Exact plasma dispersion functions for complex frequencies
On the base the theory of Cauchy type integrals is given an analytic continuation of the exact relativistic plasma dispersion functions from the real axis into the complex region and studied their analytical properties in this region. На основі теорії інтегралів типу Коші дається аналітичне продовже...
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| Zitieren: | Exact plasma dispersion functions for complex frequencies / S.S. Pavlov, F. Castejon, N.B. Dreval // Вопросы атомной науки и техники. — 2007. — № 1. — С. 66-68. — Бібліогр.: 6 назв. — англ. |
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| author | Pavlov, S.S. Castejon, F. Dreval, N.B. |
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| citation_txt | Exact plasma dispersion functions for complex frequencies / S.S. Pavlov, F. Castejon, N.B. Dreval // Вопросы атомной науки и техники. — 2007. — № 1. — С. 66-68. — Бібліогр.: 6 назв. — англ. |
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| description | On the base the theory of Cauchy type integrals is given an analytic continuation of the exact relativistic plasma dispersion functions from the real axis into the complex region and studied their analytical properties in this region.
На основі теорії інтегралів типу Коші дається аналітичне продовження точних релятивістських плазмових дисперсійних функцій з реальної осі на комплексну область та вивчаються їх аналітичні властивості в цій області.
На основе теории интегралов типа Коши дается аналитическое продолжение точных релятивистских плазменных дисперсионных функций с реальной оси в комплексную область и изучаются их аналитические свойства в этой области.
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66 Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 66-68
EXACT PLASMA DISPERSION FUNCTIONS
FOR COMPLEX FREQUENCIES
S.S. Pavlov1, F. Castejon2, N.B. Dreval1
1Institute of Plasma Physics, NSC “Kharkov Institute of Physics and Technology”,
61108, Kharkov, Ukraine;
2Asociación EURATOM-CIEMAT PARA Fusión, 28040, Madrid, Spain
On the base the theory of Cauchy type integrals is given an analytic continuation of the exact relativistic plasma
dispersion functions from the real axis into the complex region and studied their analytical properties in this region.
PACS: 52.27.Ny
1. INTRODUCTION
The basic to study linear plasma waves in hot enough
plasmas is an evaluation of the relativistic Maxwellian
plasma dielectric tensor [1]. In order to give a recipe of it
for arbitrary plasma and wave parameters the exact
plasma dispersion functions (PDFs) in the form of
Cauchy type integrals with purely real density, defined at
the real axis and tending to zero in the infinite, were
introduced [2-4]. A dielectric tensor has been presented
as a finite Larmor radius expansion in terms of those
PDFs, similarly to cases of non-relativistic and weakly
relativistic approximations, to reduce an evaluation of the
tensor to the PDFs evaluation. The exact PDFs is a
generalization of the weakly relativistic PDFs [5] on the
case of an arbitrary plasma temperature. Two ways
evaluating the exact PDFs in the real frequency region
were given and their main analytical properties were
studied.
The main scope of the present work is an analytic
continuation of the exact PDFs from the real axis to the
complex region. On the base the theory of Cauchy type
integrals we study their analytical properties in this
region.
2. EXACT PDFs IN COMPLEX REGION
Exact PDFs of half-integer index q ( 2/3≥q ) for
real frequency ω are defined by Cauchy type integrals
[2-4]
×= −
− −−
1))((2
),,(
2
2
qaK
e
za
a
qZ
µπµ
β
µ
µβ
∫
∞+
∞− −
+−−−
−
+−
zt
ttttaaqK
q
tta deβ
µβµ )2(221
1
)2(2
,
1// >N ,
×−=
−
1))((2
),,(
2
qaK
e
zaqZ
µµ
πβ
µ
βµ
( ) ( )
∫
∞+
++
−+−
−
+
0
)/1)2/((21
1
)/1)2/((
razt
dttttaqI
q
tt e ββµββµ
10 // <≤ N ,
where 2/2
//Na µ= , )/1( ωωµ cz −= , )
2
//11( Nra −−= µ ,
Tcm /2
0=µ , )1/(1 2
//N−=β ; ω///// ckN = , cω , 0m ,T are the
longitudinal refractive index, fundamental electron
cyclotron frequency, rest mass of electron and plasma
temperature; )(1 xI q− , )(1 xKq− are modified Bessel
function and Macdonald function of half-integer index
1−q ; square root means the positive branch of this
function. If an argument z takes real values (ω is real)
both integrals are divergent at the poles zt = and
zat r −= (when raz ≤ ), respectively, and must be
understood as the Principal Part of these integrals in the
sense of Cauchy. The contour of integration in that case is
chosen to pass below the pole in the expression (1) and
above the pole in the expression (2).
