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skin effect, diffusion reflection of electrons, impedance, distribution function
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REFERENCES%LITERATURE
\end{center}}




\begin{center}
SKIN EFFECT PROBLEM WITH DIFFUSION BOUNDARY CONDITIONS IN MAXWELL
PLASMA
\end{center}


\begin{center}
 {\bf Yu.F. Alabina, A.V. Latyshev, A.A. Yushkanov}
\end{center}

\begin{center}
 Moscow State Regional University\\
{\it 105005, Moscow, Radio st., 10 a\\
e-mail: yf.alabina@gmail.com, avlatyshev@mail.ru,
yushkanov@inbox.ru}
\end{center}

\begin{abstract}
%We develope
The analytical method of solving the boundary
problems for a system of equations %pertaining
describing  the behaviour
of electrons and an electric field in the Maxwell plasma half-space
is developed.
Here the diffusion reflection of electrons from the surface
as a boundary condition is supposed.
The exact solution
 for boundary value problem  of  the Vlasov -- Maxwell
equations
 through the eigenfunctions of the discrete spectrum and the
 continuous
 spectrum is obtained.
The method of decomposition by eigenfunctions of discrete
 and continuous spectrum is used.
The exact
 expression for the surface impedance also is obtained.
\end{abstract}

{\bf 1. Introduction. Statement of problem.}
 The skin effect is the
plasma's response to an external constant-amplitude variable
electromagnetic field that is tangential to the surface \cite{1}.
This problem represents essential interest \cite{1}--\cite{5},
with the focal point being on the computation of surface impedance.

The problem about skin effect is actual and is regularly
studied and last years (see, for exampe, \cite{5}).




    The paper further develops the analytical method of solving
boundary problem for a system of equations describing behaviour of
electrons and electric field in Maxwell plasma. The method is based
on the idea of the decomposition of the solution the problem
by singular
eigenfunctions of the corresponding characteristic equations.

We find the analytical solution to the boundary problem
of the skin
effect theory for electron plasma that fills the half-space. The
analytical solution has a form of the sum of an integral and
two (or
one) exponentially decreasing particular solutions of the initial
system. Depending on index the problem one of the particular
solutions disappears.

We consider a collision plasma with behaviour is described
by modelling kinetic equation with integral of collisions
in the form BGK--model (BGK=Bhatnagar, Gross, Krook) (see,
for example, \cite{4}).

Let's Maxwell plasma fills the half-space $x>0$. Here $x$ is the
orthogonal coordinate  to the plasma boundary.
Let's the external
electric field has only  $y$ component. Then the self-consistent
electric field inside in plasma also has only $y$ component
$E_y(x,t)=E(x)e^{-i\omega t}$. We consider the kinetic equation
for the
electron distribution function:

$$
\dfrac{\partial f}{\partial t}+\text{v} _{x}\dfrac{\partial
f}{\partial x}+eE(x)e^{-i\omega t}\dfrac{\partial f}{\partial
p_y}=\nu(f_0-f(t,x,\mathbf{v})).
\eqno{(1.1)}
$$

In equation (1.1) $\nu$ is the frequency of electron
collisions with
ions, $e$ is the charge of electron, $f_0(\text{v})$ is the
equilibrium Maxwell distribution function, $\textbf{p}$ is the
momentum of electron,
$$
f_0(\text{v})=n\left(\dfrac{\beta}{\pi}\right)^{3/2}
\exp(-\beta^2\text{v}^2),\quad
\beta=\dfrac{m}{2k_BT}.
$$

Here  $m$ is the mass of electron, $k_B$ is the Boltzmann constant,
$T$ is the temperature of plasma, $\text{v}$ is the velocity of the
electron, $n$ is the concentration of electrons, $c$ is the
speed of
light.

The electric field $E(x)$  satisfies to the equation:
$$
E''(x)+\dfrac{\omega^2}{c^2}E(x)= -\dfrac{4\pi i  e^{i\omega
t}\omega e}{c^2} \int v_y f(t,x,\mathbf{v})\,d^3v. \eqno{(1.2)}
$$


We assume that intensity of an electric field is such that linear
approximation is valid. Then distribution function can be
presented in the form:
$$
f=f_0\left(1+C_y\exp(-i\omega t)h(x,\mu)\right),
$$
where $\textbf{C}=\sqrt{\beta}\text{v}$ is the dimensionless
velocity of electron, $\mu=C_x$. Let $l=v_T\tau$ is the mean free
path of electrons, $v_T=1/\sqrt{\beta}$, $v_T$ is the thermal
electron velocity,\; $\tau=1/\nu$.
We introduce
the dimensionless parameters and the electric field:
$$
t_1=\nu t,\quad x_1=\dfrac{x}{l}, \quad
e(x_1)=\dfrac{\sqrt{2}e}{\nu\sqrt{mk_BT}}E(x_1).
$$

