Note on Sequence of Functions involving the Product of E γ,k
|
|
- Roland Mason
- 6 years ago
- Views:
Transcription
1 Interntionl Bulletin of Mtheticl Reserch Volue XX, Issue X, Deceber 2014 Pges 16-27, ISSN: XXXX-XXXX Note on Sequence of Functions involving the Product of E γ,k () Mehr Chnd Deprtent of Mthetics Fteh College for Woen, Rpur Phul Bthind , Indi ehr.jllndhr@gil.co Abstrct A rerkbly lrge nuber of opertionl techniques hve drwn the ttention of severl reserchers in the study of sequences of functions nd polynoils. In this sequel, here, we i to introduce new sequence of functions involving product of the generlized Mittg-Leffler function by using opertionl techniques. Soe generting reltions nd finite sution forul of the sequence presented here re lso considered. 1 Introduction The ide of representing the processes of clculus, differentition, nd integrtion, s opertors, is clled n opertionl technique, which is lso known s n opertionl clculus. Mny opertionl techniques involve vrious specil functions hve found soe significnt pplictions in vrious sub-fields of pplicble theticl nlysis. Severl pplictions of opertionl techniques cn be found in the probles of nlysis, in prticulr differentil equtions re trnsfored into lgebric probles, usully the proble of solving polynoil equtions. Since lst four decdes, nuber of workers like Chk5], Gould nd Hopper 11], Chtterje8], Singh27], Srivstv nd Singh29], Mittl15, 16, 17], Chndl6, 7], Srivstv24], Joshi nd Prjpt14], Ptil nd Thkre18] nd Srivstv nd Singh28] hve de deep reserch of the properties, pplictions nd different etensions of the vrious opertionl techniques. The key eleent of the opertionl technique is to consider differentition s n opertor D d d cting on functions. Liner differentil equtions cn then be recst in the for of n opertor vlued function F(D) of the opertor D cting on n unknown function which equls known function. Solutions re then obtined by king the inverse opertor of F cting on the known function. Indeed, rerkbly lrge nuber of sequences of functions involving vriety of specil functions hve been developed by ny uthors (see, for eple, 28]; for very recent work, see lso 1, 2, 3, 22, 23, 26]). Here we i t presenting new sequence of functions involving product of the E γ,k by using opertionl techniques. Soe generting reltions nd finite sution forul re lso obtined. For our purpose, we begin by reclling soe known functions nd erlier works. In 1971, by Mittl 15] gve the Rodrigues forul for the generlized Lgurre polynoils defined by Received: Deceber 2014 Keywords: Specil function, generting reltions, Mittg-Leffler, Sequence of function, finite sution forul, sybolic representtion. AMS Subject Clssifiction: 33E10, 44A45.
2 Note on Sequence of Functions involving the Product of E γ,k () 17 where p k () is polynoil in of degree k. T (α) kn () 1 n! α ep (p k ()) D n α+n ep ( p k ()) ], (1.1) Mittl16] lso proved the following reltion for (1.1) given by where s is constnt nd T s (s + D). T ( 1) kn () 1 n! α n ep (p k ()) T n s α ep ( p k ())], (1.2) In this sequel, in 1979, Srivstv nd Singh 28] studied sequence of functions V n (α) (;, k, s) defined by V n (α) (;, k, s) α ep {p k ()} θ n α ep { p k ()}] (1.3) n! By using the opertor θ (s + D), where s is constnt, nd p k () is polynoil in of degree k. Here, new sequence of function n (;, k, s) : 1 n! α { } n (;, k, s) is introduced s follows: pkj () ] (T,s ) n α, (1.4) where T,s (s + D), D d d, nd s re constnts, β 0, k j is finite nd non-negtive integer, p kj () re polynoils in of degree k j, where j 1, 2,..., r nd (.) is generlized Mittg-Leffler function. For the ske of copleteness, we recll the E γ,k (.). In 1903, the Swedish theticin Gost Mittg-Leffler 19] introduced the function E α (z), defined s E α (z) z n Γ(αn + 1) (1.5) where z is cople vrible nd Γ(.) is G function, α 0. The Mittg-Leffler function is direct generliztion of the eponentil function to which it reduces for α 0. For 0 < α < 1 it interpoltes between the pure eponentil nd hypergeoetric function 1 1 z. Its iportnce is relized during the lst two decdes due to its involveent in the probles of physics, cheistry, biology, engineering nd pplied sciences. Mittg-Leffler function nturlly occurs s the solution of frctionl order differentil eqution or frctionl order integrl equtions. The generliztion of E α (z) ws studied by Win 31] in 1905 nd he defined the function s
3 18 Mehr Chnd E α,β (z) z n Γ(αn + β) (α, β C; R(α) > 0, R(β) > 0) (1.6) Which is known s Win s function or generlized Mittg-Leffler function s E α,1 (z) E α (z). The forer ws introduced by Mittg-Leffler19] in connection with his ethod of sution of soe divergent series. In his ppers 19, 20], he investigted certin properties of this function. The function defined by (1.6) first ppered in the work of Win 31]. The function (1.6) is studied, ong others, by Win 31], Agrwl 4], Hubert 12] nd Hubert nd Agrwl 13] nd others. The in properties of these functions re given in the book by Erdelyi et l. (10], Section 18.1) nd ore coprehensive nd detiled ccount of Mittg-Leffler functions re presented in Dzherbshyn (9], Chpter 2). In 1971, Prbhkr 21] introduced the function E γ α,β (z) in the for of E γ α,β (z) (γ) n z n Γ(αn + β) n!, (1.7) where α, β, γ C; R(α) > 0, R(β) > 0), R(γ) > 0 nd (λ) n denotes the filir Pochher sybol or the shifted fctoril, since (1) n n! (n N 0 ) (λ) n Γ(λ + n) Γ(λ) { 1 (n 0; λ C {0}) λ(λ + 1)...(λ + n 1) (n N; λ C) (1.8) Recently generliztion of Mittge-Leffler function E γ α,β (z) of (1.7) studied by Srivstv nd Toovski 30] is defined s follows: E γ,k α,β (z) (γ) Kn z n Γ(αn + β) n!, (1.9) where α, β, γ C; R(α) > 0, R(β) > 0), R(γ) > 0; R(K) > 0 which, in the specil cse when K q(q (0, 1) N) nd in{r(α), R(β)} 0 (1.10) ws considered erlier by Shukl nd Prjpti 26]. Soe generting reltions nd finite sution forul of clss of polynoils or sequences of functions ( hve been obtined by using the properties of the differentil opertors. The opertors T,s (s + D) D d ) d is bsed on the work of Mittl 17], Ptil nd Thkre 18], Srivstv nd Singh 28].
4 Note on Sequence of Functions involving the Product of E γ,k () 19 Soe useful opertionl techniques re given below: ( ) ep (tt,s ) β f () β (1 β+s t) ) f ( (1 t) 1/), (1.11) ep (tt,s ) ( α n f () ) α (1 + t) 1+( ) f ( (1 + t) 1/), (1.12) (T,s ) n (uv) ( ) n (T,s ) n (v) ( T,1 ) (u), (1.13) nd (1 + D) (1 + + D) ( D)... (1 + ( 1) + D) β 1 ( β ) β 1 (1.14) (1 t) α β (1 t) ( ) α β (t). (1.15)! 2 Generting Reltions First generting reltion: (1 t) Second generting reltion: (1 + t) n (;, k, s) n t n (2.1) pkj () ] r ( p kj (1 t) 1/)] V (δ j,β j,γ j,k j,α n) n (;, k, s) n t n (2.2) 1+( pkj () ] r p kj ( (1 + t) 1/)]
5 20 Mehr Chnd Third generting reltion: (1 t) ( ) + n n (;, k, s) n t n (2.3) pkj () ] ( V p kj (1 t) 1/)] ( ) (,γ j,k j,α) n (1 t) 1/ ;, k, s Proof of the first generting reltion. We strt fro (1.4) nd consider n (;, k, s) t n α pkj () ] ep(tt,s ) α (2.4) Using the opertionl technique (1.11), Eqution (2.4) reduces to α (1 t) n (;, k, s) t n (2.5) pkj () ] α (1 t) pkj () ] r ( p kj (1 t) 1/)] p kj ((1 t) 1/)], which upon replcing t by t, yields (2.1). Proof of the second generting reltion. Agin fro (1.4), we hve n V (δ j,β j,γ j,k j,α n) n (;, k, s) t n α pkj () ] ep(tt,s ) α n.(2.6) Applying the opertionl technique (1.12), we get
