A short introduction to local fractional complex analysis
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1 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 short introduction to locl rctionl complex nlysis The generlied locl rctionl complex integrl ormuls, Yng-Tylor series locl rctionl Lurent s series o complex unctions in complex rctl spce, generlied residue theorems re investigted Key words:locl rctionl clculus, complex-vlued unctions, rctl, Yng- Tylor series, locl rctionl Lurent series, generlied residue theorems MS2: 28A8, 399, 3B99 Introduction Locl rctionl clculus hs plyed n importnt role in not only mthemtics but lso in physics engineers [-2] There re mny deinitions o locl rctionl derivtives locl rctionl integrls (lso clled rctl clculus) Hereby we write down locl rctionl derivtive, given by [5-7] x x x x x lim xx d x dx x x with x x x x, locl rctionl integrl o x, denoted by [5-6,8] j b Ib x t dt lim tj tj (2) t j with tj tj t j t mx t, t2, t j,, where or j,,, tj, tj is prtition o the intervlb, t, t b More recently, motivtion o locl rctionl derivtive locl rctionl integrl o complex unctions is given [] Our ttempt, in the present pper, is to continue to study locl rctionl clculus o complex unction As well, short outline o locl rctionl complex nlysis will be estblished ()
2 2 Locl rctionl clculus o the complex-vrible unctions In this section we deduce undmentls o locl rctionl clculus o the complex-vlued unctions Here we strt with locl rctionl continuity o complex unctions 2 Locl rctionl continuity o complex-vrible unctions Deinition Given, then or ny we hve [] (2) Here complex unction is clled locl rctionl continuous t, denoted by A unction lim (22) is clled locl rctionl continuous on the region, denoted by As direct result, we hve the ollowing results: lim lim g Suppose tht g lim g g the lst only i g, then we hve tht, (23) lim g g, (24) lim / g / g, (25) 22 Locl rctionl derivtives o complex unction Deinition 2 Let the complex unction derivtive o be deined in neighborhood o point The locl rctionl t is deined by the expression [] D :lim I this limit exists, then the unction by D, (26) is clled to be locl rctionl nlytic t, denoted d or d, 2
3 Remrk I the limits exist or ll in region, then nlytic in region, denoted by Suppose tht vlid [] i g ; where is constnt; is sid to be locl rctionl D g re locl rctionl nlytic unctions, the ollowing rules re d g d d g ; (27) d d d d g d d g g ; (28) d d d d I y u whereu g d d g g g d d (29) d g 2 d d, (2) d d, then g g 23 Locl rctionl uchy-riemnn equtions Deinition 3 I there exists unction dy (2) d,, uxy ivxy, (22) whereu v re rel unctions o x y The locl rctionl complex dierentil equtions, v x, y u x y x y, v x, y u x y y re clled locl rctionl uchy-riemnn Equtions Theorem Suppose tht the unction x,, (23) (24) uxy ivxy (25) 3
4 is locl rctionl nlytic in region Then we hve u x, y v x, y (26) x y u x, y v x, y (27) y x Proo Since uxy, ivxy, onsequently, the ormul (28) implies tht x y lim, we hve the ollowing identity lim (28) ux x, y y ux, y i vx x, y y vx, y lim x i y In similr mnner, setting y tking into ccount the ormul (29), we hve y such tht Hence uxy, y uxy, i vxy, y vxy, lim y i y i, v x, y u x y y y I x, rom (29) we hve such tht x ux xy, uxy, i vx xy, vxy, (29) (22) (22) lim (222) x x Thus we get the identity u x, y v x, y i (224) x x Since uxy, ivxy, is locl rctionl nlytic in region, we hve the ollowing ormul u x, y v x, y u x, y v x, y i i x x y y Hence, rom (225), we rrive t the ollowing identity (225) 4
5 This completes the proo o Theorem Remrk 2 Locl rctionl -R equtions re suicient conditions tht nlytic in The locl rctionl prtil equtions, u x, y u x y 2 2 x y 2 2, v x, y v x y x y re clled locl rctionl Lplce equtions, denoted by (226) (227) is locl rctionl (228) (229) u x, y (23) v x, y, (23) where 2 2 (232) 2 2 x y is clled locl rctionl Lplce opertor Remrk 3 Suppose tht u x, y,,, v x, y u x y x y, v x, y u x y y x u x y is locl rctionl hrmonic unction in 24 Locl rctionl integrls o complex unction Deinition 4 Let be deined, single-vlued locl rctionl continuous in region The locl rctionl integrl o deined s [] where ori,,, long the contour in rom point p to point q, is I n i i i, p n lim i n q d i (233) 5
