only on ǫ.
These and other results on interpolation and on the degree of approxima-
tion by pseudopolynomials which can be found in the vast bibliography ded-
icated to pseudoanalytic function theory (see, e.g., [7], [9]) show us that the
system of formal powers is as good for solving corresponding boundary value
problems as is the system of usual complex powers (z − z0)n, n = 0, 1, 2, . . ..
The real (or imaginary) parts of {(z − z0)n}∞n=0 are harmonic polynomials
successfully applied to the numerical solution of boundary value problems
for the Laplace equation. In a similar way the real parts of formal powers{
Z(n)(1, z0; z), Z
(n)(i, z0; z)
corresponding to the main Vekua equation
(8) (where f =
ε) can be used for the numerical solution of boundary value
problems for the conductivity equation (3) because as was shown in [14] the
system of functions
ReZ(n)(1, z0; z),
ReZ(n)(i, z0; z)
is complete in the space of solutions of (3) in the sense of theorems 4 and 5.
A formal power Z(n)(a, z0; z) related to the Vekua equation (8) corre-
sponds to a formal power ∗Z
(n)(a, z0; z) (we use the notation of L. Bers)
related to the system (7) in the following way
(n)(a, z0; z) =
ReZ(n)(a, z0; z) + if ImZ
(n)(a, z0; z).
As we will see in the meridional case it is convenient to work directly with
formal powers ∗Z
(n)(a, z0; z). Any solution ω = u+ iv of the system (7) can
be expanded into a normally convergent series of real linear combinations of
the complex functions
(n)(1, z0; z), ∗Z
(n)(i, z0; z)
3.2 Construction of generating sequences
In [15] the following result was obtained.
Theorem 6 Let F = U(u)V (v) and G = i
U(u)V (v)
where U and V are ar-
bitrary differentiable nonvanishing real valued functions, Φ = u + iv is an
analytic function of the variable z = x + iy in Ω such that Φz is bounded
and has no zeros in Ω. Then the generating pair (F,G) is embedded in the
generating sequence (Fm, Gm), m = 0,±1,±2, . . .in Ω defined as follows
Fm = (Φz)
F and Gm = (Φz)
G for even m
F and Gm = (Φz)
U2G for odd m.
This theorem opens the way for construction of generating sequences and
consequently of formal powers in a quite general situation (see [15]) and in
particular in both cases considered in the present work. In the meridional
case the theorem reduces to the result of L. Bers [2] which we use in the
next subsection while in the transverse case this and other classical results
are insufficient for constructing formal powers explicitly and theorem 6 is
indispensable.
4 Construction of formal powers
4.1 Formal powers in the meridional case
As was shown in subsection 2.1 in the meridional case the Maxwell system
reduces to the following couple of equations
xε(x)
vy, uy = −
xε(x)
which is equivalent to the system considered in [2, N18.1]
σ(x)φx = τ(y)ψy, σ(x)φy = −τ(y)ψx.
Taking σ(x) = xε(x) and τ ≡ 1 we can use the elegant formulas for the
generating powers obtained by L. Bers. Let
X(0)(x0, x) = X̃
(0)(x0, x) = 1
and for n = 1, 2, ...denote
X(n)(x0, x) = n
X(n−1)(x0, t)
tε(t)
dt for odd n
X(n)(x0, x) = n
X(n−1)(x0, t)tε(t)dt for even n
X̃(n)(x0, x) = n
X̃(n−1)(x0, t)tε(t)dt for odd n
X̃(n)(x0, x) = n
X̃(n−1)(x0, t)
tε(t)
dt for even n
Then the formal powers in the meridional case are given by the expressions
(n)(a′ + ia′′, z0; z) = a
X(n−k)ik(y − y0)k
+ ia′′
X̃(n−k)ik(y − y0)k for odd n
(n)(a′ + ia′′, z0; z) = a
X̃(n−k)ik(y − y0)k
+ ia′′
X(n−k)ik(y − y0)k for even n.
