E. A. Evangelidis

Evangelidis

G. J. J. Botha

Nadarajah

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Reply to comment on “Relativistic Landau resonances”

Botha, Gert

Evangelidis, E. A.

Northumbria University Research Portal

Citation:  Botha,  Gert  and Evangelidis,  E.  A.  (2007) Reply  to comment  on “Relativistic  Landau  resonances”. Journal  of Geophysical  Research,  112 (A5).  A05206. ISSN  0148-0227Published by: American Geophysical UnionURL: http://dx.doi.org/10.1029/2007JA012315This  version  was  downloaded  from  Northumbria  Research  Link:  http://nrl.northumbria.ac.uk/11010/Northumbria  University  has  developed Northumbria  Research  Link  (NRL)  to  enable users to access the University’s research output.  Copyright  © and moral  rights  for  items on NRL  are retained by the individual  author(s) and/or other  copyright  owners.  Single  copies of full  items can be reproduced, displayed or performed, and given to third  parties  in  any  format  or  medium  for  personal  research or  study,  educational,  or  not-for-profit  purposes without  prior  permission  or  charge,  provided  the  authors,  tit le  and  full  bibliographic  details  are  given,  as  well  as  a  hyperlink  and/or  URL  to  the  original  metadata  page. The content  must  not  be changed in  any way.  Full  items must  not  be sold commercially  in  any format  or medium  without  formal  permission of the copyright  holder.  The full  policy is available online: http://nrl.northumbria.ac.uk/policies.htmlThis document  may differ  from the final,  published version of the research and has been made available online in  accordance with  publisher  policies. To read and/or cite from the  published  version  of the  research,  please visit  the  publisher’s  website  (a subscription  may be required.)Reply to comment on ‘‘Relativistic Landauresonances’’G. J. J. Botha1 and E. A. Evangelidis2Received 1 February 2007; accepted 13 March 2007; published 12 May 2007.Citation: Botha, G. J. J., and E. A. Evangelidis (2007), Reply to comment on ‘‘Relativistic Landau resonances,’’ J. Geophys. Res.,112, A05206, doi:10.1029/2007JA012315.[1] In a comment on our paper [Evangelidis and Botha2005], with reference to the solution of integrals (63) and(64), Nadarajah [2007] pointed to an alternative formula-tion obtained from Prudnikov et al. [1986, equation(2.12.39.6)]. An alternative method of evaluating theseintegrals is to be welcomed. In order to compare theformulations of the answer in both our published paperand using the formula from Prudnikov et al. [1986], werewrite the answers to integrals (63) and (64) in the work ofEvangelidis and Botha [2005], using the formula given byPrudnikov et al [1986]. When one does this, integral (63)becomes2 mkBTð Þ2X1n¼04n 1=2ð Þn 2ð Þn1ð Þn 1ð Þnn!yð Þn ð1Þand integral (64)2 mkBTð Þ2yX1n¼04n 3=2ð Þn 2ð Þn1ð Þn 3ð Þnn!yð Þn; ð2Þwhere y is a dimensionless quantity defined by (65) in thework of Evangelidis and Botha [2005]. (x)k is thePochhammer symbol. All the other symbols have the samemeaning as in the work of Evangelidis and Botha [2005].[2] It is immediately obvious that the above formulationis as easy to implement numerically as the one obtained byEvangelidis and Botha [2005], with the ascending factorialnotation of the original replaced with ascending anddescending factorials. The expressions are equivalent andthe choice of usage is down to personal preference. Giventhe fact that the integrals reduce to the simplified expres-sions given above and in (63) and (64) of Evangelidis andBotha [2005], involving only multiplication, there is noneed to use either hypergeometric or gamma functions whencomputing the result numerically.[3] Acknowledgments. Amitava Bhattacharjee thanks Alain Brizardfor the assistance in evaluating this paper.ReferencesEvangelidis, E. A., and G. J. J. Botha (2005), Relativistic Landau reso-nances, J. Geophys. Res., 110, A02216, doi:10.1029/2004JA010756.Nadarajah, S. (2007), Comment on ‘‘Relativistic Landau resonances’’ byE. A. Evangelidis and G. J.J. Botha, J. Geophys. Res., doi:10.1029/2007JA012266, in press.Prudnikov, A. P., Y. A. Brychkov, and O. I. Marichev (1986), Integrals andSeries, vol. 2, Special Functions, Gordon and Breach, New York.G. J. J. Botha, Department of Applied Mathematics, University of Leeds,Leeds LS2 9JT, UK. (gert@maths.leeds.ac.uk)E. A. Evangelidis, Laboratory of Non-Conventional Sources of Energy,Demokritos University of Thrace, Kimeria, Xanthi, Greece. (eevangel@env.duth.gr)JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, A05206, doi:10.1029/2007JA012315, 2007ClickHereforFullArticle1Department of Applied Mathematics, University of Leeds, Leeds, UK.2Laboratory of Non-Conventional Sources of Energy, DemokritosUniversity of Thrace, Xanthi, Greece.Copyright 2007 by the American Geophysical Union.0148-0227/07/2007JA012315$09.00A05206 1 of 1

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