6 Note on Sequence of Functions involving the Product of E γ,k () 21 α n V (δ j,β j,γ j,k j,α n) n (;, k, s) t n (2.7) (1 + t) pkj () ] α (1 + t) 1 1 pkj () ] r p kj ((1 + t) 1/)] (2.8) p kj ((1 + t) 1/)]. This proves (2.2). Proof of the third generting reltion. We cn write (1.4) s (T,s ) n α () ] n! α V (δ j, β j, γ j, K j, α) n (;, k, s) pkj () ] (2.9) or ep (t (T,s )),s (T ) n α,α n! ep (tt ) α V (δj,β j,γ j,k j n,α) (;, k, s) δ j,β pkj j () ] (2.10) t! (T,s ) +n α,s n! ep (tt ) α n (;, k, s) E γ j,k j δ j,β pkj j () ]. (2.11) Using the opertionl technique (1.11), Eqution (2.10) cn be written s:
7 22 Mehr Chnd t! which, upon using (2.8), gives (T,s ) +n n! α (1 t) α pkj () ] (2.12) ) ( n (1 t) 1/ ;, k, s ) ( p kj (1 t) 1/)] Therefore, we hve t ( + n)!!n! α V (δj,β j,γ j,k j,α) +n (;, k, s) () ] n α (1 t) ) ( (2.13) p kj (1 t) 1/)] (( )) (1 t) 1/ ;, k, s. ( ) + n n +n (;, k, s) t Which, upon replcing t by t, proves the result (2.3). (1 t) pkj () ] ( (2.14) p kj (1 t) 1/)] n ( ) (1 t) 1/ ;, k, s. 3 Finite Sution Foruls First finite sution forul. n (;, k, s) Second finite sution forul. n (;, k, s) Proof of the first finite sution forul. 1 (! ( ) α ) V (δ j,β j,γ j,k j,0) n (;, k, s). (3.1) 1! ( ) ( ) α β V (δ j,β j,γ j,k j,β) n (;, k, s). (3.2)
8 Note on Sequence of Functions involving the Product of E γ,k () 23 Fro Eqution (1.4), we hve n (;, k, s) 1 n! α pkj () ] (T,s ) n α 1. (3.3) Using the opertionl technique (1.13), we hve n (;, k, s) (3.4) 1 n! α δ j,β pkj j () ] ( ) n (T,s ) n ( ) T,1 ( α 1 ) 1 n! α pkj () ] n!! (n )! (n ) (s + D) (s + + D) (s D)... (s + (n 1) + D)] pkj () ] (1 + D) (1 + + D) ( D)... (1 + ( 1) + D)] ( α 1). Using the result (1.14), we hve n (;, k, s) (3.5) δ j,β pkj j () ] n n 1 1 (n )!n (s + i + D)! ( α ). i0 Put α 0 nd replcing n by n in (3.3), we get V (δ j,β j,γ j,k j,0) n (;, k, s) 1 (n )! pkj () ] (T,s ) n. (3.6) This gives 1 (n )! (T,s ) n V (,γ,k,0) n (;, k, s) δ j,β pkj j () ]. (3.7)
9 24 Mehr Chnd n 1 1 (s + i + D) (n )! i0 Fro Equtions (3.5) nd (3.8), we hve the in result. (δj,βj,γj,kj,0) V n (;, k, s) ( n) pkj () ]. (3.8) Proof of the second finite sution forul. Eqution (1.4) cn be written s n (;, k, s) t n α Applying the (1.11) to the Eqution (3.9), we hve pkj () ] ( ep tt (,s) ) α. (3.9) α n (;, k, s) t n (3.10) pkj () ] α (1 t) α+ s ( p kj (1 t) 1/)] (1 t) pkj () ] r Using the result fro Eqution (1.15), Eqution (3.10) reduces to p kj ((1 t) 1/)]. (1 t) β+s ) ( ) α β n (;, k, s) t n (3.11) ( t)! pkj () ] r ( p kj (1 t) 1/)] ( ) α β ( t)! β pkj () ] ep (tt,s ) β
10 Note on Sequence of Functions involving the Product of E γ,k () 25 ( ) α β ( ) t n+!n! β pkj () ] (T,s ) n β n ( ) α β ( ) t n! (n )! β Now equting the coefficient of t n, we get pkj () ] (T,s ) n β. n ( ) α β n (;, k, s) (3.12) ( )! (n )! β δ j,β pkj j () ] (T,s ) n β. Using the Eqution (1.4) in (3.12), we hve the result (3.2). 4 Specil Cses (I) If we tke r 1, γ j K j 1, then the results estblished in equtions (2.1), (2.2), (2.3), (3.1) nd (3.2) reduce to the known results in 1]. (II) If we choose γ j K j δ j β j 1, the Mittge-Leffler function reduced to ep(z) i.e. E 1,1 1,1 ep(z), then the results in equtions (2.1), (2.2), (2.3), (3.1) nd (3.2) reduced to the new results involving (ep(z)) r. (III) If we choose γ j K j 1, δ j 2, β j 1, the Mittge-Leffler function reduced to cosh( z) i.e. E 1,1 2,1 cosh( z), then the results in equtions (2.1), (2.2), (2.3), (3.1) nd (3.2) reduced to the new results involving (cosh( z)) r. 5 Conclusion In this pper, we hve presented new sequence of functions involving the product of the E γ,k (.) by using opertionl techniques. With the help of our in sequence forul, soe generting reltions nd finite sution forul of the sequence re lso presented here. Our sequence forul is iportnt due to presence of E γ,k (.). On ccount of the ost generl nture of the Eγ,K (.) lrge nuber of sequences nd polynoils involving sipler functions cn be esily obtined s their specil cses but due to lck of spce we cn not ention here.
