A new method to compute the weakly relativistic plasma dispersion functions for complex argument is presented. It is demonstrated that Jacobi fractions represent those functions asymptotically for z→ ∞ in the upper semi-plane.Предлагается новый метод вычисления слаборелятивистских плазменных дисперсионных функций комплексного аргумента.Пропонується новий метод обчислення слабкорелятивістських плазмових дисперсійних функцій комплексного аргументу

Castejón, F.

Pavlov, S.S.

Tereshchenko, M.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

COMPUTATION OF THE WEAKLY RELATIVISTIC PLASMA DISPERSION FUNCTIONS USING SUPERASYMPTOTIC AND HYPERASYMPTOTIC EXPANSIONS  S.S. Pavlov1,  F. Castejón2,3, M. Tereshchenko3 1Institute of Plasma Physics NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine; 2 Laboratorio Nacional de Fusión, EURATOM / CIEMAT, Madrid, Spain; 3Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Zaragoza, Spain  A new method to compute the weakly relativistic plasma dispersion functions for complex argument is presented. It is demonstrated that Jacobi fractions represent those functions asymptotically for ∞→z  in the upper semi-plane.  PACS: 52.27.Ny 1. INTRODUCTION The evaluation of the weakly relativistic plasma dispersion functions (PDFs) also known as the Shkarofsky functions [1] is a ground of EC wave analysis in the laboratory thermonuclear plasmas. As a rule, in numerical applications these functions are calculated massively, therefore the efficiency of the involved computational algorithm is of primary importance. The theory of continued fractions of Jacobi has been proved to provide a basis for such fast calculation methods for the case of the complex error function , related to the nonrelativistic PDF by 2( ) exp( ) erfc( )w z z iz= − −( ) ( )Z z i w zπ= . The large-| |  asymptotic expansion in the region  for this function,  zIm( ) 0z ≥( )100 , odd1 , even2( ) , kkkkkk kiw zzμ μπ∞+=⎧⎪= +⎨Γ⎪⎩∼ ∑   is a divergent series, so it is undesirable to use this expansion directly for the computation of . Gautschi indicated [2] that changing the series to the continued J-fraction ( )w z10/// /1 213 2| | 1 2 | 1 | 3 2| | | |......kkk z z z zz zzzzπ πμ∞+==− − − −+ + + + =−−−−∑  can eliminate these difficulties and speed up the convergence in the region, where ( )w z  is of monotonic type. In this way, the evaluation of  requires some few first terms even if | |  is not very large. He has also shown empirically that for moderate and small | |  values one can use the continued fractions technique applied for computations at the outer boundary of this domain in combination with the Taylor expansion in the descending direction of the imaginary axis towards the target points. ( )w zzzOne can apply this method for fast computation of the weakly relativistic PDFs, as follows. The two lowest-order PDFs can be expressed in terms of  [3] and computed using the above method, and then those of higher orders are sequentially evaluated by employing the 2nd-order recursion relation between them. However, this technique lacks stability when the argument of becomes large [4]. ( )w z( )w zThe main purpose of the present work is to develop another, direct (without use of the recurrent calculations) computational algorithm for the weakly relativistic PDFs, on the basis of superasymptotic and hyperasymptotic similar to that of Ref. [2].  2. MATHEMATICAL GROUNDS From several equivalent definitions of the function  we shall exploit the Cauchy integral  ( )w z2exp( )1( ) t dtw zi t zπ+∞−∞−=−∫ ,   (1) where the contour of integration passes below the pole. According to the theory of Cauchy integral, the function  can be analytically continued to the whole complex plane. Taking the derivative of (1) with respect to ( )w zz , we will obtain another definition of    ( )w z2( ) 2 ( )iw z zw zπ′ = −     (2) with the condition (0) 1w = . Substituting the Taylor expansion  in (2) will bring us to the power series definition 0( )nnnw z a z+∞== Σ( )0 2( )( )1nnnizw z+∞==Γ +∑  .  (3) This series has infinite radius of convergence, therefore  belongs to the class of entire functions, which is the extension of a polynomial class and hence it is the class of simplest, after polynomials, analytical functions.  ( )w zThe most basic properties of a given analytical function are defined via its singularities (poles, essential singularities, branching points). In this regard the theory of entire functions can be named the theory of essential singularity because any entire function excepting a polynomial has an essential singularity at infinity. The entire functions, as more simple, were the first studied (see e.g. [5]). Particularly, according to Picard’s theorem, in the vicinity of an infinite point the modulus of an entire function attains every finite positive value. In the view of asymptotic expansion theory, both the concept of modulus of an analytical function and its behavior near the essential singularity at infinity are very important. Indeed, asymptotic properties of analytical functions are PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2010. № 6.                                                         