4.2 Formal powers in the transverse case
As was shown in subsection 2.2 the Maxwell system (1) in the transverse case
reduces to the system
vy, uy = −
where ε is a positive differentiable function of r =
x2 + y2. This system
describing ε-analytic functions is equivalent to the main Vekua equation (8)
where f =
ε. In order to apply theorem 6 we denote u = ln r and U(u) =√
ε(eu). Then taking V ≡ 1 we obtain the generating pair (F,G) for equation
(8) in the desirable form
F = U(u), G =
. (12)
The analytic function Φ (from theorem 6) corresponding to the polar coor-
dinate system has the form Φ(z) = ln z and consequently Φz(z) = 1/z. We
note that Φz has a pole in the origin and a zero at infinity. Thus, theorem
6 is applicable in any domain Ω which does not include these two points.
Moreover, as for constructing formal powers we need to use the recursive
integration defined by (10) in what follows we require Ω to be any bounded
simply connected domain not containing the origin.
From theorem 6 we have that a generating sequence corresponding to the
generating pair (12) can be defined as follows
and Gm =
for even m
and Gm =
for odd m.
As was explained in subsection 3.1 in order to have a complete system of
formal powers for each n we need to construct Z(n)(1, z0; z) and Z
(n)(i, z0; z).
For n = 0 we have
Z(0)(1, z0; z) = λ
1 F (z) + µ
1 G(z)
Z(0)(i, z0; z) = λ
i F (z) + µ
i G(z)
where λ
1 , µ
1 are real constants chosen so that
1 F (z0) + µ
1 G(z0) = 1
and λ
i , µ
i are real constants such that
i F (z0) + µ
i G(z0) = i.
Taking into account that F is real and G is imaginary we obtain that
F (z0)
1 = 0,
i = 0, µ
i = F (z0).
Thus,
Z(0)(1, z0; z) =
F (z)
F (z0)
ε(r0)
Z(0)(i, z0; z) =
iF (z0)
F (z)
ε(r0)
where r0 = |z0|.
For constructing Z(1)(1, z0; z) and Z
(1)(i, z0; z) we need first the formal
powers Z
1 (1, z0; z) and Z
1 (i, z0; z). According to definition 3 they have
the form
1 (1, z0; z) = λ
1 F1(z) + µ
1 G1(z)
1 (i, z0; z) = λ
i F1(z) + µ
i G1(z)
where λ
1 , µ
1 are real numbers such that
1 F1(z0) + µ
1 G1(z0) = 1
and λ
i , µ
i are real numbers such that
i F1(z0) + µ
i G1(z0) = i.
Thus in order to determine λ
1 , µ
1 and λ
i , µ
i we should solve two systems
of linear algebraic equations:
z0ε1/2(r0)
iε1/2(r0)
z0ε1/2(r0)
iε1/2(r0)
which can be rewritten as follows
1 + µ
1 iε(r0) = ε
1/2(r0)z0
i + µ
i iε(r0) = iε
1/2(r0)z0.
From here we obtain
1 = ε
1/2(r0)x0, µ
1 = ε
−1/2(r0)y0, λ
i = −ε1/2(r0)y0, µ
i = ε
−1/2(r0)x0.
Let us notice that in general for odd m we have
Z(0)m (1, z0; z) =
zmε1/2(r)
1/2(r)
Z(0)m (i, z0; z) =
zmε1/2(r)
1/2(r)
where
1 = ε
1/2(r0) Re z
0 = ε
1/2(r0)r
0 cosmθ0,
1 = ε
−1/2(r0) Im z
0 = ε
−1/2(r0)r
0 sinmθ0,
i = −ε1/2(r0) Im zm0 = −ε1/2(r0)rm0 sinmθ0,
i = ε
−1/2(r0) Re z
0 = ε
−1/2(r0)r
0 cosmθ0,
θ0 is the argument of the complex number z0.