6 For convenience, we ssume tht i I (234) The rules or complex integrtion re similr to those or rel integrls Some importnt results re s ollows []: Suppose tht or constnt k ; where 2; g be locl rctionl continuous long the contour in gd d gd ; k k d d (235), (236) d d d where M is n upper bound o Theorem 2 I the contour hs end points, (237) 2 d d ; (238) d d ML, (239) on p hs the primitive F on, then we hve Remrk 4 Suppose tht D L d q with orienttion p to q, i unction d FqFp (24) For k,,, n there exists locl rctionl series k k k (24) k k times k with D k, where D D This series is clled Yng-Tylor series o locl rctionl nlytic unction (or rel unction cse, see [2]) 6
7 Theorem 3 I is simple closed contour, i unction hs primitive on, then [] d (242) orollry 4 I the closed contours, 2 is such tht2 lies inside, i nlytic on, 2 between them, then we hve [] Theorem 5 2 d d is locl rctionl (443) Suppose tht the closed contours, 2 is such tht2 lies inside, i rctionl nlytic on, 2 between them, then we hve[] d d 2 is locl (244) 3 Generlied locl rctionl integrl ormuls o complex unctions In this section we strt with generlied locl rctionl integrl ormuls o complex unctions deduce some useul results Theorem 6 Suppose tht is locl rctionl nlytic within on simple closed contour is ny point interior to Then we hve Proo From(244), we rrive t the ormul where : d 2 i (3) d 2 i 2 i d, (32) Setting implies tht E i (33) 7
8 Tking (33) (34), it ollows rom (32) tht 2 E i 2 i 2 E i 2 lim E i d From (35), we get Furthermore lim (34) i E i d 2 2 E i d d Substituting (37) into (36) (33) implies tht The proo o the theorem is completed Likewise, we hve the ollowing corollry: orollry 7 Suppose tht d (35) (36) 2 (37) d 2 i (38) is locl rctionl nlytic within on simple closed contour is ny point interior to Then we hve i 2 n d (39) n Proo Tking into ccount ormul (3), we rrive t the identity Theorem 8 Suppose tht is locl rctionl nlytic within on simple closed contour is ny point interior to Then we hve d 2 i (39) Proo Tking, rom (39) we deduce the result Theorem 9 Suppose tht is locl rctionl nlytic within on simple closed contour is ny point interior to Then we hve d i E i d 8
9 Proo Tking, rom (39) we deduce the result d n, or n (3) 4 omplex Yng-Tylor s series locl rctionl Lurent s series In this section we strt with Yng-Tylor s expnsion ormul o complex unctions deduce locl rctionl Lurent series o complex unctions 4 omplex Yng-Tylor s expnsion ormul Deinition 5 Let be locl rctionl nlytic inside on simple closed contour hving its center t Then or ll points in the circle we hve the Yng-Tylor series representtion o, given by 2 k 2 2 k k (4) For: R, we hve the complex Yng-Tylor series From (344) the bove expression implies or c: R k 2 k k (42) k k d k k i Successively, it ollows rom (43) tht, (43) where k k k, (44) k 2 k d k k i, (45) 9
10 or: R Hence, the bove ormul implies the reltion (42) Theorem Suppose tht complex unction is locl rctionl nlytic inside on simple closed contour hving its center t There exist ll points in the circle such tht we hve the Yng-Tylor s series o where k 2 k d k k, (45) k k k i or: R, Proo Setting : R using (3), we hve Tking, we get i 2 d (46) q Substituting (48) into (46) implies tht n n n (47) (48)
11 d n n 2 i 2 i n n n d Tking the Yng-Tylor ormul o nlytic unction into ccount, we hve the ollowing reltion where R is reminder in the orm R n n n (49) n n, (4) R n n d 2 i n There exists Yng-Tylor series where is (4) n n n is locl rctionl nlytic t Tking into ccount the reltion n (42) n n q n M, rom (4) we get R Furthermore From (49), we hve 2 2 n 2 R M 2 n n d 2 i n n n d n M n q q n MR q q n n lim R d (43)
12 Hence d n n n 2 i (44) n 2 i Hence the proo o the theorem is completed 42 Singulr point poles Deinition 6 A singulr point o unction rctionl nlytic I interior point I, we cll d n (45) is vlue o t which ils to be locl is locl rctionl nlytic everywhere in some region except t n n isolted singulrity n (46) (47) where is locl rctionl nlytic everywhere in region including, i n is positive integer, then n hs n isolted singulrity t, which is clled pole o order I n, the pole is oten clled simple pole; i n 2, it is clled double pole, so on 43 Locl rctionl Lurent s series Deinition 7 I hs pole o order n t but is locl rctionl nlytic t every other point inside on contour with center t, then n (48) is locl rctionl nlytic t ll points inside on hs Yng-Tylor series bout so tht 2