Later we substitute $x_1$ for $x$. The substitution produces the
following form of the kinetic equation (1.1) and the equation on a
field with the displacement current (1.2):
$$
\mu\dfrac{\partial h}{\partial x}+z_0\,h(x,\mu)=e(x),  \quad
z_0=1-i\omega\tau, \eqno{(1.3)}
$$
$$
%\dfrac{d^2e(x)}{d\,x^2}
e''(x)+Q^2e(x)=-i\dfrac{\alpha}{\sqrt{\pi}}
\int\limits_{-\infty}^{\infty}\exp(-{\mu'}^2)\,h(x,\mu')\,d\mu',
\quad
Q=\dfrac{\omega l}{c},
\eqno{(1.4)}
$$
where $\delta=\dfrac{c^2}{2\pi\omega\sigma_0}$,
$\delta$ is the classical
depth of the skin layer, $\sigma_0=\dfrac{e^2n}{m\nu}$,
$\sigma_0$ is the
electric conductance, $\alpha=\dfrac{2l^2}{\delta^2}$, $\alpha$ is
the anomaly parameter.

We formulate the boundary conditions for the distribution function
of the electron in case of the diffusion electron reflection
from the
surface:
$$
h(0,\mu)=0, \qquad 0<\mu<+\infty. \eqno{(1.5)}
$$
We use the condition that function $h(x,\mu)$ vanishes far from the
surface:
$$
h(+\infty,\mu)=0, \qquad -\infty<\mu<+\infty, \eqno{(1.6)}
$$
and conditions for electric field on the interface and far from it:
$$
{e}'(0)={e_s}', \qquad e(+\infty)=0, \eqno{(1.7)}
$$
where ${e_s}'$ is the given value of the gradient of the
electric field on the plasma interface.

So, the skin effect problem is formulated completely. We seek
solution of system of the equations (1.3) and (1.4)
in this problem,
that satisfy to the boundary conditions (1.5) -- (1.7).

{\bf 2. Eigenfunctions and eigenvalues.} The separation
of variables
\cite{2}--\cite{4}
$$
h_\eta(x,\mu)=\exp(-z_0\dfrac{x}{\eta})\Phi(\eta,\mu),
\eqno{(2.1)}
$$
$$
e_\eta(x)=\exp(-z_0\dfrac{x}{\eta})E(\eta),
\eqno{(2.2)}
$$
where $\eta$ is the spectral parameter (generally,
complex-valued), reduces  system of the equation (1.3) and (1.4)
to the characteristic system:
$$
(\eta-\mu)\Phi(\eta,\mu)=\dfrac{\eta}{z_0}E(\eta),
\eqno{(2.3)}
$$
$$
\left[z_0^2+\left(\dfrac{\omega
l}{c}\right)^2\eta^2\right]E(\eta)=-\dfrac{i\alpha
\eta^2}{\sqrt{\pi}}n(\eta),
\eqno{(2.4)}
$$
where
$$
n(\eta)=\int\limits_{-\infty}^{\infty}e^{-\mu^2}
\Phi(\eta,\mu)d\mu.
$$




From equations (2.3) and (2.4) for $\eta\in(-\infty,\infty)$, we
find the eigenfunctions of the continuous spectrum \cite{2},
\cite{6}:
$$
\Phi(\eta,\mu)=\dfrac{a}{\sqrt{\pi}}\eta^3e^{-\eta^2}
P\dfrac{1}{\eta-\mu}+
\lambda(\mu)\delta(\eta-\mu),
\eqno{(2.6)}
$$
$$
E(\eta)=\dfrac{az_0}{\sqrt{\pi}}\eta^2e^{-\eta^2}, \quad
a=-i\dfrac{\alpha}{z_0^3}.
\eqno{(2.7)}
$$

In the equation (2.6) the symbol $Px^{-1}$ denotes the distribution, i.e. the
principal value of the integral of  $x^{-1}$, $\delta(x)$  is the
Dirac delta function.

