Tham khảo tài liệu 'mathematical method in science and engineering episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | OTHER DEFINITIONS OF DIFFERINTEGRALS 393 Fig. Contour C c 4- Co Li 4- L in the differintegral formula which goes to zero in the limit 60 0. For the Co integral to be zero in the limit 0 0 we have taken q as negative. Using this result we can write Equation as f f z dz r r f z dz f f z dz -Jc z -x q 1 j L z -x g 1 2 - X q I. VW Ỉ J- Li z1 -x l 1 J_ L2 z -x l l Now we have to evaluate the J .t2 integral. We first evaluate the parts of the integral for 00 0 which gives zero as r ĩtẻe - dôé r f 6ei v dSei s - x 1 - ổe e - x 1 U-e _e e l f We - dS L ij-oo -5- 1 oL 1J_OO -6-x 9 i 0. 394 FRACTIONAL DERIVATIVES AND INTEGRALS DIFFERINTEGRALS Fig. Contours for the f L and integrals Writing the remaining part of the jy. dz integral we get - 1 f 6eis eiEd6 fx 0- 1e i 1 Jo 6 -aộs Ms ritỈTr-E After taking the limit we substitute this into the definition Eq. to obtain n _ e w r q 0 and noninteger. dx 2m L J Jo 6 - x l H 5 Simplifying this we can write M1 i 7 ư 1485 To see that this agrees with the Riemann-Liouville definition we use the following relation of the gamma function and write fW Ợ 0 and noninteger. OTHER DEFINITIONS OF DIFFERINTEGRALS 395 This is nothing but the Riemann-Liouville definition. Using Equation we can extend this definition to positive values of q. Riemann Formula We now evaluate the differintegral of xp which is very useful for finding differintegrals of functions the Taylor series of which can be given. Using formula we write dqxp _ r ợ 1 dxq 7T sin 7rạ -l i Jo sp dẻ 6 rr 1 and dqxp r o l . w r p dỗ --------- sin 7T d I 9 I v dxq 7T v Jo 35 1 11 _ 1 9 1 We define I 8 so that Equation becomes _ LtA dxq 7T Jo 1 v Remembering the definition of the beta function yp 1 y qĩdy p and q 0 Jo we can write Equation as dqxp r n -I- 11 ----YL12 s q xP-qB p 1 q . dxq TĨ Also using the relation and between the .