Thus, for odd m:
Z(0)m (1, z0; z) =
cosmθ0
ε(r0)
+ i sinmθ0
ε(r0)
Z(0)m (i, z0; z) =
− sinmθ0
ε(r0)
+ i cosmθ0
ε(r0)
In a similar way we obtain the corresponding formulas for even m:
Z(0)m (1, z0; z) =
cosmθ0
ε(r0)
+ i sinmθ0
ε(r0)
Z(0)m (i, z0; z) =
− sinmθ0
ε(r0)
+ i cosmθ0
ε(r0)
In order to apply formula (10) for constructing formal powers of higher orders
we need to calculate the adjoint generating pairs (F ∗m, G
m). For odd m we
F ∗m = −
ε1/2(r)
, G∗m = ε
1/2(r)zm.
For even m we obtain
F ∗m = −izmε1/2(r), G∗m =
ε1/2(r)
Now the whole procedure of construction of formal powers can be easily
algorithmized. The obtained system of formal powers
Z(n)(1, z0; z), Z
(n)(i, z0; z)
is complete in the space of all solutions of the main Vekua equation (8) with
f = ε1/2(r), i.e., any regular solution W of (8) in Ω can be represented in
the form of a normally convergent series
W (z) =
Z(n)(an, z0; z) =
(n)(1, z0; z) + a
(n)(i, z0; z)
where a′n = Re an, a
n = Im an and z0 is an arbitrary fixed point in Ω.
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ematics 66, No. 4, 1270-1300.
	Introduction
	Reduction of the static Maxwell system to p-analytic functions
	The meridional case
	The transverse case
	p-analytic functions and formal powers
	The main Vekua equation
	Construction of generating sequences
	Construction of formal powers
	Formal powers in the meridional case
	Formal powers in the transverse case

  We consider the static Maxwell system with an axially symmetric dielectric
permittivity and construct complete systems of its solutions which can be used
for analytic and numerical solution of corresponding boundary value problems.

Introduction
Consider the static Maxwell system
div(εE) = 0, rotE = 0 (1)
where we suppose that ε is a function of the cylindrical radial variable r =√
x21 + x
2: ε = ε(r). Two important situations are usually studied: the
meridional field and the transverse field.
∗Research was supported by CONACYT, Mexico
http://arxiv.org/abs/0704.1492v1
The first case is characterized by the condition that the vector E is inde-
pendent of the angular coordinate θ and the component Eθ of the vector E in
cylindrical coordinates vanishes identically. The vector of such field belongs
to a plane containing the axis x3 and depends only on the distance r to this
axis as well as on the coordinate x3. The field then is completely described
by a two-component vector-function in the plane (r, x3).
The second case is characterized by the condition that the vector E is in-
dependent of x3 and the component E3 is identically zero. The vector of such
field belongs to a plane perpendicular to the axis x3 and the corresponding
model reduces to a two-component vector-function in the plane (x1, x2).
In the present work in both cases we construct a complete system of
solutions of the corresponding model. We use the fact that in both cases the
system (1) reduces to a system describing so-called p-analytic functions [1],
[4], [8], [10], [11], [13], [17], [18]
vy, uy = −
vx. (2)
In the first case the function p is a function of one Cartesian variable x
meanwhile in the second it is a function of r =
x2 + y2. In both cases
we construct an infinite system of so-called formal powers [2], [5]. This is a
complete system of exact solutions of equations (2) generalizing the system
of usual complex powers (z− z0)n, n = 0, 1, 2, . . .. Locally, near the center z0
the formal powers behave asymptotically like powers. Nevertheless in general
their behaviour can be arbitrarily different from that of powers but with a
guarantee of their completeness in the sense that any solution of the consid-
ered equations can be represented as a uniformly convergent series of formal
powers. The general theory of formal powers was developed by L. Bers [2] as
a part of his pseudoanalytic function theory. Its application was restricted
by the fact that only for a quite limited class of pseudoanalytic functions the
explicit construction of formal powers was possible. L. Bers’ results allow
us to construct a complete system of formal powers in the meridional case.
