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 , so as to make it possible analytical integration.
where [q.sub.n] is the coefficient of expansion of the integrand
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