11 26 Mehr Chnd References 1] Agrwl, P. nd Chnd, M. (2013). Grphicl Interprettion of the New Sequence of Functions Involving Mittge-Leffler Function Using Mtlb, Aericn Journl of Mthetics nd Sttistics 2013, 3(2): DOI: /j.js ] Agrwl, P. nd Chnd, M. (2013), On new sequence of functions involving p F q, South Asin Journl of Mthetics, Vol. 3(3): ] Agrwl, P. nd Chnd, M. (ccepted 2013), A NEW SEQUENCE OF FUNCTIONS INVOLVING A PRODUCT OF THE p F q, Mtheticl Sciences And Applictions E-Notes Volue 1. 4] Agrwl, R.P., (1953). A propos dune note de M. Pierre Hubert, C.R. Acd. Sci. Pris, 236, ] Chk, A. M., (1956) A clss of polynoils nd generliztion of stirling nubers, Duke J. Mth., 23, ] Chndel, R.C.S., (1973) A new clss of polynoils, Indin J. Mth., 15(1), ] Chndel, R.C.S., (1974) A further note on the clss of polynoils Tn α,k (, r, p), Indin J. Mth.,16(1), 8] Chtterje, S. K., (1964) On generliztion of Lguerre polynoils, Rend. Mt. Univ. Pdov, 34, ] Dzherbshyn, M.M., (1966). Integrl Trnsfors nd Representtions of Functions in the Cople Plne, Nuk, Moscow, (in Russin). 10] Erdelyi, A., Mgnus,W., Oberhettinger, F. nd Tricoi, F. G. (1955). Higher Trnscendentl Functions, Vol. 3, McGrw - Hill, New York, Toronto nd London. 11] Gould, H. W. nd Hopper, A. T., (1962) Opertionl foruls connected with two generliztions of Herite polynoils, Duck Mth. J., 29, ] Hubert, P., (1953). Quelques resultnts retifs l fonction de Mittg-Leffler, C.R. Acd. Sci. Pris, 236, ] Hubert, P. nd Agrwl, R.P., (1953). Sur l fonction de Mittg-Leffler et quelques unes de ses generliztions, Bull. Sci. Mth., (Ser.II), 77, ] Joshi, C. M. nd Prjpt, M. L., (1975) The opertor T,k, nd generliztion of certin clssicl polynoils, Kyungpook Mth. J., 15, ] Mittl, H. B., (1971) A generliztion of Lguerre polynoil, Publ. Mth. Debrecen, 18, ] Mittl, H. B., (1971), Opertionl representtions for the generlized Lguerre polynoil, Glsnik Mt.Ser III, 26(6), ] Mittl, H. B., (1977), Biliner nd Bilterl generting reltions, Aericn J. Mth., 99, ] Ptil, K. R. nd Thkre, N. K., (1975), Opertionl foruls for function defined by generlized Rodrigues forul-ii, Sci. J. Shivji Univ. 15, 1-10.
12 Note on Sequence of Functions involving the Product of E γ,k () 27 19] Mittg-Leffler, G.M., (1903). Une generlistion de lintegrle de Lplce-Abel, C.R. Acd. Sci. Pris (Ser. II), 137, ] Mittg-Leffler, G.M., (1905). Sur l representtion nlytiqie dune fonction onogene (cinquiee note), Act Mthetic, 29, ] Prbhkr, T.R. (1971). A Singulr integrl eqution with generlized Mittg-Leffler function in the kernel, Yokoh Mth. J., Vol. 19, pp ] Prjpti, J.C. nd Ajudi, N.K., (2012), On sequence of functions nd their MATLAB ipleenttion, Interntionl Journl od Physicl, Cheicl nd Mtheticl Sciences, Vol. No. 2, ISSN: , p.p ] Slebhi, I.A., Prjpti, J.C. nd Shukl, A.K., On sequence of functions, Coun. Koren Mth. 28(2013), No.1,p.p ] Shrivstv, P. N., (1974), Soe opertionl foruls nd generlized generting function, The Mth. Eduction, 8, ] Srivstv,H. M. nd Choi,J., (2012), Zet nd q-zet Functions nd Associted Series nd Integrls, Elsevier Science Publishers, Asterd, London nd New York. 26] Shukl, A. K. nd Prjpti J. C., (2007) On soe properties of clss of Polynoils suggested by Mittl, Proyecciones J. Mth., 26(2), ] Singh, R. P., (1968), On generlized Truesdell polynoils, Rivist de Mthetic, 8, ] Srivstv, A. N. nd Singh, S. N., (1979) Soe generting reltions connected with function defined by Generlized Rodrigues forul, Indin J. Pure Appl. Mth., 10(10), ] Srivstv, H. M. nd Singh, J. P., (1971) A clss of polynoils defined by generlized, Rodrigues forul, Ann. Mt. Pur Appl., 90(4), ] Srivstv, H.M. nd Toovski, Z.(2009). Frctionl clculus with n integrl opertor contining generlized Mittg-Leffler function in the kernel. Appl. Mth. Coput. 211(1), ] Win, A., (1905). Über den Fundentl stz in der Theorie der Funcktionen, E α(), Act Mthetic, 29,
Lyapunov-type inequalities for Laplacian systems and applications to boundary value problems
Avilble online t www.isr-publictions.co/jns J. Nonliner Sci. Appl. 11 2018 8 16 Reserch Article Journl Hoepge: www.isr-publictions.co/jns Lypunov-type inequlities for Lplcin systes nd pplictions to boundry
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationMUHAMMAD MUDDASSAR AND AHSAN ALI
