When [rho] [right arrow] 1+, the ellipse [[epsilon].sub.[rho]] shrinks to the interval [-1,1], and when [rho] [right arrow] 00, the interior of [[epsilon].sub.[rho]] approaches the whole complex plane (which is useful when we deal with entire

integrands such as those in Section 6).

To the end, we focus our attention on the

integrands in it.

The index x of the radius vector is uniquely determined by the

integrands and the Dirac convolution rules.

Next, the remaining part of the magnetic vector potential is written as the sum of branch-cut integrals and closed- contour integrals around the poles of the

integrand. Finally, the hyperbolic branch cuts are extracted from the

integrands of the branch-cut integrals and replaced with equivalent pole sets [23], so as to make it possible analytical integration.

Using the same method as in Ito's classical theory, this integral can be extended to random

integrands, that is, to the class of predictable processes X = {X(t, x); t [member of] [0, T], x [member of] R}, such that E [[integral].sup.T.sub.0] [[integral].sub.R] [[absolute value of (X(t, x))].sup.2] dx dt < [infinity].

By designing the function relation between the weights of two neural networks, one neural network is able to approximate

integrands, whereas the other approximates original function.

We can continue calculating derivatives with respect to y in (12) in order to get new integrals, but the

integrands we get are increasingly complex and we omit these results here.

In this section, we discuss the uniqueness of their corresponding parameters by two useful integral functions, whose

integrands can be used to obtain solutions of QVI.

Since the semigroup is compact, and the

integrands are dominated by an integrable process (due to Corollary 3.2 and growth properties (A2)-(A3)), and [mathematical expression not reproducible] it is clear that for each fixed t [member of] I, both [e.sup.n.sub.1](t) and [e.sup.n.sub.2](t) converge strongly in E to zero P- as.

The

integrands [f.sub.i], [g.sub.i], i = 1, 2 at small values of the argument are irregular.

The tables are organized in a logical manner with standard forms of

integrands arranged in increasing order of complexity, ranging from algebraic forms to special functions and combinations.