For real argument z those integrals can be evaluated
by means of the next nonsingular integral forms [4]
∫
∞+
∞− −
−
+=
−
zt
dtaftaf
zafiqZ
tz ),,(),,(
),,(
211
1
µµ
µπ , (3)
∫
−
−+
−
+−−=
−−zra
razt
dtaftaf
zraafiqZ
tzra
0
),,(),,(
),,(
)(222
2
µµ
µπ
∫
∞− −+
−
−−0 ),,( )(22
razt
dtaf tzra µ ,
respectively, where
×+−
−
−= −
−
−−
12
1
2
2
1 ))2/((
))((2
),,( q
q
a
tta
aK
etaf µ
µπµ
β
µ
µβ
× ( ) t
q eK ttaa
β
µβ )2/(221 +−−− ,
( )
( )
≤
>
−
+−×
×
−
+−
−
=
.0,0
0
,)/1)2/((21
1
)/1)2/((1
))((22
),,(2
t
t
t
ettaqI
q
ttq
aK
e
taf
β
βµβ
βµ
µ
βµ
µ
πβ
µ
Calculating the exact PDFs on the basis of integrals
(1), (2) and using integrals forms (3), (4) allows one also
to continue analytically those PDFs on total complex
region on the base some facts from the theory of Cauchy
type integrals.
(1)
,(2)
(4)
67
0,04
0,07
0,1
0,5
0,04
0,04
3
0,04
10872,0
72,0
108
108
72,0
108
72,0
72,0
72,0
72,0108
108
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Y
X
Fig.1. Module-argument diagram of function
),,(2 12/5 µzaZa for 1.1// =N and =iT 100keV
0,04
0,08
0,2
0,7
0,04
0,04
1,4E2
45
45
1,4E2
1,4E2
45
1,4E2
1,4E2
45
45
45
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Y
X
Fig.2. Module-argument diagram of function
),,(2 12/5 µzaZa for 1.1// =N and =iT 1000keV
We start from the case 1// >N which is rather similar
to analytical continuation of the non-relativistic PDF
since in this case the integrand in (1) is also the entire
function at the contour of integration. Then for iyxz +=
in the upper semi-plane those PDFs are defined by the
expressions
∫
+∞
∞−
+∫
+∞
∞−
−
=
+−+−
+
22
1
22
1
)()(
),,(),,()(
),,(
yxtyxt
q
dttayf
i
dttafxt
zaZ
µµ
µ ,
0>y ,
),,(
),,(
),,( 1
1 µπ
µ
µ xafi
dttaf
xaZ
xt
q +∫
+∞
∞−
=
−
+ , 0=y , (6)
where formula (5) follows from expression (1) by means
of analytical continuation of integrand into upper semi-
plane and formula (6) is obtained from (5) by limit
passing 0→y . Divergent expression (6) can be evaluated
on the base nonsingular integral form (3). Then from (5)
and formulas of Sokhotskii-Plemelj it follows analytical
continuation of functions ),,( µzaZq from upper semi-
plane into the low one
),,(2),,(),,( 1 µπµµ zaifzaqZzaqZ +
+
=
+ ∗ , 0<y , (7)
where asterisk denotes complex conjugation. This branch
of ),,( µzaZq corresponds to Landau rule of passing the
pole. If to start from analytical continuation of (1) first
into the low semi-plane and then into the upper one we
will obtain the second branch ),,(),,( * µµ zaZzaZ qq
∗− =
which has a sense for negative values of //N [6].