NEW INTEGRAL INEQUALITIES THROUGH GENERALIZED CONVEX FUNCTIONS WITH APPLICATION rxiv:138.3954v1 [th.ca] 19 Aug 213 MUHAMMAD MUDDASSAR AND AHSAN ALI Abstrct. In this pper, we estblish vrious inequlities
More informationResearch Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation
Journl of Applied Mthemtics Volume 2011, Article ID 743923, 7 pges doi:10.1155/2011/743923 Reserch Article On Existence nd Uniqueness of Solutions of Nonliner Integrl Eqution M. Eshghi Gordji, 1 H. Bghni,
More informationA General Dynamic Inequality of Opial Type
Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn
More informationSolutions of Klein - Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 6 Ver. IV (Nov. - Dec. 2017), PP 19-24 www.iosrjournls.org Solutions of Klein - Gordn equtions, using Finite Fourier
More informationNew Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity
Punjb University Journl of Mthemtics (ISSN 116-56) Vol. 45 (13) pp. 33-38 New Integrl Inequlities of the Type of Hermite-Hdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science,
More informationUNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY
UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY YIFEI PAN, MEI WANG, AND YU YAN ABSTRACT We estblish soe uniqueness results ner 0 for ordinry differentil equtions of the
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
More informationTWO DIMENSIONAL INTERPOLATION USING TENSOR PRODUCT OF CHEBYSHEV SYSTEMS
Proceedings of the Third Interntionl Conference on Mthetics nd Nturl Sciences (ICMNS ) TWO DIMENSIONAL INTERPOLATION USING TENSOR PRODUCT OF CHEYSHEV SYSTEMS Lukit Abrwti, nd Hendr Gunwn Anlsis nd Geoetr
More informationFormulae For. Standard Formulae Of Integrals: x dx k, n 1. log. a dx a k. cosec x.cot xdx cosec. e dx e k. sec. ax dx ax k. 1 1 a x.
Forule For Stndrd Forule Of Integrls: u Integrl Clculus By OP Gupt [Indir Awrd Winner, +9-965 35 48] A B C D n n k, n n log k k log e e k k E sin cos k F cos sin G tn log sec k OR log cos k H cot log sin
More informationLYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN
Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR
More informationOn New Inequalities of Hermite-Hadamard-Fejer Type for Harmonically Quasi-Convex Functions Via Fractional Integrals
X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey On New Ineulities of Hermite-Hdmrd-Fejer Type for Hrmoniclly Qusi-Convex Functions Vi Frctionl Integrls Mehmet Kunt * nd İmdt İşcn Deprtment
More informationASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 3, September 2010 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II TIBERIU TRIF Dedicted to Professor Grigore Ştefn
More informationProc. of the 8th WSEAS Int. Conf. on Mathematical Methods and Computational Techniques in Electrical Engineering, Bucharest, October 16-17,
Proc. of the 8th WSEAS Int. Conf. on Mtheticl Methods nd Coputtionl Techniques in Electricl Engineering, Buchrest, October 6-7, 006 Guss-Legendre Qudrture Forul in Runge-utt Method with Modified Model
More informationBailey [1] established a simple but very useful identity: If
itlin journl of pure nd pplied mthemtics n 7 010 (179 190) 179 CERTAIN TRANSFORMATION AND SUMMATION FORMULAE FOR q-series Remy Y Denis Deprtment of Mthemtics University of Gorkhpur Gorkhpur-73009 Indi
More informationThe presentation of a new type of quantum calculus
DOI.55/tmj-27-22 The presenttion of new type of quntum clculus Abdolli Nemty nd Mehdi Tourni b Deprtment of Mthemtics, University of Mzndrn, Bbolsr, Irn E-mil: nmty@umz.c.ir, mehdi.tourni@gmil.com b Abstrct
More informationOn Inequality for the Non-Local Fractional Differential Equation
Globl Journl of Pure nd Applied Mthemtics. ISSN 0973-178 Volume 13, Number 3 2017), pp. 981 993 Reserch Indi Publictions http://www.ripubliction.com/gjpm.htm On Inequlity for the Non-Locl Frctionl Differentil
More informationLyapunov-Type Inequalities for some Sequential Fractional Boundary Value Problems
Advnces in Dynmicl Systems nd Applictions ISSN 0973-5321, Volume 11, Number 1, pp. 33 43 (2016) http://cmpus.mst.edu/ds Lypunov-Type Inequlities for some Sequentil Frctionl Boundry Vlue Problems Rui A.
