NUMERICAL EVALUATION OF H-FUNCTION BY CONTINUED FRACTIONS ABSTRACT INTRODUCTION

Transcription

1 NMERICAL EVALATION OF H-FNCTION BY CONTINED FRACTIONS B.S. Rn & H.S. Dhmi Deptt. Of Mthemtics niversity of Kumun S.S.J. Cmpus, Almor (ttrnchl INDIA ABSTRACT In the present pper n ttempt hs been mde to evlute for different vlues of prmeters m, n, p, q nd the vrible in the rnge 0. to 0.0 by the ppliction of continued frctions INTRODCTION Fox H-function is one of the most generlized function mong ll specil functions. It covers wider rnge nd gives deeper, more generl nd more useful results directly pplicble in vrious problems rising in physicl nd biologicl sciences, engineering nd sttistics. With n im to provide ssistnce to those mthemticins, engineers, scientists nd sttisticins who discover tht they need to generte numericl vlue of the specil function in the course of solving their problems, Lozier nd Olver [3] hve prepred report which consist reviews of pcges, librries nd systems usble in the evlution of clssicl functions but not for more generlized functions lie E-function, G-function, H-function etc. Lozier [] hs developed softwre test device for use in

2 testing the ccurcy or numericl precision of mthemticl softwre for elementry specil functions. Specil functions cn be evluted by continued frctions using qd scheme. Such type of wor ws initited long bc in 95. Detiled ccount of which cn be seen to the boo of P. Henrici [6]. Historicl survey of continued frctions nd Pde pproximtion hs been compiled in the wor of Brezinsi []. A code of evlute modified Bessel s functions bsed on the continued frction method hs been investigted by Segur et. l [] nd for higher order function, the device hs been developed by Rtis nd Crdob [8]. Rtios of contiguous hypergeometric functions of the type 3 F hve been represented in the form of continued frctions by Singh [7]. A set of fst codes to clculte Bessel functions of integrl nd frctionl orders, modified Bessel functions etc., bsed on the continued frctions method hs been presented by Monrel et. l [5]. We re ming n ttempt to evlute more generlized function from mong specil functions, the H- function by the ppliction of continued frction for vlues of prmeters in the rnges :- m vrying from to n hs been fixed s 0 p vrying from 0 to q vrying from to nd vrible vrying in the rnges 0. to 0.0

3 3. GENERAL FORMLATION Appliction of initil conditions to the qd schemes ssocited with confluent hypergeoemtric function, Guss hypergeometric function nd Bessel s function hve yielded results, which hs been compiled in the boo of Henirici [6] but does not throw ny light on the evlution of generlized function lie G-function, H-function etc. We hve generted following results under qd scheme for the H- function q e q ( n ( n ( n ( n γ ( n n 03,,,, ( n( n γ ( n ( n (,, ( γ ( n 3( n ( 3,, V W (. q q e ( n ( n ( n ( n( n δ ( γ n n 03,,,, ( n ( γ δ ( γ n 3( γ n ( 3,, ( γ n δ ( γ n ( γ n (,, V W (.

4 Corresponding contined frction for the function cn be obtined with the help of following results:- q 0 e 0 q 0 The division lgorithm for the H-function bsed hs been produced s Jν( Jν ( ( ν ( ν ( ν ( v 6 ( v 8 V W (. 3

5 5. NMERICAL EVALATION OF THE FNCTION In the present section we hve defined vrious H- functions by RIT method nd then qd - scheme hs been utilized in order to tbulte the function for vlue of, nd. Defining the function L NM H 3 [ /(,0 0, 3 O QP,] V W by RIT method (. nd then ting qd Schemes ( ( where,,3, V W (. we cn tbulte the functions by ming the substitutings 0., 0., 0.3, 0.,0.5,0.6,0.7,0.8,0.9,.0 0., 0.3,0.,0.5,.0 in the from of tble (Appendix-I

6 6 The function H cn be defined with the help of RIT 0 0,, (,,(, - method in the following form:- e e e / π π (.3 we cn tbulted the function by ming the substitutings 8 V W b gb g b g ( (, , (. 0.,0.,0.3,.0.,0.5,0.6,0.7,0.8,0.9,.0,.0,3.0,.0,5.0,,,3,, in the form of tble (Appendix-II

7 7 The function H (,,(,,(, (, 0, 0, the help of RIT - method in the following form cn be defined with bπg L NM e 3 L NM bπg e F e I H K O QP O P Q 5 P (. nd then ting qd schemes b 8 gb g 6 ( 6 b ( g where 3,,, V W ( 6. we cn tbulte the functions by ming the substituting 0.,0.,0.3,0.,0.5,0.6,0.7,0.8,0.9,.0,, In the form of tble (Appendix-III

