Poissons Integral

Now we will establish a relationship between the potential U(p) at any point p of the volume V and its values on the spherical surface, surrounding all masses. Fig. 1.11. The reason why we consider this problem is very simple it plays the fundamental role in Stokes s theorem, which allows one to determine the elevation of the geoid with respect to the reference ellipsoid. 

As in the previous section it is natural to start from the Green s formula 

Here the unit vector n and radius vector R have opposite directions. The volume V is surrounded by the surface S as well as a spherical surface with infinitely large radius. In deriving this equation we assume that the potential U p) is a harmonic function, and the Green s function is chosen in such a way that allows us to neglect the second integral over the surface when its radius tends to an infinity. The integrand in Equation (1.117) contains both the potential and its derivative on the spherical surface S. In order to carry out our task we have to find a Green s function in the volume V that is equal to zero at each point of the boundary surface  

In the ease of the plane boundary surface we represented the Green s function as a combination of potentials due to point masses located at points p and s, which are mirror reflection of each other with respect to the plane S. By analogy, let us attempt to describe the Green s function as a difference 

s is a point located somewhere inside the sphere along the line Op, p and sq the distances between points q, p and q, s, respectively, and a. a constant. Now we demonstrate that for each observation point p it is possible to find such a point and coefficient a, that the Green s function is equal to zero at all points of the sphere. In order to prove it, we first choose the position of the point from the condition  

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The resulting electrostatic potential can be obtained from the charge density distribution via the Poisson integral,... 

The remaining proof concerns the check of the Bom normalization condition at the level of the whole wave-function packet. Using the same Poisson integral mle, one computes the wave-function anal5dical expression in successive steps ... 

A.2 EULER S BETA AND GAMMA FUNCTIONS APPLICATIONS ON POISSON INTEGRALS AND STIRLING S APPROXIMATION... 

From the previous section and, specifically, from inspection of the Poisson integral in Eq. (2.141) it is clear that the scalar potential is transmitted instantaneously retardation effects do not show up in Coulomb gauge for the scalar potential (however, this is not true for the vector potential). We will later utilize this fact in section 3.5 to derive the retardation-free scalar potential in a special frame of reference in which Coulomb gauge holds. 

The classical electrostatic electron-nucleus interaction energy Eg,nuc of a positive nuclear charge distribution PnC ") with the surrounding electronic density Pe(r) is given by the Poisson integral [cf. Fq. (2.141)],... 

As 7 is harmonic above the geoid, this linear combination, multiplied by r, is also harmonic above the geoid. As such it can be continued downward to the geoid by using the standard Poisson integral. 

We established equation [5.29], which gives the diffusion flux with the electric field. Because of the impossibihty of integrating the relation (Poisson integration) to evaluate the flux, two simplifying but contradictory assumptions were introduced to express the electric field in the layer. These two assumptions provide comparison for the layer of product MG to a plane condenser (we are working only with plates) and it is known that the capacity of such a condenser is inversely proportional to its thickness. 

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