A common misconception by students is failing to take into account the axis of rotation in the
integrand. Even though it does not affect the formula in this case, it is worth noticing that the axis of rotation could change, and it always has a place in the
integrand.
Since the
integrand is periodic, this expression reduces to
In particular, for the case when the
integrand is a product of an arbitrary function g(x) of the integration variable x and a Gaussian PDF N(x; [[mu].sub.x], [C.sub.x]) an integration rule of the form
If, however, the diffusion coefficients or the
integrand of the integral over the Poisson measure has a more complicated form, then the apparatus proposed in the aforementioned papers can hardly be used.
In the
integrand, the coefficients [A.sub.i] (i = 1, 2, 3, 4) and [B.sub.i] (i = 1, 2) are weight constants used to balance the size of concerned population and the importance of control cost.
When regarding the
integrand in (15) as the Lagrangian, we can define a conserved quantity associated with translations in [theta], that is,
The value of t which maximizes the
integrand in Equation (20) is called the saddlepoint and is denoted by ts.
Unfortunately, if the same procedure is applied for higher Pr numbers, the polynomial approximations yield coefficients that do not permit integration as the square root expression in the
integrand (equation (17)) takes a negative value.
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.
where [q.sub.n] is the coefficient of expansion of the
integrand function [4],
Proof: Noting that [mathematical expression not reproducible] do not depend on ph, the partial derivative of the
integrand in the definition of [??], (see Relation (36)), is
