Fold both left and right sides together.
Moreover, left and right conformable integrals were extended to higher order in [16] so that for (alpha=n+1) we have (( _{a}I^{alpha}f ) (x)= ( _{a}mathbf{I}^{alpha}f ) (x)) and (( I_{b}^{alpha}f ) (x)= ( mathbf{I}_{b}^{alpha}f ) (x)).
The function (g_{tilde{A}_{p}}^{rm M}[h_{rm L},h_{rm R}]({rm or}, [h_{rm R},h_{rm L}])longrightarrow [b,c]) is strictly increasing (decreasing) when (h_{rm L}<h_{rm R} (h_{rm R}<h_{rm L}).) Convex combination of right and left integral values through an index of optimism is called the total integral value [8, 11].
These type of methods of ranking fuzzy numbers are based on the convex combination of right and left integral values through an index of optimism found in Liou and Wang [11] and Kim and Park [8].
The left Riemann Liouville integral (({mathcal{I}_{a^^{alpha}}x)(t)) of order (alpha>0) is defined by bigl({mathcal{I}_{a^^{alpha}}x bigr) (t)= frac{1}{Gamma alpha)} int ^{t}_{a} t-zeta)^{alpha-1} t-zeta,d zeta, quad t>a, (2.1) where (Gamma(cdot)) is the gamma function.
[15] The left Riemann Liouville fractional integral (left forward) of order (alpha >0) of a function (f:(0,infty )rightarrow mathrm{I}!mathrm{R}) is given by begin{aligned} I_{0+}^{alpha }f(t)=frac{1}{Gamma (alpha )}int _0^t t-s)^{alpha -1}f(s)mathrm{{d}}s, quad t>0 end{aligned}provided that the right-hand side exists.
We will deeply miss everything about Kate; especially her irreverent humor and spontaneous wit which never left her and was integral to her joie de vivre.
The variable z and the two parameters of the distribution, α and β, can be matched with the terms in Eq. (11) and the density integrated out, leaving the integral proportional to a single non-constant term: begin{aligned} I propto c^{-frac{n-1}{2}} = left[ frac{(n+1)left(x^{2} + sum_{i=1}^{n} {x_{i}}^{2}right -left(x+sum_{i=1}^{2}right -left)^{2}}{n+1} right]^{-frac{n-1}{2}}. end{aligned} (14).
The left Hadamard fractional integral of order α, (Re alpha) >0) has the following form: bigl(_{a}mathcal{J}^{alpha}f bigr) (t)= frac{1}{Gamma alpha)} int_{a}^{t} (log t-log u)^{alpha-1}f u), frac{du}{u}.
The left Riemann Liouville fractional integral of order α, (Re (alpha)>0) is defined by bigl(_{a}I^{alpha}f bigr) (t)= frac{1}{Gamma alpha)} int_{a}^{t} (t-u)^{alpha-1}f u),du. (3).
The discrete (left) generalized fractional integral operator is defined by bigl( mathbf{ E}^{1}_{overline{rho, mu}, omega,a^varphibigr) (t)=sum _{s=a+1}^{t} bigl t-rho(s)bigl t-rhorline{mu-1}} E_{overline{rho,mu}} bigl(omega, t-rho(s bigrr)varphi(s), quad tin mathbb{N}_{a}.