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl
More informationSome Improvements of Hölder s Inequality on Time Scales
DOI: 0.55/uom-207-0037 An. Şt. Univ. Ovidius Constnţ Vol. 253,207, 83 96 Some Improvements of Hölder s Inequlity on Time Scles Cristin Dinu, Mihi Stncu nd Dniel Dănciulescu Astrct The theory nd pplictions
More informationResearch Article On the Definitions of Nabla Fractional Operators
Abstrct nd Applied Anlysis Volume 2012, Article ID 406757, 13 pges doi:10.1155/2012/406757 Reserch Article On the Definitions of Nbl Frctionl Opertors Thbet Abdeljwd 1 nd Ferhn M. Atici 2 1 Deprtment of
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationTHIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions
THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, Hui-Hsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.
More informationResearch Article On New Inequalities via Riemann-Liouville Fractional Integration
Abstrct nd Applied Anlysis Volume 202, Article ID 428983, 0 pges doi:0.55/202/428983 Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationINEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei
Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV
More informationImprovement of Ostrowski Integral Type Inequalities with Application
Filomt 30:6 06), 56 DOI 098/FIL606Q Published by Fculty of Sciences nd Mthemtics, University of Niš, Serbi Avilble t: http://wwwpmfnicrs/filomt Improvement of Ostrowski Integrl Type Ineulities with Appliction
More informationExact solutions for nonlinear partial fractional differential equations
Chin. Phys. B Vol., No. (0) 004 Exct solutions for nonliner prtil frctionl differentil equtions Khled A. epreel )b) nd Sleh Omrn b)c) ) Mthemtics Deprtment, Fculty of Science, Zgzig University, Egypt b)
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationDYNAMICS OF TUPLES OF MATRICES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volue 37, Nuber 3, Mrch 2009, Pges 025 034 S 0002-9939080977-7 Article electroniclly published on October 7, 2008 DYNAMICS OF TUPLES OF MATRICES G. COSTAKIS,
More informationExistence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions
Applied Mthetic 0 9-96 doi:0.436/.0.079 Pulihed Online Octoer 0 (http://www.scirp.org/journl/) Eitence nd Uniquene of Solution for Frctionl Order Integro-Differentil Eqution with Non-Locl nd Glol Boundry
More informationSelberg s integral and linear forms in zeta values
Selberg s integrl nd liner forms in zet vlues Tnguy Rivol Abstrct Using Selberg s integrl, we present some new Euler-type integrl representtions of certin nerly-poised hypergeometric series. These integrls
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationWHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) =
WHEN IS A FUNCTION NOT FLAT? YIFEI PAN AND MEI WANG Abstrct. In this pper we prove unique continution property for vector vlued functions of one vrible stisfying certin differentil inequlity. Key words:
More informationGeneralized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral
DOI 763/s4956-6-4- Moroccn J Pure nd Appl AnlMJPAA) Volume ), 6, Pges 34 46 ISSN: 35-87 RESEARCH ARTICLE Generlized Hermite-Hdmrd-Fejer type inequlities for GA-conve functions vi Frctionl integrl I mdt
More informationApproximation of functions belonging to the class L p (ω) β by linear operators
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 3, 9, Approximtion of functions belonging to the clss L p ω) β by liner opertors W lodzimierz Lenski nd Bogdn Szl Abstrct. We prove
More informationA New Generalization of Lemma Gronwall-Bellman
Applied Mthemticl Sciences, Vol. 6, 212, no. 13, 621-628 A New Generliztion of Lemm Gronwll-Bellmn Younes Lourtssi LA2I, Deprtment of Electricl Engineering, Mohmmdi School Engineering Agdl, Rbt, Morocco
More informationThe Hadamard s inequality for quasi-convex functions via fractional integrals
Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationA Brief Note on Quasi Static Thermal Stresses In A Thin Rectangular Plate With Internal Heat Generation
Americn Journl of Engineering Reserch (AJER) 13 Americn Journl of Engineering Reserch (AJER) e-issn : 3-847 p-issn : 3-936 Volume-, Issue-1, pp-388-393 www.jer.org Reserch Pper Open Access A Brief Note
More informationInternational Jour. of Diff. Eq. and Appl., 3, N1, (2001),
Interntionl Jour. of Diff. Eq. nd Appl., 3, N1, (2001), 31-37. 1 New proof of Weyl s theorem A.G. Rmm Mthemtics Deprtment, Knss Stte University, Mnhttn, KS 66506-2602, USA rmm@mth.ksu.edu http://www.mth.ksu.edu/
More informationAMATH 731: Applied Functional Analysis Fall Some basics of integral equations
AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationRIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17 (2009, 3 12 RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE GEORGE A. ANASTASSIOU Abstrct. Here re estblished vrious tight probbilistic