Nevertheless they are not applicable to the model arising from the trans-
verse case. In the recent works [12] and [15] the class of solvable in this sense
systems (2) was substantially extended. In the present work we use these
results for solving the static Maxwell system in the transverse case. This
combination of the relation between the static Maxwell system (1) and the
system (2) together with the classical results of L. Bers on pseudoanalytic
formal powers and new developments in [12] and [15] allow us to obtain a
general solution of the static Maxwell system in the axially symmetric case
in the sense that we construct a complete system of its solutions for both the
meridional and the transverse fields.
2 Reduction of the static Maxwell system to
p-analytic functions
2.1 The meridional case
Introducing the cylindrical coordinates and making the assumptions that
E is independent of the angular variable θ and that the component Eθ is
identically zero we obtain that (1) can be written as follows
∂(rεEr)
∂(εE3)
Denote x = r, y = x3, u = E3 and v = rεEr. Then the system takes the
xε(x)
vy, uy = −
xε(x)
where the subindices denote the derivatives with respect to the corresponding
variables. Thus, in the case of a meridional field the vector E is completely
described by an xε(x)-analytic function ω = u+ iv.
2.2 The transverse case
We assume that E is independent of the longitudinal variable x3 and E3 ≡ 0.
Then from (1) we have that the vector (E1, E2)
T is the gradient of a function
u = u(x1, x2) which satisfies the two-dimensional equation
div(ε∇u) = 0. (3)
Denote x = x1, y = x2, z = x+ iy and consider the system
vy, uy = −
vx. (4)
It is easy to see that if the function ω = u + iv is its solution then u is
a solution of (3), and vice versa [14], if u is a solution of (3) in a simply
connected domain Ω then choosing
v = A(iεuz), (5)
where
A [Φ] (x, y) = 2
ReΦdx+ ImΦdy
+ c, (6)
c is an arbitrary real constant, Γ is an arbitrary rectifiable curve in Ω leading
from (x0, y0) to (x, y) we obtain that ω = u + iv is a solution of (4). Here
the subindex z means the application of the operator ∂z =
(∂x + i∂y). Note
that due to the fact that u is a solution of (3) the function Φ = iεuz satisfies
the condition ∂y ReΦ − ∂x ImΦ = 0 and hence the integral in (6) is path-
independent. For a convex domain then expression (6) can be written as
follows
A [Φ] (x, y) = 2
ReΦ(η, y)dη +
ImΦ(x0, ξ)dξ
Note that v is a solution of the equation
∇v) = 0.
Thus, equation (3) (and hence the system (1) in the case under consideration)
is equivalent to the system (4) in the sense that if ω = u+ iv is a solution of
(4) then its real part u is a solution of (3) and vice versa, if u is a solution
of (3) then ω = u+ iv, where v is constructed according to (5) is a solution
of (4).
We reduced both considered cases the meridional and the transverse to
the system describing p-analytic functions. In the first case p = xε(x) is a
function of one Cartesian variable and in the second p = ε(r), r =
x2 + y2.
As we show below in both cases we are able to construct explicitly a complete
system of formal powers and hence a complete system of exact solutions of
the corresponding Maxwell system. Let us notice that equation (3) with ε
being a function of the variable r was considered in the recent work [6] whith
applications to electrical impedance tomography. The algorithm proposed
in that work implies numerical solution of a number of ordinary differential
equations arising after a standard separation of variables. Our construction
of a complete system of solutions of (3) is based on essentially different ideas
and does not require solving numerically any differential equation.
3 p-analytic functions and formal powers
3.1 The main Vekua equation
Consider the system describing p-analytic functions
vy, uy = −
vx, (7)
where we suppose that p is a positive and continuously differentiable function
of x and y. Together with tis system we consider the following Vekua equation
which due to its importance in relation to second-order elliptic equations of
mathematical physics is called [14], [15] the main Vekua equation
W, (8)
where f =
p. The function ω = u + iv is a solution of (7) iff [15] W =
uf + iv/f is a solution of (8).