At the Figs. 1,2 there are presented plots of module-
argument diagram for function ),2,(2 12/5 µzaaZa for
1.1// =N , which corresponds ICR frequency range, and
=iT 200, 2000 keV, respectively, for | 1z |=| )2/( az | ≤
≤ constant, obtained using formulas (5)-(7) (module is
presented in logarithm scale). It can be concluded from
these plots that exact PDFs are loosing module symmetry
which there is in non-relativistic PDF respectively of
imaginary axis. This anti-symmetry becomes more and
more essential with growing of ion plasma temperature.
0,1
0,2
0,4
1
3
0,1
6
0,1
1,3E2
54
54
1,3E2
1,3E2 54
1,3E2
1,3E2
54
54
54
1,3E2
54
54 54
-4 -3 -2 -1 0 1 2 3 4
-3
-2
-1
0
1
2
3
Y
X
Fig.3. Module-argument diagram of function
),,(2 12/5 µzaZa for =eT 2keV and 6.0// =N
For the case 10 // <≤ N on the same way we will have
next formulas for analytical continuation of formula (2)
for ),,( µzaZq on the whole complex plane with cutting
along the line raz =Re ( 0Im <z )
∫∫
+∞+∞
+
+−+
−
+−+
−+
=
0
22
2
0
22
2
)(
),,(
)(
),,()(),,(
yaxt
dttayfi
yaxt
dttafaxtzaZ
rr
r
q
µµ
µ ,
0>y ,
),,(
),,(
),,( 2
0
2 µπ
µ
µ xaafi
axt
dttaf
xaZ r
r
q −−
−+
= ∫
+∞
+ , 0=y , (9)
<>
<<−−
= ∗+
∗+
+
.0,Re),,(
0,Re),,,(2),,(
),,( 2
yazzaZ
yazzaaifzaZ
zaZ
rq
rrq
q µ
µπµ
µ
The cutting line is uniquely defined by PDFs behavior for
the case 1// >N while plasma temperature is increasing
(fig. 1,2). We will call this continuation first branch of the
function ),,( µzaZq in this case.
If to start from analytical continuation of expression
(2) into low semi-plane and then by similar way into
upper semi-plane we will obtain the function
(5)
(10)
(8)
68
),,(),,( * µµ zaZzaZ qq
∗− = with cutting along the line
raz =Re ( 0Im >z ). We will call this continuation which
has a sense for negative values of //N by second branch of
the function ),,( µzaZq in the case 10 // <≤ N . Obviously,
both those branches are identical for raz >Re .
Thus, we have obtained a double-valued analytical
function defined on the whole complex plane excepting
the points raz = and ∞=z , since it is known [5] that
function ),,( µzaZq and its derivatives till ( 2/1−q )th are
continues at the point raz = , and the ( 2/1−q )th
derivative has a single pole at that point. These branches
are separating when ∞→µ since +∞→ra in this case.
0,03 0,05
0,03
0,1
0,5
0,03
0,9
21,2E2
60
60
1,2E2
60
60
-15 -10 -5 0 5 10 15
-10
-5
0
5
10
Y
X
Fig.4. Module-argument diagram of function
),,(2 12/5 µzaZa for =eT 2keV and 06.0// =N
At the Figs. 3,4 there are presented the same plots as
in Figs. 1,2 for =eT 2 keV and =//N 0.6, 0.06,
respectively, which are relevant to ECR frequency range,
obtained using formulas (8)-(10). It can be conclude from
these plots that exact PDFs are loosing module symmetry
which there is in non-relativistic PDF respectively of
imaginary axis with decreasing of //N . This anti-
symmetry becomes more and more essential (similar to
the case 1// >N with increasing of T ) with decreasing of
//N . It worth to note that the number of zeroes in region
oz >1Re in this case is finite and defined by the cutting
line, in difference with the case 1// >N where the number
of such zeroes is infinite.