Since the distribution and electric field functions
decline as we go
further from the boundary, we regard the continuous
as positive real
half-axis:  $0<\eta<+\infty$. The eigen solution of
the continuous
spectrum $h_\eta(x,\mu)$, $e_\eta(x)$ are decreasing
from value $x$,
since $\Re z_0>0$.

In the equations (2.6) and (2.7) there is the normalization
condition:
$$
\int\limits_{-\infty}^{\infty}e^{-\mu^2}\Phi(\eta,\mu)d\mu=
\left[1+\left(\dfrac{\omega
l}{c}\right)^2\eta^2\right]e^{-\eta^2},
$$
$\lambda(z)$ is the dispersion function,
$$
\lambda(z)=1+\left(\dfrac{\omega
l}{cz_0}\right)^2z^2+\dfrac{az^3}{\sqrt{\pi}}
\int\limits_{-\infty}^{\infty}
\dfrac{e^{-\mu^2}d\mu}{\mu-z}.
\eqno{(2.8)}
$$

Using the method developed in \cite{2, 4}, we determine the
structure of the discrete spectrum of the characteristic system
equations (2.3) -- (2.4). By definition, this spectrum consists of
the zeros of the dispersion function that do not belong to the cut
$(-\infty,\infty)$.

Let $N$  is the number of such zeros. Taking into account the fact
that the dispersion function has a double pole at $z=\infty$, the
number of its zeros is
$$
N=2+\dfrac{1}{2\pi}[\arg \lambda(z)]_{\gamma_\varepsilon},
\eqno{(2.9)}
$$
where $\gamma_\varepsilon$ is the contour that encompasses the cut
that is at the distance  $\varepsilon$ from the former cut
$(-\infty,\infty)$ and does not contain zeros of the dispersion
function.

Passing to the limit in the equation (2.9) as
$\varepsilon\rightarrow0$, we obtain
$$
N=2+\dfrac{1}{2\pi}\left[\arg\dfrac{\lambda^+(\tau)}{\lambda^-
(\tau)}\right]_{(-\infty,\infty)}=2+\dfrac{1}{\pi}
\left[\arg\dfrac{\lambda^+(\tau)}{\lambda^-
(\tau)}\right]_{(0,\infty)}.
$$

Here $\lambda^\pm(\tau)$ are the boundary values of the function
$\lambda(z)$ above and below the cut $(-\infty,\infty)$,
$$
\lambda^\pm(\mu)=\lambda(\mu)\pm i\pi a\mu^3 \exp(-\mu^2),
$$
where
$\lambda(\mu)$ it is defined by (2.8), and the integral here is
understood in sense of a principal value (according Cauchy).

The fact that $\lambda(\mu)$  has a double pole at $z=\infty$
follows from the asymptotic expansion in the Laurent series:
$$
\lambda(z)=1+az^2\sqrt{\pi}+a\dfrac{\sqrt{\pi}}{2}+
a\int\limits_{-\infty}^\infty
\dfrac{\tau^3\exp(-\tau^2)}{\tau-z}d\tau,
\quad z\rightarrow\infty.
$$

We consider the domain  $D^+$ (we denote its exterior by
$D^-$) on the
$\Lambda$-plane (fig. 1) with the boundary given by
the parametric
equations:
$$
\partial D^+=\{\delta=\delta_1+i\delta_2: \Re \lambda^+(\mu)=0,
\Im \lambda^-(\mu)=0,
\quad -\infty \leqslant\mu \leqslant+\infty\}.
$$

Here $\delta$ is a new parameter,
$$
\delta=\delta_1+i\delta_2=-\dfrac{4c^2}{\pi
v_T^2\omega_p^2}\cdot \dfrac{(\omega+i \nu)^2}{\omega},
$$
$$
\delta_1=-\dfrac{\omega^3-3\omega \nu^2}{v_c^2 \omega
\omega_p^2},\quad
\delta_2=-\dfrac{3\omega^2 \nu-\nu^3}{v_c^2\omega
\omega_p^2}, \quad
v_c=\sqrt{\dfrac{\pi}{4}}\dfrac{v_T}{c}.
$$

Besides, the boundary of the domain $D^+$ can be written in
the explicit form:
$$
\partial D^+=\{\delta=\delta_1+i\delta_2: \delta_1=p(\mu) ,
\delta_2=q(\mu),
\quad -\infty \leqslant\mu \leqslant+\infty\},
$$
where
$$
p(z)=-\dfrac{z^3}{\sqrt{\pi}}\int\limits_{-\infty}^{\infty}
\dfrac{\exp(-\mu^2)\,d\mu}{\mu-z}, \qquad
q(z)=\sqrt{\pi}z^3 \exp(-z^2).
$$

As in \cite{4}, it can be shown that:

1) if $\delta\in D^+$, then
$N=4$ and the problem index $\varkappa(G)=1$;

2) if  $\delta\in D^-$,
then $N=2$ and $\varkappa(G)=0$.