In [15] there was proposed a method for explicit construction of the system
of formal powers corresponding to the main Vekua equation under a quite
general condition on f . Here we briefly describe the method for which we
need first to recall some basic definitions from L. Bers’ theory of formal
powers [2].
Let F and G be a couple of solutions of a Vekua equation
Wz = a(F,G)W + b(F,G)W in Ω (9)
such that Im(FG) > 0. Then (F,G) is said to be a generating pair corre-
sponding to (9). The complex functions a(F,G) and b(F,G) are called charac-
teristic coefficients of the pair (F,G) and it can be seen that
a(F,G) = −
FGz − FzG
FG− FG
, b(F,G) =
FGz − FzG
FG− FG
Together with these characteristic coefficients another pair of characteristic
coefficients is introduced in relation to the notion of the (F,G)-derivative:
A(F,G) = −
FGz − FzG
FG− FG
, B(F,G) =
FGz − FzG
FG− FG
where the z means the application of the operator ∂z =
− i ∂
. As in
the present work we do not use explicitly the notion of the (F,G)-derivative,
we refer the interested reader to [2] for its definition and properties. However
we do need the concept of characteristic coefficients for defining the following
important object.
Definition 1 Let (F,G) and (F1, G1) - be two generating pairs in Ω corre-
sponding to the Vekua equations with coefficients a(F,G), b(F,G) and a(F1,G1)
and b(F1,G1) respectively. Then (F1, G1) is called successor of (F,G) and
(F,G) is called predecessor of (F1, G1) if
a(F1,G1) = a(F,G) and b(F1,G1) = −B(F,G).
Definition 2 A sequence of generating pairs {(Fm, Gm)}, m = 0,±1,±2, . . .
, is called a generating sequence if (Fm+1, Gm+1) is a successor of (Fm, Gm).
If (F0, G0) = (F,G), we say that (F,G) is embedded in {(Fm, Gm)}.
For any generating pair (F,G) the corresponding (F,G)-integral is defined
as follows
wd(F,G)ζ = F (z) Re
G∗wdζ +G(z) Re
F ∗wdζ
where Γ is a rectifiable curve leading from z0 to z and (F
∗, G∗) is an adjoint
generating pair defined by the equations
F ∗ = −
FG− FG
, G∗ =
FG− FG
If w is an (F1, G1) - pseudoanalytic function (i.e., it is a solution of the Vekua
equation with the coefficients a(F1,G1) and b(F1,G1)) then its (F,G)-integral is
path-independent.
Now we are ready to introduce the definition of formal powers.
Definition 3 The formal power Z
m (a, z0; z) with center at z0 ∈ Ω, coeffi-
cient a and exponent 0 is defined as the linear combination of the genera-
tors Fm, Gm with real constant coefficients λ, µ chosen so that λFm(z0) +
µGm(z0) = a. The formal powers with exponents n = 1, 2, . . . are defined by
the recursion formula
Z(n)m (a, z0; z) = n
(n−1)
m+1 (a, z0; ζ)d(Fm,Gm)ζ. (10)
This definition implies the following properties.
m (a, z0; z) is an (Fm, Gm)-pseudoanalytic function of z.
2. If a′ and a′′ are real constants, then Z
′+ia′′, z0; z) = a
m (1, z0; z)+
m (i, z0; z).
3. The asymptotic formulas
Z(n)m (a, z0; z) ∼ a(z − z0)n, z → z0 (11)
hold.
Writing Z(n)(a, z0; z) we indicate that the formal power corresponds to
the generating pair (F,G).
