Volum Issu July 0 ISSN: X Intrntionl Journl of Advnd Rsrh in Computr Sin nd Softwr Enginring Rsrh Ppr Avill onlin t: www.ijrss.om Dominting Funtion Thory from Nwton to Linitz s Approh of Indfinit Intgrtion Dhrmndr Kumr Ydv * Assistnt Profssor of Mthmtis Glgotis Univrsity Grtr Noid UP Rsrh Sholr Vino Bhv Univrsity Hzrig Jhrhnd Dip Kumr Sn Assoit Profssor of Mthmtis R. S. Mor Collg Govindpur Dhnd-09 Ph. D. Suprvisor Astrt In th ppr w hv summrizd th d-funtion thory its proprtis nd pplitions in th indfinit intgrls of som indfinit nonintgrl funtions disussd y Ydv & Sn nd lso to som lssil nonlmntry funtions. Th intgrls not oming undr ny of th dominting funtions form hv n lft in sris nd thos whih r not d-l hv n lft with ommnt tht thy r not intgrl. Kywords Nonlmntry funtions Eistn thorm on indfinit intgrility sd on d-funtions Dominting Squntil nd Dominting squntil funtions t. 00 AMS Sujt Clssifition-- 0E0 0D0 0K0 I. INTRODUCTION Th first rportd study of indfinit nonintgrl funtions llipti intgrls ws du to John Wllis. Suh intgrls nnot vlutd in trms of lmntry funtions ws provd y Josph Liouvill in. Liouvill lso rtd frmwor for onstrutiv intgrtion y finding out whn indfinit intgrls of lmntry funtions r gin lmntry funtions whih lid th foundtion of modrn intgrl lulus Glois diffrntil qutions Symoli intgrtion Computr lgr t. But no ttmpts wr md to m thm intgrl or to ovrom th prolms of thir nottions. Ydv & Sn [-] introdud si stndrd forms of indfinit nonintgrl funtions nd provd thm y th hlp of strong Liouvill s thorm nd its spil ss. Thy [-0] pplid Nwton s pproh to find thir intgrls without ny mthmtil nottions. To giv mthmtil symols of intgrls in sris disussd in [-0] thy [] introdud stndrd dominting funtion. Thn thy [-] introdud diffrnt d-funtions sd on trigonomtri hyproli ponntil nd rithmi funtions. Applying Nwton s pproh of intgrtion in sris thy [ ] stlishd nw istn thorm on indfinit intgrility nd found th indfinit intgrls of mny lssil nonlmntry funtions. Th omplt wor of d-funtion thory nd its pplitions hs n prsntd in dtil in th Ph. D. thsis sumittd y Ydv [] undr th suprvision of Dr. Sn. II. PRELIMINARY IDEAS Nwton nd Linitz s Approh of Intgrtion: Th two prinipl invntors of lulus Nwton nd Linitz hd vry distint pprohs to intgrtion. Nwton llowd for infinit sris solutions nd vlutd intgrls y pnding funtions in powr sris nd intgrting trm-y-trm. In ontrst Linitz fvourd solutions in finit trms nd srhd for losd form prssions of intgrls. Wll into th th ntury mthmtiins prssd diffrnt prfrns finit vs. infinit sris for rprsnttions of indfinit intgrls. In dominting funtion thory w hv followd oth of th pprohs. Dominting Funtion Thory: Ydv & Sn [-] hv dvlopd this thory in sris of pprs whih hs n wll plind nd prsntd in Ydv s Ph. D. thsis []. Thy hv propoundd th following trms nd proprtis: Dominting Funtion: An infinit sris of th form Bn n 0 n n C A hs n lld dominting funtion nd thos funtions whih n writtn in this form hv n lld domintl funtions y Ydv nd Sn []. Thorm.: Dominting funtion is lwys ontinuous diffrntil nd indfinit intgrl pt t th point of disontinuity. Thorm.: Nssry Condition for Indfinit Intgrility: Evry indfinit intgrl funtion is domintl funtion. Empl: h ++ + t. 0 IJARCSSE All Rights Rsrvd Pg n n
Ydv t l. Intrntionl Journl of Advnd Rsrh in Computr Sin nd Softwr Enginring July - 0 pp. -9 Thorm.: Suffiint Condition for Indfinit Intgrility: If f domintl funtion it is indfinit intgrl. Empl: t. n n os Thorm.: Nssry nd Suffiint Conditions for Indfinit Intgrility: A funtion f is indfinit intgrl if nd only if it is domintl. In othr words if funtion f hs n indfinit intgrl thn it is D-l funtion nd onvrsly if f is D-l funtion it hs n indfinit intgrl. III. APPLICATIONS To find th indfinit intgrls w first find th Tylor s or Lurnt s sris pnsion of th intgrnd to show tht whthr thy r d-l funtion or not to tst th onditions of istn thorm of intgrility. Thrftr w find thir intgrls in sris nd dnot thm in diffrnt forms of dominting funtions if possil. In othr words w strt with Nwton s pproh of indfinit intgrtion nd nd with Linitz s losd form prssions for th intgrls. Sin th funtions d-funtions form s follows: r d-l funtions thy r indfinit intgrl nd n prssd in. d!!!... d 0. d...!!!! 0. d... d!!!!! 0. d 0! Whrs th funtions tn h os os tn s s s tn h osh osh h s tnh h sh sh tnh os tn os tn s h osh tnh sh osh tn tn d r d-l funtions nd r indfinit intgrl ut thir intgrls nnot prssd in trms of d-funtions form. Suh intgrls n only prssd in sris s follow: 0 IJARCSSE All Rights Rsrvd Pg