Here $\varkappa=\varkappa(G)$ is the index of the problem,
$G(\mu)=\dfrac{\lambda^+(\mu)}{\lambda^-(\mu)}$,
$$
\varkappa(G)={\rm ind}_{[-\infty,\infty]}G(\mu)=\dfrac{1}{2\pi}
\int\limits_{-\infty}^{\infty}d \arg \dfrac{\lambda^+(\mu)}
{\lambda^-(\mu)}.
$$

The boundary mode, i.e.,
the case
$\delta\in \partial D$, is not considered in this paper,
since it
was analyzed in details in \cite{2}.

We find the general solution to
system (1.3), (1.4) in the form of an expansion in the
eigenfunctions of the discrete and continuous spectrum; this
solution automatically satisfies the boundary conditions
at infinity:
$$
h(x,\mu)=\dfrac{a}{\sqrt{\pi}}\sum\limits_{k=0}^{1}
\dfrac{A_k\eta_k^3}{\eta_k-\mu}\exp\Big(-\eta_k^2-
\dfrac{z_0x}{\eta_k}\Big)+
$$
$$
+
\int\limits_{0}^{\infty}\exp\Big(-\dfrac{z_0x}{\eta}\Big)
A(\eta)\Phi(\eta,\mu)\,d\eta, \eqno{(2.10)}
$$
$$
e(x)=\dfrac{az_0}{\sqrt{\pi}}\sum\limits_{k=0}^{1}
A_k\eta_k^2\exp\Big(-\eta_k^2-\dfrac{z_0x}{\eta_k}\Big)+
$$
$$
+\dfrac{az_0}{\sqrt{\pi}}\int\limits_{0}^{\infty}
\exp\Big(-\eta^2-\dfrac{z_0x}{\eta}\Big)\eta^2\,A(\eta)\,d\eta.
\eqno{(2.11)}
$$

Here  $A_k (k=0,1)$ are the unknown coefficients corresponding to
the discrete spectrum,  $A_1=0$, if  $\delta\in D^-$; and $A(\eta)$
is an unknown function called the coefficient of the continuous
spectrum $\Re(z/\eta_k)>0 \quad (k=0,1)$, $\Re z_0=1$.
\begin{figure}[h]
\begin{center}
\includegraphics[height=0.4\textheight]{fig1.EPS}
\end{center}
\begin{center}
  Fig. 1.
\end{center}
\end{figure}



{\bf 3. The index of the problem equals zero.}
We consider the case when the
problem index is equal to zero. Substituting expressions (2.10) and
(2.11) into (1.5), we obtain the singular integral
equation with the
Cauchy kernel:
$$
\dfrac{a}{\sqrt{\pi}}\int\limits_{0}^{\infty}\dfrac{\eta^3
\exp(-\eta^2)A(\eta)}{\eta-\mu}\,d\eta+\lambda(\mu)A(\mu)+
a\varphi(\mu)=0, \quad 0<\mu<\infty, \eqno{(3.1)}
$$
where
$$
\varphi(\mu)=\dfrac{A_0}{\sqrt{\pi}}\dfrac{\eta_0^3\exp(-\eta_0^2)}
{\eta_0-\mu}.
$$

We define the auxiliary function in the complex plane
$$
N(z)=\dfrac{1}{\sqrt{\pi}}\int\limits_{0}^{\infty}
\dfrac{\eta^2\exp(-\eta^2)A(\eta)}{\eta-z}\,d\eta.
$$

Then the equation (3.1) is possible to transform
to the Riemann --- Hilbert
boundary value problem:
$$
\lambda^+(\mu)\Big[N^+(\mu)+\varphi(\mu)\Big]=
\lambda^-(\mu)\Big[N^-(\mu)+\varphi(\mu)\Big],\quad 0<\mu<\infty.
\eqno{(3.2)}
$$

We consider the Riemann --- Hilbert boundary value problem
on the half-axis:
$$
\dfrac{X^+(\mu)}{X^-(\mu)}=G(\mu),\qquad 0<\mu<+\infty.
$$
where
$$
G(\mu)=\dfrac{\lambda^+(\mu)}{\lambda^-(\mu)},
$$
$G(\mu)$ is the coefficient of the
boundary value problem.

