1- I. M. ALGHROUZ: A New Approach To Fractional Derivatives, J. AOU, V. 10, (2007), pp

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1 Jourl o Al-Qus Op Uvrsy or Rsrch Sus - No.4 - Ocobr 8 Rrcs: - I. M. ALGHROUZ: A Nw Approch To Frcol Drvvs, J. AOU, V., 7, pp K.S. Mllr: Drvvs o or orr: Mh M., V 68, 995 pp I. PODLUBNY: Frcol Drl Equos, Acmc Prss, B.OLDHAM, J.SPANIER: Th Frcol Clculus, Nw York: Acmc Prss, S. SAMKO, A. KILBAS: Frcol Irls Drvvs: Thory Applco. GBS Publshrs, Lhor, PA, 993

2 NEW PROOFS OF FRACTIONAL DERIVATIVES OF THE EXPONENTIAL AND TRIGNOMETRIC FUNCTIONS Ibrhm M. Alhrouz A, Hc w ob, cos cos s / / Us h c h: W hv: cos / / s / cos cos / s / cos / s / cos / s / cos / cos s / s cos Smlr rum vs: s s

3 Jourl o Al-Qus Op Uvrsy or Rsrch Sus - No.4 - Ocobr 8 Thus, / L 3 Us Lmm 3, w : Bu, h c whch mpls h: c. Now, suppos h:. L k b y r smllr h, h k us h quly: To ob: Thus or y, R o, k k k o k k k,. Us horms, h ollow horm s ob: Thorm 3: For y, R,. Corollry : Th rcol rvvs o orr o h ucos Proo: s cos r v by s s, cos cos Us h y cos s ccor o Thorm 3: cos s 9

4 NEW PROOFS OF FRACTIONAL DERIVATIVES OF THE EXPONENTIAL AND TRIGNOMETRIC FUNCTIONS Ibrhm M. Alhrouz Mulply by w : L L us h y: L! For y posv r, h ollow quvl quly s ob: L 3 3 L 3 Now, l h o h rh s o h prvous quly, clrly sss h h, h h c, whr c s cos. Hc s suc o prov h: c. Now, l h: L L 3 Two css r cosr s ollows: Cs I: L >,, h h srs s covr or y {,,3,L } L 3 Upo us Lmm, quly o h orm s ob: / L k 8 8

5 Jourl o Al-Qus Op Uvrsy or Rsrch Sus - No.4 - Ocobr / L Thus, L / Hc, L / Thorm : For y R,. Proo: I s suc o prov h horm or Z, bcus or Z, h horm s obvous. So, ssum h Z. Us h pso L L!!!! Apply Do,w hv: L L hc h horm c b prov h ollow y s prov: L L L L L

6 NEW PROOFS OF FRACTIONAL DERIVATIVES OF THE EXPONENTIAL AND TRIGNOMETRIC FUNCTIONS Ibrhm M. Alhrouz Proo: Us h o o Gmm uco, w hv For h rs rl Lmm mpls 3 L For h sco rl, us rp ro by prs w hv: Bu, / L L L 3 Sc > wh >, h: / L L Thror, L / Now, rom, 3 w hv: 6

7 Jourl o Al-Qus Op Uvrsy or Rsrch Sus - No.4 - Ocobr 8 Proo: Sc!, h!!!.. L!! 3 3! 4 A proo o h ollow y ws rpor p.38 o R. [4]: 3 L L!! 3! 3 Subsu, quo 3 bcoms: 4 L L 3!! 3 3! 4 From 4, w hv: L 3 Thus, L. 3 Lmm 3: For y >, / L 3 5

8 NEW PROOFS OF FRACTIONAL DERIVATIVES OF THE EXPONENTIAL AND TRIGNOMETRIC FUNCTIONS Ibrhm M. Alhrouz I hs ppr s prov h h rcol rvvs o orr, pol uco lmms: s h pol uco R, o h. Ths s o us h ollow Lmm For y >, h Proo: 3 k / k. Th uco k sss h ollow rl quo.. k k k 3 L y y, y k Also, k bcus >.Hc h soluo o hs rl quo s h ollow uco: So Lmm : For y >, / k / / L k. 3 L 4

9 Jourl o Al-Qus Op Uvrsy or Rsrch Sus - No.4 - Ocobr I [], s prov h, by us h o s Equo., c c Also, [] h w o o h rcol rvvs o h cos uco h uco vs h sm rsuls. Thorm : I, h Proo: I s suc o prov h horm or Z, bcus or Z, h horm s obvous. So, ssum h Z. Wr!, h! Th Bu So Thror,.

10 NEW PROOFS OF FRACTIONAL DERIVATIVES OF THE EXPONENTIAL AND TRIGNOMETRIC FUNCTIONS Ibrhm M. Alhrouz INTRODUCTION AND DEFINITIONS orr A rcol rvv s so o h mlr h rvv o h uco. Th mos commo o or h rcol rvv o R o h uco s h Rm-Louvll rl, s [],[3],[5]:. whr s h Eulr s Gmm Fuco. For mpl, us quo., h /h rvvs o h ucos c b vlu s: / / / / I [], w o o h rcol rvv o h uco o h orr, R s s: I : L ±, h c b wr h orm!.! Th rcol rvvs o orr, R, o h uco s by:. Nos: For ch >,!. For ch,! or,, L m 3, m < m, m s ov r. L m 3 For ch, s o-r umbr.

11 Jourl o Al-Qus Op Uvrsy or Rsrch Sus - No.4 - Ocobr 8 hk R. cos.[] cos / R s cos s s / ABSTRACT

12 NEW PROOFS OF FRACTIONAL DERIVATIVES OF THE EXPONENTIAL AND TRIGNOMETRIC FUNCTIONS Ibrhm M. Alhrouz Absrc I hs ppr w proo o h wll kow c h h rcol rvv o o orr R s qul o s v ccor o h mo o []. Also, s prov h s s / cos cos /.

13 NEW PROOFS OF FRACTIONAL DERIVATIVES OF THE EXPONENTIAL AND TRIGNOMETRIC FUNCTIONS Ibrhm M. Alhrouz* * Dprm o Mhmcs, Al-Qus Uvrsy, Jruslm, Pls. 9