Integration theorem

Showing that T(p) is the proper fourier transform of T(x) suggests that the fourier integral theorem should hold for the two wavefunetions T(x) and T(p) we have obtained, e.g. 

Hydrodynamic Equations.—Before deriving the hydro-dynamic equations, some integral theorems that are useful in the solution of the Boltzmann equation will be proved. Consider a function of velocity, G(Vx), which may also be a function of position and time let... 

Using the integral theorem, a volume integral V(ro) in (5.1.26) for J-dimensional sphere of the radius ro is transformed into a surface one, from where one gets... 

Expression (24) reduces to the standard 1 = 0 for q —> n, due to the divergence of the gamma function T(z) for nonpositive integers. The fractional Riemann-Liouville integral operator oDJq fulfills the generalized integration theorem of the Laplace transformation ... 

Fj rtoft, R. (1950). Application of integral theorems in deriving criteria of stability for laminar flows and for baroclinic circular vortex. Geofys. Publ. Oslo 17(6), 1-51. 

Thus, the local statement of the virial theorem is, term for term, the differential form of the integrated theorem in eqn (6.22). Because of this correspondence, one can define the density corresponding to the total virial T(n) as... 

As shown in equation (22.10), the real part of the impedance tends toward a finite value as frequency tends toward infinity. The transfer function Z x) — Zr,oo tends toward zero with increasing frequency. As Z(x) is analytic, Cauchy s integral theorem, given in Appendix A as Theorem (A.2), can be written as... 

The integral which extends over the area of the region has been converted to the volume integral of the divergence of q according to Gauss integral theorem. 

Equation (69) may be further simplified if we recall the integration theorem of Laplace transformation as generalized to fractional calculus, [31], namely,... 

Here, as above, y is the Sack inertial parameter. Noting the initial condition, Eq. (238), all the cn j (0) in Eq. (256) will vanish with the exception n = 0. On using the integration theorem of Laplace transformation as generalized to fractional calculus, we have from Eq. (256) the three-term recurrence relation [cf. Eq. (240)] for the only case of interest q — 1 (since the linear dielectric response is all that is considered) ... 

In dielectric relaxation l = 1 so that by taking the Laplace transform of Eqs. (273)-(275) over the time variables and noting the generalized integral theorem for Laplace transforms, we then have a system of algebraic recurrence relations for the Laplace transform of cln m(t) (m = 0, 1) governing the dielectric response, namely,... 

Applying the integration theorem of one-sided Fourier transformation generalized to fractional calculus, we have from Eq. (A2.1)... 

Here y = x/r[ = =C, /2/IkT is chosen as the inertial effects parameter (y = /2/y is effectively the inverse square root of the parameter y used earlier in Section I). Noting e initial condition, Eq. (134), all the < (()) in Eq. (136) will vanish with the exception of n = 0. Furthermore, Eq. (136) is an example of hoyv, using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics. The normalized complex susceptibility /(m) = x ( ) z"( ) is given by linear response theory as... 

It follows from the continuum assumption that the integrands in (3.415) are continuous and differentiable functions, so the integral theorems of Leibnitz and Gauss (see app. A) can be applied transforming the system description into an Eulerian control volume formulation. The governing mixture... 

To obtain a general integral representation for solutions of the creeping-flow equations, it is necessary first to derive a general integral theorem reminiscent of the Green s theorem from vector calculus. 

Another consequence of the integral theorem (8-111) is that we can calculate inertial and non-Newtonian corrections to the force on a body directly from the creeping-flow solution. Let us begin by considering inertial corrections for a Newtonian fluid. In particular, let us recall that the creeping-flow equations are an approximation to the full Navier-Stokes equations we obtained by taking the limit Re -> 0. Thus, if we start with the ftdl equations of motion for a steady flow in the form... 

Surprisingly, however, it is not actually necessary to solve for the velocity and pressure fields, ui and p, in order to determine the first correction to the force Fj. Instead, by manipulating the integral theorem (8 111), we can determine Fj based on the solution of creeping-flow problems only. 

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