The index in this problem is $\varkappa(G)=0$, then
the solution of
problem (3.2) has the form \cite{4}:
$$
X(z)=\exp V(z), \qquad V(z)=\dfrac{1}{2\pi i}
\int\limits_{0}^{\infty}\dfrac{\ln G(\tau)\,d\tau}{\tau-z}.
\eqno{(3.3)}
$$

We use (3.3) to reduce problem (3.2) for determining an analytic
function $N(z)$ from its zero jump problem on the cut:
$$
X^+(\mu)\Big[N^+(\mu)+\varphi(\mu)\Big]=
X^-(\mu)\Big[N^-(\mu)+\varphi(\mu)\Big],\quad 0<\mu<\infty.
\eqno{(3.4)}
$$

A general solution of (3.4) is given by the formula
$$
N(z)=\dfrac{A_0}{\sqrt{\pi}}\dfrac{\eta_0^3\exp(-\eta_0^2)}
{z-\eta_0}+\dfrac{C_0}{(z-\eta_0)X(z)}.\eqno{(3.5)}
$$
where  $C_0$ is an arbitrary constant.

Let's eliminate poles at the decision (3.5), we have:
$$
C_0=-\dfrac{A_0}{\sqrt{\pi}}\exp(-\eta_0^2)\eta_0^3X(\eta_0).
\eqno{(3.6)}
$$

Employing the Sokhotsky formula for difference from the boundary
values of function  $N(z)$, we find the coefficient of the
continuous spectrum:
$$
2\sqrt{\pi}\,i\,\eta^3\,\exp(-\eta^2)\,A(\eta)=\dfrac{C_0}
{\eta-\eta_0}\Big[\dfrac{1}{X^+(\eta)}-\dfrac{1}{X^-(\eta)}\Big].
$$

From the definition of the auxiliary function  $N(z)$  we have:
$$
N(0)=\dfrac{1}{\sqrt{\pi}}\int\limits_0^{\infty}
\eta^2\exp(-\eta^2)A(\eta)d\eta.
\eqno{(3.7)}
$$

Considering equality (3.7), we write (2.11) in form:
$$
-\dfrac{C_0}{\eta_0X(\eta_0)}+N(0)=\dfrac{1}{az_0}.
$$

From general solution (3.5) the constant $C_0$  is found:
$$
C_0=-\dfrac{\eta_0}{az_0}X(0).
\eqno{(3.8)}
$$

From comparison (3.6) and (3.8) we get the unknown coefficients
corres\-pon\-ding to the discrete spectrum:
$$
A_0=\dfrac{\sqrt{\pi}\,X(0)}
{a\,z_0\,X(\eta_0)\,\eta_0^2\,\exp(-\eta_0^2)}.
$$


The derivative of expression (2.11) leads us to equality
$$
\dfrac{e'(x)}{az_0^2}=\dfrac{C_0}{\eta_0^2X(\eta_0)}
\exp(-\dfrac{z_0x}{\eta_0})-
\dfrac{1}{\sqrt{\pi}}\int\limits_0^\infty
\exp(-\eta^2-\dfrac{z_0x}{\eta})\eta
A(\eta)d\eta.
$$

From here we find the electric field derivative on plasma boundary
$x=0$:
$$
\dfrac{e'(0)}{az_0^2}=\dfrac{C_0}{\eta_0X(\eta_0)}-
\dfrac{1}{\sqrt{\pi}}\int\limits_0^\infty
\exp(-\eta^2)\eta A(\eta)d\eta.
\eqno{(3.9)}
$$

For calculation of integral from the expression (3.9) we shall
use a derivative of auxiliary function
$$
N'(\mu)=\dfrac{1}{\sqrt{\pi}}\int\limits_{0}^{\infty}
\dfrac{\eta^3\exp(-\eta^2)A(\eta)
\,d\eta} {(\eta-\mu)^2}.\eqno{(3.10)}
$$

Using the expression (3.10) we find the derivative in zero:
$$
N'(0)=\dfrac{1}{\sqrt{\pi}}\int\limits_{0}^{\infty}
\eta\exp(-\eta^2)A(\eta)
\,d\eta.\eqno{(3.11)}
$$