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationSUPERSTABILITY OF DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS
Electronic Journl of Differentil Equtions, Vol. 01 (01), No. 15, pp. 1. ISSN: 107-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu SUPERSTABILITY OF DIFFERENTIAL
More informationON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 9811 015, 43 49 DOI: 10.98/PIM15019019H ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS
More informationMAC-solutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationApplication of Exp-Function Method to. a Huxley Equation with Variable Coefficient *
Interntionl Mthemticl Forum, 4, 9, no., 7-3 Appliction of Exp-Function Method to Huxley Eqution with Vrible Coefficient * Li Yo, Lin Wng nd Xin-Wei Zhou. Deprtment of Mthemtics, Kunming College Kunming,Yunnn,
More informationGENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED s-convex IN THE SECOND SENSE
Journl of Alied Mthemtics nd Comuttionl Mechnics 6, 5(4), - wwwmcmczl -ISSN 99-9965 DOI: 75/jmcm64 e-issn 353-588 GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES
More informationParametrized inequality of Hermite Hadamard type for functions whose third derivative absolute values are quasi convex
Wu et l. SpringerPlus (5) 4:83 DOI.8/s44-5-33-z RESEARCH Prmetrized inequlity of Hermite Hdmrd type for functions whose third derivtive bsolute vlues re qusi convex Shn He Wu, Bnyt Sroysng, Jin Shn Xie
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu
More informationAsymptotic behavior of intermediate points in certain mean value theorems. III
Stud. Univ. Bbeş-Bolyi Mth. 59(2014), No. 3, 279 288 Asymptotic behvior of intermedite points in certin men vlue theorems. III Tiberiu Trif Abstrct. The pper is devoted to the study of the symptotic behvior
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge
More informationf (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)
TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationSOME HARDY TYPE INEQUALITIES WITH WEIGHTED FUNCTIONS VIA OPIAL TYPE INEQUALITIES
SOME HARDY TYPE INEQUALITIES WITH WEIGHTED FUNCTIONS VIA OPIAL TYPE INEQUALITIES R. P. AGARWAL, D. O REGAN 2 AND S. H. SAKER 3 Abstrct. In this pper, we will prove severl new ineulities of Hrdy type with
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationA unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term
An. Ştiinţ. Univ. Al. I. Cuz Işi. Mt. (N.S. Tomul LXIII, 07, f. A unified generliztion of perturbed mid-point nd trpezoid inequlities nd symptotic expressions for its error term Wenjun Liu Received: 7.XI.0
More informationNEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX
Journl of Mthemticl Ineulities Volume 1, Number 3 18, 655 664 doi:1.7153/jmi-18-1-5 NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX SHAHID
More informationOn Error Sum Functions Formed by Convergents of Real Numbers
3 47 6 3 Journl of Integer Sequences, Vol. 4 (), Article.8.6 On Error Sum Functions Formed by Convergents of Rel Numbers Crsten Elsner nd Mrtin Stein Fchhochschule für die Wirtschft Hnnover Freundllee
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationON THE WEIGHTED OSTROWSKI INEQUALITY
ON THE WEIGHTED OSTROWSKI INEQUALITY N.S. BARNETT AND S.S. DRAGOMIR School of Computer Science nd Mthemtics Victori University, PO Bo 14428 Melbourne City, VIC 8001, Austrli. EMil: {neil.brnett, sever.drgomir}@vu.edu.u
More informationResearch Article Analytical Solution of the Fractional Fredholm Integrodifferential Equation Using the Fractional Residual Power Series Method
Hindwi Compleity Volume 7, Article ID 457589, 6 pges https://doi.org/.55/7/457589 Reserch Article Anlyticl Solution of the Frctionl Fredholm Integrodifferentil Eqution Using the Frctionl Residul Power
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationRELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE
TJMM 10 018, No., 141-151 RELATIONS ON BI-PERIODIC JACOBSTHAL SEQUENCE S. UYGUN, H. KARATAS, E. AKINCI Abstrct. Following the new generliztion of the Jcobsthl sequence defined by Uygun nd Owusu 10 s ĵ
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationIndefinite Integral. Chapter Integration - reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More informationarxiv: v1 [math.ca] 28 Jan 2013
ON NEW APPROACH HADAMARD-TYPE INEQUALITIES FOR s-geometrically CONVEX FUNCTIONS rxiv:3.9v [mth.ca 8 Jn 3 MEVLÜT TUNÇ AND İBRAHİM KARABAYIR Astrct. In this pper we chieve some new Hdmrd type ineulities
More informationDYNAMICAL SYSTEMS SUPPLEMENT 2007 pp Natalija Sergejeva. Department of Mathematics and Natural Sciences Parades 1 LV-5400 Daugavpils, Latvia
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 920 926 ON THE UNUSUAL FUČÍK SPECTRUM Ntlij Sergejev Deprtment of Mthemtics nd Nturl Sciences Prdes 1 LV-5400
More informationUNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY
UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY YIFEI PAN, MEI WANG, AND YU YAN ABSTRACT We study ordinry differentil equtions of the type u n t = fut with initil conditions
