Following this strategy, previously developed in [6], let us consider the random

improper integralFurthermore, the concept of the absolute convergence of a fuzzy

improper integral does not make sense in the fuzzy literature.

where the integral is the usual Riemann

improper integral and [alpha] [member of] (0,1].

then the Ito integral [bar.Y](0, [omega]) can be used to approximate the

improper integral (26), where

The initial IPS might show an obvious gradient in range direction, which is caused by the

improper integral constant.

Or, if asked to determine whether the

improper integral [[integral].sup.[infinity].sub.0] 1/[e.sup.x] + [e.sup.-x] is convergent, we might start by using technology to evaluate [[integral].sup.10.sub.0] 1/[e.sup.x] + [e.sup.-x] dx and [[integral].sup.15.sub.0] 1/[e.sup.x] + [e.sup.-x].

it is possible to conclude that the first integral at the second member of (17) is an

improper integral of a non-oscillating function which decays asymptotically as [mathematical expression not reproducible], while the last one is a proper integral.

If a [member of] T, sup T = ot and f is rd-continuous on [a, [infinity]), then one defines the

improper integral by

for t [greater than or equal to] T and the

improper integral of the larger function converges for [absolute value of v] < k, then by the comparison test, the integral [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges for [absolute value of v] < k.

The function [g.sub.1] must guarantee the convergence of the

improper integral (27) (i.e., the existence of [u.sub.1]), and the fact that this function [u.sub.1] is solution of the Neumann problems with f = 0.

approaches a finite limit as A [right arrow] [infinity], then we call that limit the

improper integral of first kind of f from a to[infinity]and write

In contrast with the usual convolution algorithm, the nonhistory-dependent algorithm described in the previous section needs the replacement of the

improper integral by a finite sum.