Using expression for coefficient of the continuous spectrum by
(3.11) we find:
$$
N'(z)=-\dfrac{C_0}{(z-\eta_0)^2}\Big(\dfrac{1}{X(z)}-
\dfrac{1}{X(\eta_0)}\Big)-\dfrac{C_0}{z-\eta_0}\dfrac{X'(z)}
{X^2(z)},
$$
whence
$$
N'(0)=-\dfrac{C_0}{\eta_0^2}\Big(\dfrac{1}{X(0)}-
\dfrac{1}{X(\eta_0)}\Big)-\dfrac{C_0}{\eta_0}
\dfrac{X'(0)} {X^2(0)}.
\eqno{(3.12)}
$$

Substituting (3.12) into (3.9) we obtain the equation:
$$
e'(0)=\dfrac{az_0^2C_0}{\eta_0X(0)}\bigg[\dfrac{1}{\eta_0}-
\dfrac{X'(0)}{X(0)}\bigg]=z_0\bigg[\dfrac{X'(0)}{X(0)}-
\dfrac{1}{\eta_0}\bigg]. \eqno{(3.13)}
$$

In \cite{2} the following formula for the impedance is given:
$$
Z=\dfrac{4\pi i \omega l}{c^2z_0}\cdot
\dfrac{e(0)}{e'(0)}.\eqno{(3.14)}
$$

Substituting (3.13) into (3.14), we obtain the
exact expression for
the impedance:
$$
Z=\dfrac{4\pi i \omega l}{c^2 \,z_0}\bigg[\dfrac{X'(0)}{X(0)}-
\dfrac{1}{\eta_0}\bigg]^{-1}. \eqno{(3.15)}
$$

The equation (3.15) is the exact expression for calculation
of the surface
impedance. This equation expresses value of the impedance in terms
of the function  $X(z)$ and zeros of  the
dispersion function of the problem.


{\bf 4. The index of the problem equals one.}
%{\bf 4. Problem index is equal one.}
Now we consider case when the
problem index is equal one, i.e.  $\varkappa(G)=1$.

Substituting expressions (2.10) and (2.11) into (1.5),
we obtain the
singular integral equation with the Cauchy kernel:
$$
a\varphi(\mu)+\dfrac{a}{\sqrt{\pi}}\int\limits_{0}^{\infty}
\dfrac{\eta^3\exp(-\eta^2)A(\eta)}{\eta-\mu}\,d\eta+
\lambda(\mu)A(\mu)=0, \quad 0<\mu<\infty, \eqno{(4.1)}
$$
where
$$
\varphi(\mu)=\dfrac{1}{\sqrt{\pi}}\sum\limits_{k=0}^{1}
\dfrac{A_k\eta_k^3\exp(-\eta_k^2)}{\eta_k-\mu}.
$$

We define the auxiliary function in the complex plane:
$$
N(z)=\dfrac{1}{\sqrt{\pi}}\int\limits_{0}^{\infty}
\dfrac{\eta^3\exp(-\eta^2)A(\eta)}{\eta-z}\,d\eta,
$$
for which boundary values from above and from below on the
valid axis formulas of Sokhotskii are carried out:
$$
N^+(\mu)-N^-(\mu)=2\sqrt{\pi}\,i\,\mu^3\exp(-\mu^2)A(\mu).
$$

Using the boundary values  $N^\pm(\mu)$ and
$\lambda^\pm(\mu)$, we
pass from the singular equation (4.1) to the
Riemann --- Hilbert boundary value
problem:
$$
\lambda^+(\mu)[N^+(\mu)+\varphi(\mu)]=\lambda^-(\mu)[N^-(\mu)+
\varphi(\mu)], \quad 0<\mu<\infty. \eqno{(4.2)}
$$

Let's solve a corresponding Riemann --- Hilbert
boundary value problem:
$$
\dfrac{X^+(\mu)}{X^-(\mu)}=G(\mu),\qquad 0<\mu<+\infty,
\eqno{(4.3)}
$$
where
$$
G(\mu)=\dfrac{\lambda^+(\mu)}{\lambda^-(\mu)},\qquad
0<\mu<\infty.
$$

Since problem index is equal one $\varkappa(G)=1$  as the solution
of (4.3) we take function
$$
X(z)=\dfrac{1}{z}\exp V(z),\quad V(z)=\dfrac{1}{2\pi
i}\int\limits_{0}^{\infty} \dfrac{\ln G(\tau)-2\pi
i}{\tau-z}\,d\tau. \eqno{(4.4)}
$$