More information1 2-D Second Order Equations: Separation of Variables
Chpter 12 PDEs in Rectngles 1 2-D Second Order Equtions: Seprtion of Vribles 1. A second order liner prtil differentil eqution in two vribles x nd y is A 2 u x + B 2 u 2 x y + C 2 u y + D u 2 x + E u +
More informationFRACTIONAL INTEGRALS AND
Applicble Anlysis nd Discrete Mthemtics, 27, 3 323. Avilble electroniclly t http://pefmth.etf.bg.c.yu Presented t the conference: Topics in Mthemticl Anlysis nd Grph Theory, Belgrde, September 4, 26. FRACTONAL
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationON SOME RESULTS INVOLVING STATISTICAL CHARACTERISTICS, OPERATIONAL FORMULAE OF SPECIAL FUNCTIONS AND ANOMALOUS DIFFUSION
ROMAI J., v.13, no.1(2017, 41 55 ON SOME RESULTS INVOLVING STATISTIAL HARATERISTIS, OPERATIONAL FORMULAE OF SPEIAL FUNTIONS AND ANOMALOUS DIFFUSION Hemnt Kumr 1, Mhmood Ahmd Pthn 2 1 Deprtment of Mthemtics,
More informationIntegral inequalities for n times differentiable mappings
JACM 3, No, 36-45 8 36 Journl of Abstrct nd Computtionl Mthemtics http://wwwntmscicom/jcm Integrl ineulities for n times differentible mppings Cetin Yildiz, Sever S Drgomir Attur University, K K Eduction
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationHyers-Ulam and Hyers-Ulam-Aoki-Rassias Stability for Linear Ordinary Differential Equations
Avilble t http://pvu.edu/ Appl. Appl. Mth. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015, pp. 149 161 Applictions nd Applied Mthetics: An Interntionl Journl (AAM Hyers-Ul nd Hyers-Ul-Aoki-Rssis Stbility
More informationON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS
ON ALTERNATING POWER SUMS OF ARITHMETIC PROGRESSIONS A. BAZSÓ Astrct. Depending on the prity of the positive integer n the lternting power sum T k n = k + k + + k...+ 1 n 1 n 1 + k. cn e extended to polynomil
More informationON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES
Volume 1 29, Issue 3, Article 86, 5 pp. ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES SOUMIA BELARBI AND ZOUBIR DAHMANI DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MOSTAGANEM soumi-mth@hotmil.fr zzdhmni@yhoo.fr
More informationA short introduction to local fractional complex analysis
A short introduction to locl rctionl complex nlysis Yng Xio-Jun Deprtment o Mthemtics Mechnics, hin University o Mining Technology, Xuhou mpus, Xuhou, Jingsu, 228, P R dyngxiojun@63com This pper presents
More informationOrthogonal Polynomials and Least-Squares Approximations to Functions
Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationA Compound of Geeta Distribution with Generalized Beta Distribution
Journl of Modern Applied Sttisticl Methods Volume 3 Issue Article 8 5--204 A Compound of Geet Distribution ith Generlized Bet Distribution Adil Rshid University of Kshmir, Sringr, Indi, dilstt@gmil.com
More informationGeneralization of Quasi-Differentiable Maps
Globl Journl of Mtheticl Sciences: Theory nd Prcticl. ISSN 0974-300 Volue 4, Nuber 3 (0),. 49-55 Interntionl Reserch Publiction House htt://www.irhouse.co Generliztion of Qusi-Differentible Ms Sushil Kur
More informationDually quasi-de Morgan Stone semi-heyting algebras II. Regularity
Volume 2, Number, July 204, 65-82 ISSN Print: 2345-5853 Online: 2345-586 Dully qusi-de Morgn Stone semi-heyting lgebrs II. Regulrity Hnmntgoud P. Snkppnvr Abstrct. This pper is the second of two prt series.
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationMultiple Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems on Time Scales
Electronic Journl of Qulittive Theory of Differentil Equtions 2009, No. 32, -3; http://www.mth.u-szeged.hu/ejqtde/ Multiple Positive Solutions for the System of Higher Order Two-Point Boundry Vlue Problems
More informationHermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave
Applied Mthemticl Sciences Vol. 9 05 no. 5-36 HIKARI Ltd www.m-hikri.com http://d.doi.org/0.988/ms.05.9 Hermite-Hdmrd Type Ineulities for the Functions whose Second Derivtives in Absolute Vlue re Conve
More informationResearch Article Moment Inequalities and Complete Moment Convergence
Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 2009, Article ID 271265, 14 pges doi:10.1155/2009/271265 Reserch Article Moment Inequlities nd Complete Moment Convergence Soo Hk
More informationPositive Solutions of Operator Equations on Half-Line
Int. Journl of Mth. Anlysis, Vol. 3, 29, no. 5, 211-22 Positive Solutions of Opertor Equtions on Hlf-Line Bohe Wng 1 School of Mthemtics Shndong Administrtion Institute Jinn, 2514, P.R. Chin sdusuh@163.com
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationHERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α, m)-convex
HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α -CONVEX İMDAT İŞCAN Dertent of Mthetics Fculty of Science nd Arts Giresun University 8 Giresun Turkey idtiscn@giresunedutr Abstrct:
More information