73 Series: Plasma Physics (16), p. 73-75. associated not only with the principle of modulus maximum but also with that of modulus minimum, which is valid except for the vicinities of the function’s zeroes, generally located at the Stokes lines [6]. According to these principles, the modulus of an entire function has equal number of regions of maximum and minimum in any vicinity of infinity, with the Stokes lines lying between them. If the modulus between two adjacent Stokes lines tends to zero (or constant) in some region at infinity, the asymptotic properties of the function depend on the value of its decrement. In this regard  is the simplest case of a single maximum and minimum, which are separated by two Stokes lines corresponding to the bisectors of the 3rd and 4th quadrant and pass through the zeroes of . In the more wide upper sector, which corresponds to the minimum, the function  has a unique asymptotic expansion in the form of divergent series because otherwise (convergent series) there would be no essential singularity at infinity. This series converges in asymptotic sense and the speedy of this convergence has the exponential character. If it is necessary the further convergence acceleration can be provided on the base the continued fractions or Pade approximants techniques. Nevertheless, such common reasoning certainly needs a clear proof.  ( )w z( )w z( )w zSubstituting the expansion at infinity 0( )nnnw z a z+∞== Σ  in (2) and using the condition , which is again the result of (1), we obtain the real-axis asymptotic expansion: ( ) 0w ∞ =( ) 2 1021( ) nniw z n zπ+∞+=≈ Γ +∑  .  (4) Firstly, let us show that (4) is valid not only for the real valued , but at least for the whole upper half-plane. As it is known from the theory of Cauchy integral, analytical continuation of the real-axis integral (1) to the domain of zIm 0z >  can be made by corresponding analytical continuation of the subintegral function 1 ( )t z− . We shall use this fact in the following proof. It is also known that the functions , 2( ) Re ( ) exp( )u z w z z≡ = −74 2( ) Im ( ) ( )z w z D zπν ≡ = , where 2 20( ) exp( ) exp( )zD z z t dt= − ∫  is the Dawson integral, are connected through the Hilbert transform formulas: ∫∫+∞∞−+∞∞− −−=−−=zdvPzuzduPzivτττπτττπ)(1)(,)(1)(   (5) (P  denotes the Cauchy principal value of the integral). Relations (5) are valid as far as  and ( )u z ( )zν  satisfy both the Holder condition on the real axis and the conditions , ( ) 0u z → ( ) 0zν →  . One more representation of  can be stated using (1) and (5): (z → ±∞)( )w z( )1( ) D t dtw zt zπ+∞−∞=−∫ .   (6) Now it follows from (1) and (6) that ( )1( )2w t dtw zi t zπ+∞−∞=−∫ .   (7) On the other hand,  is an analytical function at the domain of Im( )w z0z ≥ , therefore one can apply the Cauchy integral formula to any internal finite contour. Let this contour be a semicircle of radius R  traversed counterclockwise, then the following representation is valid inside it: ( ) ( )1 1( )2 2RR arcw t dt w t dtw zi t z i t zπ π+−= +− −∫ ∫ . (8) Comparison of (7) and (8) in the limit R  yields → ∞( ) 0arcw t dtt z→−∫     for any contour with R → ∞ . From this result it follows that  at ( ) 0w z → z → ∞  in the upper half-plane.  It is known from the theory that an analytical function, which is given by the Cauchy integral defined on the real axis, can be continued from the upper half-plane to the lower one by the following relation: 2( ) ( ) 2 exp( )w z w z i zπ∗ ∗= + −  .   Therefore, the asymptotic behavior of  at infinity in the lower half-plane is the same as in the upper one only in the sectors ( )w z4 arg 0zπ− < ≤  and arg 5 4zπ π≤ < , where 2exp( ) 0z →−  at z → ∞ . It is clear from the above shown proof that at a fixed x  and y → +∞  in the upper half-plane )exp()(Re 22 yxiyxw +→+ , )exp()(Im 22 yxiyxwi +−→+ , eliminating each other exactly and providing . As a consequence, either integration or Taylor expansion of the function  for its calculation in the region of small and moderate | |  values has to be directed downward the imaginary axis in order to preserve the accuracy, as it was done in the algorithm of Gautschi.  0)( →zw( )w zzWith a few modifications, this proof can be used for wide class of functions capable of being represented as a Cauchy-type integral defined on the real axis, provided that the density of the corresponding integral vanishes at infinity. This is the case of the weakly relativistic PDFs. Really, the weakly relativistic PDF of index 2/3+q  can be written in the following form [7]: 1/ 21/ 23/ 2 1/ 20( ) (2 )( , )( )q uaqq qu e I aueF z a dua u z a+ −∞−++ += + −∫, (9) where 1/ 2qI +  is the modified Bessel function and the integration contour passes above the pole.  