The definition of formal powers shows us that in order to obtain Z(n)(a, z0; z)
we need to have first the formal power Z
(n−1)
1 (a, z0; z) for which it is necessary
to calculate Z
(n−2)
2 (a, z0; z) and so on. Thus, the problem of construction of
formal powers of any order for a given generating pair (F,G) reduces to the
construction of a corresponding generating sequence. Then definition 3 gives
us a simple algorithm for constructing the formal powers. In other words,
one needs a pair of exact solutions for each of the infinite number of Vekua
equations corresponding to a generating sequence.
In the next subsection we show how this seemingly difficult task can be
accomplished in a quite general situation. Meanwhile here we recall some well
known results in order to explain that the system of formal powers in fact
represents a complete system of solutions of a corresponding Vekua equation.
First of all, let us notice that due to the property 2 of formal powers for every
n (and for a fixed z0) it is sufficient to construct only two formal powers:
Z(n)(1, z0; z) and Z
(n)(i, z0; z), then for any coefficient a the corresponding
formal power Z(n)(a, z0; z) is a linear combination of the former two.
An expression of the form
n=0 Z
(n)(an, z0; z) is called a formal polyno-
mial. Under the conditions imposed in this work on the function ε and on
the domain of interest Ω (see section 4 and for more details [15]) the fol-
lowing Runge-type theorem is valid where following [2] we say that a series
converges normally in a domain Ω if it converges uniformly on every bounded
closed subdomain of Ω.
Theorem 4 [3] A pseudoanalytic function defined in a simply connected do-
main can be expanded into a normally convergent series of formal polynomi-
In other words a pseudoanalytic function can be represented as an infinite
linear combination of the functions
Z(n)(1, z0; z), Z
(n)(i, z0; z)
Moreover, if we know that a pseudoanalytic function W satisfies the Hölder
condition on the boundary of a domain of interest Ω (a common require-
ment when a boundary value problem is considered) then, e.g., the following
estimate in the C(Ω)-norm is available.
Theorem 5 [16] Let W be a pseudoanalytic function in a domain Ω bounded
by a Jordan curve and satisfy the Hölder condition on ∂Ω with the exponent
α (0 < α ≤ 1). Then for any ǫ > 0 and any natural n there exists a
pseudopolynomial of order n satisfying the inequality
|W (z)− Pn(z)| ≤
Const
for any z ∈ Ω
where the constant does not depend on n, but only on ǫ.
These and other results on interpolation and on the degree of approxima-
tion by pseudopolynomials which can be found in the vast bibliography ded-
icated to pseudoanalytic function theory (see, e.g., [7], [9]) show us that the
system of formal powers is as good for solving corresponding boundary value
problems as is the system of usual complex powers (z − z0)n, n = 0, 1, 2, . . ..
The real (or imaginary) parts of {(z − z0)n}∞n=0 are harmonic polynomials
successfully applied to the numerical solution of boundary value problems
for the Laplace equation. In a similar way the real parts of formal powers{
Z(n)(1, z0; z), Z
(n)(i, z0; z)
corresponding to the main Vekua equation
(8) (where f =
ε) can be used for the numerical solution of boundary value
problems for the conductivity equation (3) because as was shown in [14] the
system of functions
ReZ(n)(1, z0; z),
ReZ(n)(i, z0; z)
is complete in the space of solutions of (3) in the sense of theorems 4 and 5.
A formal power Z(n)(a, z0; z) related to the Vekua equation (8) corre-
sponds to a formal power ∗Z
(n)(a, z0; z) (we use the notation of L. Bers)
related to the system (7) in the following way
(n)(a, z0; z) =
ReZ(n)(a, z0; z) + if ImZ
(n)(a, z0; z).
As we will see in the meridional case it is convenient to work directly with
formal powers ∗Z
(n)(a, z0; z). Any solution ω = u+ iv of the system (7) can
be expanded into a normally convergent series of real linear combinations of
the complex functions
(n)(1, z0; z), ∗Z
(n)(i, z0; z)
3.2 Construction of generating sequences
In [15] the following result was obtained.