Using (4.4) we transform the boundary value problem (4.2) to
determining an analytic function from its zero jump problem
on the cut:
$$
X^+(\mu)[N^+(\mu)+\varphi(\mu)]=X^-(\mu)[N^-(\mu)+ \varphi(\mu)],
\quad 0<\mu<\infty. \eqno{(4.5)}
$$

The general solution of (4.5) is following
$$
N(z)=\dfrac{1}{\sqrt{\pi}}\sum\limits_{k=0}^{1}
\dfrac{A_k\eta_k^3\exp(-\eta_k^2)}{z-\eta_k}+
\sum\limits_{k=0}^{1}\dfrac{C_k}{(z-\eta_k)X(z)}, \eqno{(4.6)}
$$
where  $C_0$, $C_1$  is the arbitrary values.

 Since $N(\infty)=0$,  from here we receive that $C_0+C_1=0$.

Let's eliminate poles at the decision (4.6), we get:
$$
C_0=-\dfrac{A_0}{\sqrt{\pi}}\eta_0^3\exp(-\eta_0^2)X(\eta_0),
\quad
 C_1=-\dfrac{A_1}{\sqrt{\pi}}\eta_1^3\exp(-\eta_1^2)X(\eta_1).
\eqno{(4.7)}
$$

Substituting the general solution (4.6) in the Sokhotskii formula
for (4.6) we find unknown function called the coefficient of the
continuous spectrum:
$$
\eta^3\exp(-\eta^2)A(\eta)=\dfrac{1}{2\sqrt{\pi}\,i}
\bigg[\dfrac{1}{X^+(\eta)}-\dfrac{1}{X^-(\eta)}\bigg]
\sum\limits_{k=0}^{1}\dfrac{C_k}{\eta-\eta_k}. \eqno{(4.8)}
$$

Using (4.7) we reduce (2.10) to the form:
$$
-\dfrac{C_0}{\eta_0X(\eta_0)}-\dfrac{C_1}{\eta_1X(\eta_1)}+
\dfrac{1}{\sqrt{\pi}}\int\limits_0^\infty
\eta^2\exp(-\eta^2)A(\eta)d\eta=
\dfrac{1}{az_0},
\eqno{(4.9)}
$$


From equation (4.9) we will find last integral
using definition of function
$N(z)$:
$$
N(0)=\dfrac{1}{\sqrt{\pi}}\int\limits_0^\infty\eta^2
\exp(-\eta^2)A(\eta)d\eta.
$$

Using the common decision (4.6), we found  $N(0)$:
$$
N(0)=C_0\left[\dfrac{1}{\eta_0X(\eta_0)}-
\dfrac{1}{\eta_1X(\eta_1)}-\dfrac{1}
{\eta_0X(0)}+\dfrac{1}{\eta_1X(0)}\right]
$$

We substitute this expression into  the equation (4.9):
$$
C_0=\dfrac{X(0)}{az_0\Big(\dfrac{1}{\eta_1}-
\dfrac{1}{\eta_0}\Big)}.
\eqno{(4.10)}
$$

Differentiating expansion for electric field (2.11) and using (4.7),
where $x=0$  we found:
$$
e'(0)=\dfrac{C_0az_0^2}{\eta_0^2X(\eta_0)}-
\dfrac{C_0az_0^2}{\eta_1^2X(\eta_1)}-\dfrac{az_0^2}{\sqrt{\pi}}
\int\limits_{0}^{\infty}\eta \exp(-\eta^2)A(\eta)\,d\eta.
\eqno{(4.11)}
$$

Integral from (4.11) is the auxiliary function derivative
$N'(0)$:
$$
N'(0)=\dfrac{1}{\sqrt{\pi}} \int\limits_{0}^{\infty}\eta
\exp(-\eta^2)A(\eta)\,d\eta.
$$

The common solution (4.6) we present in form:
$$
N(z)=\dfrac{1}{z-\eta_0}\bigg[\dfrac{C_0}{X(z)}+
\dfrac{1}{\sqrt{\pi}}A_0\eta_0^3\exp(-\eta_0^2)\bigg]+
$$
$$
+\dfrac{1}{z-\eta_1}\bigg[\dfrac{C_1}{X(z)}+
\dfrac{1}{\sqrt{\pi}}A_1\eta_1^3\exp(-\eta_1^2)\bigg].
$$

