Sentence examples for provided that the integral from inspiring English sources

Accordingly, for any random variable X, E X) is defined to be the Lebesgue integral of X with respect to the probability measure P, provided that the integral exists.

provided that the integral in (2) exists.

(2.2) provided that the integral in (2.2) exists.

provided that the integral exists on, where is the Gamma function.

System (3.1) is equivalent to the fractional integral equation (3.4) provided that the integral in (3.4) exists.

But we can choose any order of moment β provided that the integral in Equation (9) exists.

provided that the integrals on the right-hand side are convergent.

Then the non-integer order integral of order (qin mathbf{R_) of the function (z(t)) is defined as mathbf{I}_{a+}^{q} z(t)=frac{1}{Gamma (q)} int_{a}^{t}frac{z( tau )}{ t-tau )^{ t-taudtau, provided that integral on the right is pointwise defined on ((0, infty )).

When ( t, z ) ∈ S, we provide that the integral on the right-hand side of (8) represents Cauchy's principal value.

In this paper, we introduce multilinear fractional integrals and its commutators on non-homogeneous metric spaces, then we study the boundedness in Lebesgue spaces for these operators, provided that fractional integral is bounded from (L^{r}(mu)) to (L^{s}(mu)), for some (rin 1, 1/beta)) and (1/s=1/r-beta) with (0<beta<1).

where u ˆ i = ( u 1, …, u i − 1, 1, u i + 1, …, u n ), d ˆ i u = d u 1 ⋯ d u i − 1 d u i + 1 ⋯ d u n, and provided that the above integral converges.