Ydv t l. Intrntionl Journl of Advnd Rsrh in Computr Sin nd Softwr Enginring July - 0 pp. -9 0 IJARCSSE All Rights Rsrvd Pg. d K... 00 0 0. d K... 0 9 0 9. tn d.... 90. h d... 0 0 0 9. os d... 0 0. os d... 900 0. tn s d... 00 0 0. s s tn d... 0900 900 0. h osh d... 0 0 0. osh h d... 900 0. tnh s d h... 0 0 0. s s tnh h d h... 0900 900 0. d... 0 0. os d... 9. d... 0 90 0 0. os d... 00 0
Ydv t l. Intrntionl Journl of Advnd Rsrh in Computr Sin nd Softwr Enginring July - 0 pp. -9 tn. d... 0 0. s d... 00. h d..... 0 0. osh d... 0 00. tnh d... 0 0. sh d... 00. 9 os d os... 0. osh d osh h osh h... 9. d tn tn tn tn tn.. K tn 0. tn d d tn d s d s d tn d... K. tn d... 00. d 9os... But th funtions h os oth os ot os ot os os ot os oth h osh osh oth r not d-l thrfor thy r not indfinit intgrl. IV. CONCLUSIONS From ov w find tht thr r still mny funtions whih r indfinit intgrl in sris y d-funtion thory ut thy nnot prssd in dominting squntil nd dominting squntil funtions form s wll s thr r lots of funtions whih r not indfinit intgrl in sris lso. This indits tht lot of rsrh is still ndd to ovrom th prolms of indfinit nonintgrl funtions nd mny nw funtions ndd to introdud. Anowldgmnt I D. K. Ydv m highly grtful to Dr. D. K. Sn who trnsformd m from n vrg studnt into mthmtiin. Rrly thr li him ts orn on this rth in mny nturis hving suh llnt tlnt who ms suh 0 IJARCSSE All Rights Rsrvd Pg
Ydv t l. Intrntionl Journl of Advnd Rsrh in Computr Sin nd Softwr Enginring July - 0 pp. -9 mirl. I got his rgulr guidn for study s wll s rsrh lst 9 yrs. Whtvr th rsrh I hv don ws not possil without his innovtiv pproh of dmi hlp. In ft I m lwys opying him in thing th studnts nd whnvr I m omplld to thin rgrding rsrh thing or ny prolm in mthmtis I lwys follow his ody lngug nd th wy h thins. At lst I would li to mntion hr tht I nnot gt rid of his dt of guru in this lif nd I m fortunt tht I ould find him s my thr nd guid. REFERENCES [] D. K. Ydv Gnrl Study on Non-Intgrl Funtion In Indfinit intgrls s At Cini Indi Mrut Indi -0 00 [] D. K. Ydv & D. K. Sn Rvisd Ppr on Indfinit Non-intgrl Funtions At Cini Indi Mrut Indi - 00 [] D. K. Ydv & D. K. Sn Proof of First Stndrd Form of Nonlmntry Funtions Int. J. of Advnd Rsrh in Sin & Enginring Junpur UP Indi F. 0 [] D. K. Ydv & D. K. Sn Proof of Sond Stndrd Form of Nonlmntry Funtions Int. J. of Advnd Rsrh in Comp. Sin & Soft. Enginring Junpur UP Indi 0-0 F. 0 [] D. K. Ydv & D. K. Sn Proof of Third & Sith Stndrd Forms of Nonlmntry Funtions Int. J. of Advnd Rsrh in Comp. Sin & Soft. Enginring - April 0 [] D. K. Ydv & D. K. Sn Proof of Fourth Stndrd Form of Nonlmntry Funtions Int. J. of Advnd Rsrh in Comp. Sin & Soft. Enginring Junpur UP Indi - April 0 [] D. K. Ydv & D. K. Sn Proof of Fifth Stndrd Form of Nonlmntry Funtions Int. J. of Advnd Rsrh in Comp. Sin & Soft. Enginring Junpur UP Indi 9- April 0 [] D. K. Ydv & D. K. Sn t l Proposd Possil Solutions of Non-intgrl Funtions Byond th Limits of Prtil Frtions At Cini Indi Mrut Indi - 00 [9] D. K. Ydv & D. K. Sn t l Continution of Possil Solutions of Indfinit Non-intgrl Funtions At Cini Indi Mrut Indi 09-0 00 [0] D. K. Ydv & D. K. Sn t l Rviw of th Intgrls of Indfinit Intgrl Funtions At Cini Indi Mrut Indi -0 009 [] D. K. Ydv & D. K. Sn t l Introdution of Dominting Funtion Intrntionl Journl of Mthmtil Sins & Enginring Applitions Pun Indi - 009 [] D. K. Ydv & D. K. Sn Proprtis of Dominting & Indfinit Intgrl Funtions Intrntionl Journl of Mthmtil Sins & Enginring Applitions Pun Indi - 00 [] D. K. Ydv & D. K. Sn Dominting Funtion Thory With Rfrn to Powr Sris Intrntionl Journl of Mthmtil Sins & Enginring Applitions Pun Indi -9 0 [] D. K. Ydv & D. K. Sn Dominting Squntil Funtions Intrntionl Journl of Mthmtil Sins & Enginring Applitions Pun Indi 9-0 0 [] D. K. Ydv & D. K. Sn Gnrl Intgrls of Dominting Squntil Funtions Intrntionl Journl of Mthmtil Sins & Enginring Applitions Pun Indi - 0 [] D. K. Ydv A Study on Indfinit Nonintgrl Funtions Ph. D. Thsis Vino Bhv Univrsity Hzrig Jhrhnd Indi Sptmr 0 0 IJARCSSE All Rights Rsrvd Pg 9