Theorem 6 Let F = U(u)V (v) and G = i
U(u)V (v)
where U and V are ar-
bitrary differentiable nonvanishing real valued functions, Φ = u + iv is an
analytic function of the variable z = x + iy in Ω such that Φz is bounded
and has no zeros in Ω. Then the generating pair (F,G) is embedded in the
generating sequence (Fm, Gm), m = 0,±1,±2, . . .in Ω defined as follows
Fm = (Φz)
F and Gm = (Φz)
G for even m
F and Gm = (Φz)
U2G for odd m.
This theorem opens the way for construction of generating sequences and
consequently of formal powers in a quite general situation (see [15]) and in
particular in both cases considered in the present work. In the meridional
case the theorem reduces to the result of L. Bers [2] which we use in the
next subsection while in the transverse case this and other classical results
are insufficient for constructing formal powers explicitly and theorem 6 is
indispensable.
4 Construction of formal powers
4.1 Formal powers in the meridional case
As was shown in subsection 2.1 in the meridional case the Maxwell system
reduces to the following couple of equations
xε(x)
vy, uy = −
xε(x)
which is equivalent to the system considered in [2, N18.1]
σ(x)φx = τ(y)ψy, σ(x)φy = −τ(y)ψx.
Taking σ(x) = xε(x) and τ ≡ 1 we can use the elegant formulas for the
generating powers obtained by L. Bers. Let
X(0)(x0, x) = X̃
(0)(x0, x) = 1
and for n = 1, 2, ...denote
X(n)(x0, x) = n
X(n−1)(x0, t)
tε(t)
dt for odd n
X(n)(x0, x) = n
X(n−1)(x0, t)tε(t)dt for even n
X̃(n)(x0, x) = n
X̃(n−1)(x0, t)tε(t)dt for odd n
X̃(n)(x0, x) = n
X̃(n−1)(x0, t)
tε(t)
dt for even n
Then the formal powers in the meridional case are given by the expressions
(n)(a′ + ia′′, z0; z) = a
X(n−k)ik(y − y0)k
+ ia′′
X̃(n−k)ik(y − y0)k for odd n
(n)(a′ + ia′′, z0; z) = a
X̃(n−k)ik(y − y0)k
+ ia′′
X(n−k)ik(y − y0)k for even n.
4.2 Formal powers in the transverse case
As was shown in subsection 2.2 the Maxwell system (1) in the transverse case
reduces to the system
vy, uy = −
where ε is a positive differentiable function of r =
x2 + y2. This system
describing ε-analytic functions is equivalent to the main Vekua equation (8)
where f =
ε. In order to apply theorem 6 we denote u = ln r and U(u) =√
ε(eu). Then taking V ≡ 1 we obtain the generating pair (F,G) for equation
(8) in the desirable form
F = U(u), G =
. (12)
The analytic function Φ (from theorem 6) corresponding to the polar coor-
dinate system has the form Φ(z) = ln z and consequently Φz(z) = 1/z. We
note that Φz has a pole in the origin and a zero at infinity. Thus, theorem
6 is applicable in any domain Ω which does not include these two points.
Moreover, as for constructing formal powers we need to use the recursive
integration defined by (10) in what follows we require Ω to be any bounded
simply connected domain not containing the origin.
From theorem 6 we have that a generating sequence corresponding to the
generating pair (12) can be defined as follows
and Gm =
for even m
and Gm =
for odd m.
As was explained in subsection 3.1 in order to have a complete system of
formal powers for each n we need to construct Z(n)(1, z0; z) and Z
(n)(i, z0; z).
For n = 0 we have
Z(0)(1, z0; z) = λ
1 F (z) + µ
1 G(z)
Z(0)(i, z0; z) = λ
i F (z) + µ
i G(z)
where λ
1 , µ
1 are real constants chosen so that
1 F (z0) + µ
1 G(z0) = 1
and λ
i , µ
i are real constants such that
i F (z0) + µ
i G(z0) = i.