Derivative of this expression is equal:
$$
N'(z)=-\dfrac{1}{(z-\eta_0)^2}
\bigg[\dfrac{1}{X(z)}-\dfrac{1}{X(\eta_0)}\bigg]-
\dfrac{C_0}{z-\eta_0}
\dfrac{X'(z)}{X^2(z)}+
$$
$$
+\dfrac{C_0}{(z-\eta_1)^2}\bigg[\dfrac{1}{X(z)}-
\dfrac{1}{X(\eta_1)}\bigg]+\dfrac{C_0}{z-\eta_1}\dfrac{X'(z)}
{X^2(z)}.
$$

Hence, we obtain:
$$
N'(0)=-\dfrac{1}{\eta_0^2}
\bigg[\dfrac{1}{X(0)}-\dfrac{1}{X(\eta_0)}\bigg]+
\dfrac{C_0}{\eta_0}
\dfrac{X'(0)}{X^2(0)}+
$$
$$
+\dfrac{C_0}{\eta_1^2}\bigg[\dfrac{1}{X(0)}-
\dfrac{1}{X(\eta_1)}\bigg]-\dfrac{C_0}{\eta_1}\dfrac{X'(0)}
{X^2(0)}.
$$



Using this equality for the electric field derivative we have:
$$
e'(0)=\dfrac{C_0az_0^2}{X(0)}\Bigg[\bigg(\dfrac{1}{\eta_0}-
\dfrac{1}{\eta_1}\bigg)\bigg(\dfrac{1}{\eta_0}+
\dfrac{1}{\eta_1}\bigg)+\dfrac{X'(0)}{X(0)}\bigg(\dfrac{1}{\eta_0}-
\dfrac{1}{\eta_1}\bigg)\Bigg].
$$

Therefore according to (4.10) we receive that
$$
e'(0)=z_0\Bigg[\dfrac{X'(0)}{X(0)}-\dfrac{1}{\eta_0}-
\dfrac{1}{\eta_1}\Bigg].
$$

Hence, expression for the impedance is equal:
$$
Z=\dfrac{4\pi i \omega l}{c^2 z_0}\cdot
\Bigg[\dfrac{X'(0)}{X(0)}-\dfrac{1}{\eta_0}-
\dfrac{1}{\eta_1}\Bigg]^{-1}.
$$

This equation expresses value of the impedance in terms of factor
function $X(z)$  and zeros of dispersive function of the problem.

\begin{center}
  CONCLUSIONS
\end{center}

A closed form solution of the system of two equations Boltzmann ---
Vlasov and Maxwell arising in the skin effect problem for a
Maxwellian plasmas is presented. The kinetic Boltzmann --- Vlasov
equation with a $\tau$--model collision operator is considered.
Case's \cite{7} method and Riemann --- Hilbert boundary value
problem \cite{8} with the coefficient $G(\mu)=\dfrac{\lambda^+(\mu)}
{\lambda^-(\mu)}$ is utilized, here $\lambda(z)$ is the dispersion
function of the problem. The exact solution of the initial boundary
value problem with diffusion scattering of electrons from plasma
boundary is constructed. The exact formula for calculation of the
surface impedance is obtained.

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\bibitem{5}{\sl I.D. Kaganovich, O.V. Polomarov and C.E.
Theodosiou.}
Resisting the anomalous  rf  field penetration into a warm
plasma// ArXiv: physics/0506135.


\bibitem{6} {\sl V.S. Vladimirov and  V.V. Zharinov.}
Equations of mathematical physics.
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\bibitem{7} {\sl K.M. Case and P.M. Zweifel.} Linear
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\bibitem{8} {\sl F.D. Gakhov.} Boundary Value Problems.
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\end{thebibliography}
\end{document}


\bibitem{9}{\it Lisanti M.} Observation of the skin--depth
effect on the Casimir force between metallic surfaces//
ArXiv:quant-ph/0502123.

\bibitem{10}{\it Nikitin A.Yu.,
L\'{o}pez-Tejeira F., Mart\'{i}n-Moreno L.} Scattering of surface
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ArXiv:cond-mat/0609278.

\bibitem{11}{\it Tomchuk P.M., Grigorchuk N.I.} Shape and size
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%\bibitem{4}{\it Zimbovskaya N.A.} Local Features of the Fermi
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%cond-mat/0506269.

%\bibitem{4'}{\sl Zimbovskaya N.A. }
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%normal magnetic field // ArXiv:cond-mat / 0601600.
In \cite{4} showed that electrons can transport the plasma
current away from the skin layer due their thermal motion
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numerical study provides through specific numerical
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find a compromise in handling the yet unresolved problem of
how to unify the kinetic response and collisions in wave
phenomena.
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