8 8 The function H,, cn be defined with the help of RIT method in the following form Γ( 0,, Γ( Γ( 5 6 V W ( 7. nd then ting qd schemes s ( ( ( ( ( 3,, V W ( 8. we cn tbulte the function by ming the substitutings ccording condition z< 0.,0.,0.3,0.,0.5,0.6,0.7,0.8,0.9,3/, /, in the form of tble (Appendix-IV

9 9 The function H 0,, method in the following form (, (, (, cn be defined with the help of RIT π o L NM cos( π( Γ( cos( π πγ( 3 5 πγ( t O QP V W ( 9. nd then ting qd schemes s ( ( ( 3,, V W ( 0. we cn tbulte the function by ming the substituting 0.,0.,0.6,0.8,.0,.0,3.0,.0,5.0,6.0,7.0,8.0,9.0,0.0 0,, in the form of tble (Appendix-V

10 0 0, (, - The function H, (, (, cn be defined with the help of RIT method in the following form e L NM e O QP V W e (. nd then ting qd schemes s F HG I K JF HG ( ( ( 3,, we cn tbulte the function by ming the substituting 0.,0.,0.6,0.8,.0,.0,3.0,.0,5.0,6.0,7.0,8.0,9.0,0.0 0,, in the form of tble (Appendix-VI I K J V W (.

11 Appliction of division lgorithm to the expnsion of H 0 0,, (,,(, by RIT method yields. 3 ( ( ( ( ( ( ( ( ( 6 (( (( (( (( ( 6 (( (( (( (( ( J J J ----(.3 The tble cn be constructed by ming the substitutings 0.,0.,0.3,0.,0.5,0.6,0.7,0.8,0.9,.0,.0,3.0,.0,5.0,,3, s compiled in (ppendix - VII

12 Above defined proceedure, when pplied to the H (,,(,,(, (, 0, 0, produces following expression :- J ( ( L NM l q l q l q l q ( ( ( ( 6 O QP J ( ( ( ( ( ( 3 l q l q l q l q --(. we cn tbulted the function by ming the substitutings 0.,0.,0.3,.0.,0.5,0.6,0.7,0.8,0.9,.0,, in the form of tble (Appendix-VIII

13 3 Appedix-I

14 Appedix-II

15 5 Appedix-III,,,, 0. (0 0. ( ( ( ( ( (0 0.7 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (00.0 ( (0 0.0

16 6 Appedix-IV,, 3, 3,,

17 7 Appedix-V ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (0.9 ( ( (0.68 ( ( ( ( (0.757 ( 0.05 ( (0 8. ( ( ( 0. ( ( 0.8 ( ( ( ( ( ( ( ( 0.80 ( ( 0.07

18 8 Appedix-VI (0.387 ( ( ( (0.056 ( 0.86 ( (0.770 ( (0 0. ( (0.550 ( ( (0.583 ( (0 0.9 ( ( ( ( ( ( ( ( (- 0.3 ( ( (- 0.7 (-0.77 ( ( ( ( ( ( ( ( ( (- 0.0 ( (-0.057

19 9 Appedix-VII,,, 3,

20 0 Appedix-VIII,,,, 0. ( ( ( ( (- 0. ( ( ( ( ( ( ( (0-0.7 ( ( ( ( ( (0-0.7 ( ( ( ( ( ( ( ( ( ( (

21 REFRENCES [] C. Brezinsi, History of continued frctions nd pde pproximnts Springer series in computtionl mthemtics, Vo.. Springer Verlg Berlin, 99 [] D.W. Lozier (997 A proposed softwre test service for specil functions in the boo of R. Boisvert (ed.. The qulity of Numericl Softwre : Assessment nd Enhncement Chpmn nd Hll. Londn. [3] D.W. Lozier nd F.W.J. Olver (Dec. 000 Numericl evlution of specil function in the boo of Wlter Gurtchi (ed. Mthemtics of computtion : A Hlf century of computtionl mthemtics, proceedings of symposi in Applied mthemtics, Americn mthemticl society province Rhode Islnd 090. []. J. Segur P Fermndez de Cord b nd Yu. C. rtis (997 : A code evlute modified Bessel functions bsed on the continued frction method Compute. Phy. Comm. 05 No. -3, [5]. L. Monrel, P. Frenndez de cord J. A. Monsoriu nd S. Abrhm Ibrhim (003 specil function evlution using continued frctions methods nd Engineering AIP Conference Proceedings, 66 (I PP [6]. P. Henrici (977 Specil function, Intregl trnsform symptotics nd continuted frctions Applied nd computtionl Complex Anlysis-, Wiley New Yor. [7]. R.Y. Denis, S.N. Singh (000 Representtion of rtios of contiguous hypergeometric functions in the form of continued frctions, Fr. Est J. Mth Sci. (FJMS (000 03, [8]. Yu. L. Rtis nd P. Fremndez de cordobe, A code to clculte (high order Bessel functions bsed on the continued frctions method, Compute Phys. Comm. 76 (993, ****