Taking into account that F is real and G is imaginary we obtain that
F (z0)
1 = 0,
i = 0, µ
i = F (z0).
Thus,
Z(0)(1, z0; z) =
F (z)
F (z0)
ε(r0)
Z(0)(i, z0; z) =
iF (z0)
F (z)
ε(r0)
where r0 = |z0|.
For constructing Z(1)(1, z0; z) and Z
(1)(i, z0; z) we need first the formal
powers Z
1 (1, z0; z) and Z
1 (i, z0; z). According to definition 3 they have
the form
1 (1, z0; z) = λ
1 F1(z) + µ
1 G1(z)
1 (i, z0; z) = λ
i F1(z) + µ
i G1(z)
where λ
1 , µ
1 are real numbers such that
1 F1(z0) + µ
1 G1(z0) = 1
and λ
i , µ
i are real numbers such that
i F1(z0) + µ
i G1(z0) = i.
Thus in order to determine λ
1 , µ
1 and λ
i , µ
i we should solve two systems
of linear algebraic equations:
z0ε1/2(r0)
iε1/2(r0)
z0ε1/2(r0)
iε1/2(r0)
which can be rewritten as follows
1 + µ
1 iε(r0) = ε
1/2(r0)z0
i + µ
i iε(r0) = iε
1/2(r0)z0.
From here we obtain
1 = ε
1/2(r0)x0, µ
1 = ε
−1/2(r0)y0, λ
i = −ε1/2(r0)y0, µ
i = ε
−1/2(r0)x0.
Let us notice that in general for odd m we have
Z(0)m (1, z0; z) =
zmε1/2(r)
1/2(r)
Z(0)m (i, z0; z) =
zmε1/2(r)
1/2(r)
where
1 = ε
1/2(r0) Re z
0 = ε
1/2(r0)r
0 cosmθ0,
1 = ε
−1/2(r0) Im z
0 = ε
−1/2(r0)r
0 sinmθ0,
i = −ε1/2(r0) Im zm0 = −ε1/2(r0)rm0 sinmθ0,
i = ε
−1/2(r0) Re z
0 = ε
−1/2(r0)r
0 cosmθ0,
θ0 is the argument of the complex number z0.
Thus, for odd m:
Z(0)m (1, z0; z) =
cosmθ0
ε(r0)
+ i sinmθ0
ε(r0)
Z(0)m (i, z0; z) =
− sinmθ0
ε(r0)
+ i cosmθ0
ε(r0)
In a similar way we obtain the corresponding formulas for even m:
Z(0)m (1, z0; z) =
cosmθ0
ε(r0)
+ i sinmθ0
ε(r0)
Z(0)m (i, z0; z) =
− sinmθ0
ε(r0)
+ i cosmθ0
ε(r0)
In order to apply formula (10) for constructing formal powers of higher orders
we need to calculate the adjoint generating pairs (F ∗m, G
m). For odd m we
F ∗m = −
ε1/2(r)
, G∗m = ε
1/2(r)zm.
For even m we obtain
F ∗m = −izmε1/2(r), G∗m =
ε1/2(r)
Now the whole procedure of construction of formal powers can be easily
algorithmized. The obtained system of formal powers
Z(n)(1, z0; z), Z
(n)(i, z0; z)
is complete in the space of all solutions of the main Vekua equation (8) with
f = ε1/2(r), i.e., any regular solution W of (8) in Ω can be represented in
the form of a normally convergent series
W (z) =
Z(n)(an, z0; z) =
(n)(1, z0; z) + a
(n)(i, z0; z)
where a′n = Re an, a
n = Im an and z0 is an arbitrary fixed point in Ω.
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	Introduction
	Reduction of the static Maxwell system to p-analytic functions
	The meridional case
	The transverse case
	p-analytic functions and formal powers
	The main Vekua equation
	Construction of generating sequences
	Construction of formal powers
	Formal powers in the meridional case
	Formal powers in the transverse case