CONCLUSIONS
The next conclusions can be drawn from this study.
1. On the base the theory of Cauchy type integrals it
was studied analytical properties of exact PDFs in
complex frequency region.
2. For the case | //N | 1< , relating to the ECR frequency
range, it was shown that every exact PDF is two
branched analytic function for rn az <Re (one branch
has a sense for 0// >N and second branch for 0// <N )
with cutting line rn az =Re (these branches coincides
for rn az >Re ).
3. In the alternative case, | //N | 1> , relating to ICR
frequency range, these branches are separating, as in
non-relativistic approximation.
These results can be useful to study the properties of
plasma wave instabilities and collisionless dumping in the
frame of the initial value problem in relativistic regimes.
REFERENCES
1. B.A. Trubnikov // Plasma Physics and the problem of
Controlled Thermonuclear Reactions/ ed.
M.A. Leontovich, Pergamon, 1959, v. III, p.122.
2. S.S. Pavlov, F. Castejon // Probl .of Atom. Scien. and
Techn. Series “Plasma Physics” (11). 2005, N 2, p. 55.
3. S.S. Pavlov, F. Castejon // Probl. of Atom. Scien. and
Techn. Series “Plasma Physics” (11). 2005, N 2, p. 61.
4. F. Castejon, S.S. Pavlov. Relativistic plasma dielectric
tensor based on the exact plasma dispersion function
concept // Physics of Plasmas. 2006, v. 13, p.072105.
5. I.P. Shkarofsky // Phys. Fluids. 1966, v. 9, p. 561.
6. L.D. Landau // J. Phys. USSR, 1946, N10, p. 25.
C.C. , . , .
.
C.C. , . , .
.
|
| id | nasplib_isofts_kiev_ua-123456789-110414 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:18:59Z |
| publishDate | 2007 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Pavlov, S.S. Castejon, F. Dreval, N.B. 2017-01-04T12:19:05Z 2017-01-04T12:19:05Z 2007 Exact plasma dispersion functions for complex frequencies / S.S. Pavlov, F. Castejon, N.B. Dreval // Вопросы атомной науки и техники. — 2007. — № 1. — С. 66-68. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.27.Ny https://nasplib.isofts.kiev.ua/handle/123456789/110414 On the base the theory of Cauchy type integrals is given an analytic continuation of the exact relativistic plasma dispersion functions from the real axis into the complex region and studied their analytical properties in this region. На основі теорії інтегралів типу Коші дається аналітичне продовження точних релятивістських плазмових дисперсійних функцій з реальної осі на комплексну область та вивчаються їх аналітичні властивості в цій області. На основе теории интегралов типа Коши дается аналитическое продолжение точных релятивистских плазменных дисперсионных функций с реальной оси в комплексную область и изучаются их аналитические свойства в этой области. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Exact plasma dispersion functions for complex frequencies Точні плазмові дісперсійні функції для комплексних частот Точные плазменные дисперсионные функции для комплексных частот Article published earlier |
| spellingShingle | Exact plasma dispersion functions for complex frequencies Pavlov, S.S. Castejon, F. Dreval, N.B. Basic plasma physics |
| title | Exact plasma dispersion functions for complex frequencies |
| title_alt | Точні плазмові дісперсійні функції для комплексних частот Точные плазменные дисперсионные функции для комплексных частот |
| title_full | Exact plasma dispersion functions for complex frequencies |
| title_fullStr | Exact plasma dispersion functions for complex frequencies |
| title_full_unstemmed | Exact plasma dispersion functions for complex frequencies |
| title_short | Exact plasma dispersion functions for complex frequencies |
| title_sort | exact plasma dispersion functions for complex frequencies |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110414 |
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