The functions 3/ 2qF +  defined by expressions (9) have the real branching points at a . In such cases the integration contour has to be added with the semicircle of radius ε  passing around the branching point in the upper half-plane. Therefore the respective elaboration of the formula (8) will contain an extra term in the right hand side, which is vanishing at 0ε →  (due to continuity of 3/ 2qF +  at a  point), and hence has no effect onto the further reasoning.  Note that the Stokes lines of the weakly relativistic PDFs are not coincided with that of , as far as the densities of integral (9)  determine them. But evidently, these two lines are still located in the 3rd and 4th quadrants, and this important fact will be of further use.  ( )w zwith the starting values  and ),( 00 aiyxFq +),(),(),( 0010000 aiyxFaiyxFaiyxF qqq +−+=+′ −  at the boundary between regions R and Q obtained from continued fraction expansions. Calculations show that the application of this method gives an average gain in time in comparison with the technique of Cauchy-type integrals approximately of a factor 20.  In comparison with the method [3] this technique gives an average loss in time with a factor 5 but nevertheless it allows to avoid instability of computation  for high enough q . ),( azFq3. EVALUATION OF THE WEAKLY RELATIVISTIC PDFs We’ll assume iyxz +=  to lie in the region  of the plane. This is no restriction of generality, since 0≥y⎪⎩⎪⎨⎧><−+= ∗∗∗,Re,),(Re),,(2),(),(azazFazazaifazFazFqqqπ   4. CONCLUSIONS where )2()/(),( 12/)1( auIaueauf qqua −−−−= , can be used to continue  into the remaining half-plane. ),( azFq1. It is demonstrated that for the weakly relativistic PDFs asymptotic expansions in the vicinity of infinity totally define those PDFs in the whole complex plane. Computation of  in the region Q of large ),( azFqaz −  values i.e. superasymptotic part of calculations, was performed on the basis of asymptotic expansion 2. On this basis a new method to compute the weakly relativistic plasmas dispersion functions for complex argument is presented. 1,11,,0,01,)1(,1,)(),(−+−+∞=++−+==−−= ∑jqjqjqqjjjqqaCCqjCCzaCazF  3. It is shown that the method of Gautschi for fast evaluation of non-relativistic plasma dispersion function can be generalized to the weakly relativistic case directly, without usage of the unstable, for large | | values, recursive equation for the weakly relativistic functions.  zand use of Jacobi continued fractions in total correspondence with [8] and the chain length 9=n  to provide 10 correct decimal digits after the decimal point.  REFERENCES In the region S of small az −  values we use the method of [3] since in this region it is stable and most fast. 1. I.P. Shkarofsky // Phys. Fluids. 1966, v. 9, p. 561. 2. W. Gautschi //SIAM J. Numer. Anal. 1970, v. 7, p. 187. In the region R of moderate az −  there simultaneously used both the J-continued fraction expansion with higher the values of  and the Taylor expansion of  in the direction of negative imaginary semi-axis  n),( azFq3. V. Krivenski  and A. Orefice // J. Plasma Physics. 1983, v. 30, p. 125.  4. M. Brambilla. Kinetic theory of plasma waves in homogeneous plasmas. Oxford: “University Press”, 1998. 5. B.Ya. Levin, Yu. Lyubarskii, M. Sodin, V. Tkachenko. Lectures on Entire Functions. Providence: “AMS”, 1996. [ ] ...,)(),(21)(),(),(),(20000000+−+′′+−+′++=+yyiaiyxFyyiaiyxFaiyxFaiyxFqqqqwhich is most favorable as was shown in the previous section. In this expansion the derivatives of  were defined from the equation ),( azFq6. J.P. Boyd. The devil's invention: asymptotic, superasymptotic and hyperasymptotic series // Acta Appl. Math. 1999, v. 56, p. 1. 7. P.A. Robinson // J. Math. Phys. 1987, v. 28, p. 1203. [ ] [ ].0)2()1(2)(2)()3()2()1()(=−−−−++−−−−−−−nqnqnqnqFnFnqzFnqzaFza 8. S.S. Pavlov, F. Castejon, A. Cappa, M. Tereshchenko // Problems of Atomic Science and Technology. Series “Plasma Physics” (15). 2009, N 1, p. 69.  Article received 6.10.10  ВЫЧИСЛЕНИЕ СЛАБОРЕЛЯТИВИСТСКИХ ПЛАЗМЕННЫХ ДИСПЕРСИОННЫХ ФУНКЦИЙ  НА ОСНОВЕ СУПЕРАСИМПТОТИЧЕСКИХ И ГИПЕРАСИМПТОТИЧЕСКИХ РАЗЛОЖЕНИЙ  C.C. Павлов, Ф. Кастехон, М. Терещенко  Предлагается новый метод вычисления слаборелятивистских плазменных дисперсионных функций комплексного аргумента.   ОБЧИСЛЕННЯ СЛАБКОРЕЛЯТИВІСТСЬКИХ ПЛАЗМОВИХ ДІСПЕРСІЙНИХ ФУНКЦІЙ  НА ОСНОВІ СУПЕРАСИМПТОТИЧНИХ ТА ГІПЕРАСИМПТОТИЧНИХ РОЗКЛАДАНЬ C.C. Павлов, Ф. Кастехон, М. Терещенко  Пропонується новий метод обчислення слабкорелятивістських плазмових дисперсійних функцій комплексного аргументу.  75 

Computation of the weakly relativistic plasma dispersion functions using superasymptotic and hyperasymptotic expansions

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Computation of the weakly relativistic plasma dispersion functions using superasymptotic and hyperasymptotic expansions

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)