13 n n n n This is clled locl rctionl Lurent series or More generlly, it ollows tht n n (49) s locl rctionl Lurent series k k (42) i For: r R we hve locl rctionl Lurent series From (344), the bove expression implies tht where: r R k k k (42) k 2 k d i, (422) Setting : r 2 : R, rom (244) we hve i k i k d d 2 2 Successively, it ollows rom the bove tht 2 where k k k, (423) k 2 k d i, (424) or: r R Theorem I then hs locl rctionl nlytic t every other point inside contour with center t, so tht hs locl rctionl Lurent series bout where or: r R we hve k k, i, (425) 3
14 k 2 k d i (426) Proo Setting : r 2 : R, rom (244) we hve tht d d 2 i 2 i (427) 2 Tking the right side o (427) into ccount implies tht or 2 q R (428) M (429) By using (429) it ollows rom (427) tht From (427) we get where 2 i 2 d n d 2 i n 2 2 i d n n n d R 2 i n n lim R lim d 2 i n is reminder n (43) (43) M, tking Since n q n, we hve R 2 n d 2 i n n n d n 4
15 M 2 n 2 n M n n d d Furthermore Hence M 2 n 2 n M M q q 2 i n 2 lim R q n d q n d n d 2 i n ombing the ormuls (43) (433), we hve the result Hence, the proo o the theorem is inished n (432) (433) 5 Generlied residue theorems In this section we strt with locl rctionl Lurent series study generlied residue theorems Deinition 8 Suppose tht is n isolted singulr point o series Then there is locl rctionl Lurent vlid or R The coeicient t, is requently written s k k (5) i o is clled the generlied residue o Re s One o the coeicients or the Yng-Tylor series corresponding to 5 (52)
16 n the coeicient is the residue o, (53) t the pole It cn be ound rom the ormul n d Re s lim n d where n is the order o the pole n n (54) Setting i We know tht this is k k, the expression (53) yields k k (55) i n which is the coeicient o n n n n n, (56) n The generlied residue is thus n where Re s n, (57) n orollry 2 I is locl rctionl nlytic within on the boundry o region except t number o poles within, hving residue, then d Re s 2 i (58) Proo Tking into ccount the deinitions o locl rctionl nlytic unction the pole we hve locl rctionl Lurent s series thereore k k (59) i n n n n (5) Hence we hve the ollowing reltion k d k d (5) i 6
17 urthermore d d (52) From (39), it is shown tht (53) d d i i 2 2 Hence we hve the ormul d 2 i (54) Tking into ccount the deinition o generlied residue, we hve the result This proo o the theorem is completed From (58), we deduce the ollowing corollry: orollry 3 I is locl rctionl nlytic within on the boundry o region inite number o poles,, 2 within then i 2 i k It sys tht the locl rctionl integrl o, hving residues b c n,, except t respectively, d Re s b c (55) residues t the singulr points enclosed by the contour is simply2 i times the sum o the 6 Applictions: Guss ormul o complex unction Theorem 4 Suppose tht R E i we hve is locl rctionl nlytic is ny point, then or the circle RE i d 2 2 (6) Proo By using (3) there exists simple closed contour is ny point interior to such tht i 2 d (62) When cn been tken to be R E i or, 2 R E i, substituting the reltions (63) 7
18 in (62) implies tht d i R E i d, (64) RE i i R E i d 2 i (65) R E i some cncelling gives the result Reerences [] KMKolwnkr, ADGngl Frctionl dierentibility o nowhere dierentible unctions dimensions hos, 6 (4), 996, [2] Arpinteri, Pornetti A rctionl clculus pproch to the description o stress strin loclition in rctl medi hos, Solitons Frctls,3, 22,85 94 [3] FBAdd, Jresson About non-dierentible unctions J Mth Anl Appl, 263 (2), [4] ABbkhni, VDGejji On clculus o locl rctionl derivtives J Mth Anl Appl,27, 22, [5] F Go, XYng, Z Kng Locl rctionl ewton s method derived rom modiied locl rctionl clculus In: Proc o the second Scientiic Engineering omputing Symposium on omputtionl Sciences Optimition (SO 29), , IEEE omputer Society,29 [6] XYng, F Go The undmentls o locl rctionl derivtive o the one-vrible nondierentible unctions World Sci-Tech R&D, 3(5), 29, [7] XYng, FGo Fundmentls o Locl rctionl itertion o the continuously nondierentible unctions derived rom locl rctionl clculus In: Proc o the 2 Interntionl onerence on omputer Science Inormtion Engineering (SIE2), , Springer, 2 [8] XYng, LLi, RYng Problems o locl rctionl deinite integrl o the one-vrible nondierentible unction World Sci-Tech R&D, 3(4), 29, [9] JH He A new rctionl derivtion Therml Science5,, 2, [] W hen Time spce bric underlying nomlous disusion hos, Solitons Frctls, 28, 26, [] XYng Frctionl trigonometric unctions in complex-vlued spce: Applictions o complex number to locl rctionl clculus o complex unction ArXiv:62783v [mth-ph] [2] XYng Generlied locl rctionl Tylor's ormul or locl rctionl derivtives ArXiv:62459v [mth-ph] 8
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