desarrollo multipolar

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Como $latex Delta u = f leftrightarrow u = Delta^{-1} f$ entonces $latex mathcal{M}(u) = mathcal{M}(Delta^{-1} f)$ y

$latex mathcal{M}(Delta^{-1}f) = frac{1}{r} M(f) + frac{1}{r^2}n_i D^i(f) + frac{3}{2} frac{1}{r^3} n_{langle i} n_{j rangle} Q^{ij}(f) + O(frac{1}{r^4}) + $

$latex + Delta_0^{-1} mathcal{M}(f)$

con

$latex M(f) = – frac{1}{4 pi} int f$,

$latex D^i(f) = – frac{1}{4 pi} int x^i f$,

$latex Q^{ij}(f) = – frac{1}{4 pi} int x^i x^j f$

y $latex mathcal{M}(f) = 0$ si $latex f$ es de soporte compacto.

 1.- $latex boxed{Delta Theta_X = frac{3}{4} 8 pi mathcal{D}^i S_i^*}$ donde $latex Theta_X := mathcal{D}_j X^j$

En este caso, $latex f_{Theta_X} := frac{3}{4} 8 pi mathcal{D}^i S_i^*$ y, por tanto, $latex mathcal{M}(f_{Theta_X})=0$. De esta manera, tenemos:

$latex M(f_{Theta_X}) =$,

$latex D^i(f_{Theta_X}) = – frac{1}{4 pi} int x^i frac{3}{4} 8 pi mathcal{D}^i S_i^* = -frac{3}{2} (int mathcal{D}^j(x^i S_j^*) d^3x’ – int S_j^* mathcal{D}^j x^i d^3x’)$,

$latex Q^{ij}(f_{Theta_X}) = – frac{1}{4 pi} int x^i x^j frac{3}{4} 8 pi mathcal{D}^i S_i^*$

$latex mathcal{M}(Delta^{-1}f_{Theta_X}) = + O()$

2.- $latex boxed{Delta X^i = 8 pi f^{ij} S_j^* – frac{1}{3} mathcal{D}^i Theta_X}$

Ahora hacemos $latex f_{X^i} := 8 pi f^{ij} S_j^* – frac{1}{3} mathcal{D}^i Theta_X$

$latex M(f_{X^i}) = – frac{1}{4 pi} int 8 pi f^{ij} S_j^* – frac{1}{3} mathcal{D}^i Theta_X$,

$latex D^i(f_{X^i}) = – frac{1}{4 pi} int x^i (8 pi f^{ij} S_j^* – frac{1}{3} mathcal{D}^i Theta_X)$,

$latex Q^{ij}(f_{X^i}) = – frac{1}{4 pi} int x^i x^j (8 pi f^{ij} S_j^* – frac{1}{3} mathcal{D}^i Theta_X)$

$latex mathcal{M}(Delta^{-1}f_{X^i}) = + O()$

3.- $latex boxed{Delta psi = -2 pi E^* psi^{-1} – frac{1}{8} ( f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij}) psi^{-7}}$

En esta ocasión, $latex f_psi := -2 pi E^* psi^{-1} – frac{1}{8} ( f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij}) psi^{-7}$

$latex M(f_psi) = – frac{1}{4 pi} int -2 pi E^* psi^{-1} – frac{1}{8} ( f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij}) psi^{-7}$,

$latex D^i(f_psi) = – frac{1}{4 pi} int x^i (-2 pi E^* psi^{-1} – frac{1}{8} ( f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij}) psi^{-7})$,

$latex Q^{ij}(f_psi) = – frac{1}{4 pi} int x^i x^j (-2 pi E^* psi^{-1} – frac{1}{8} ( f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij}) psi^{-7})$

$latex mathcal{M}(Delta^{-1}f_psi) = + O()$

4.- $latex boxed{ Delta (alpha psi) = big( 2 pi (E^* + 2S^*) psi^{-2} + frac{7}{8} (f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij} ) psi^{-8} big) (alpha psi) }$

Definimos $latex f_{alpha psi}:=big( 2 pi (E^* + 2S^*) psi^{-2} + frac{7}{8} (f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij} ) psi^{-8} big) (alpha psi)$

$latex M(f_{alpha psi}) = – frac{1}{4 pi} int big( 2 pi (E^* + 2S^*) psi^{-2} + frac{7}{8} (f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij} ) psi^{-8} big) (alpha psi)$,

$latex D^i(f_{alpha psi}) = – frac{1}{4 pi} int x^i (big( 2 pi (E^* + 2S^*) psi^{-2} + frac{7}{8} (f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij} ) psi^{-8} big) (alpha psi))$,

$latex Q^{ij}(f_{alpha psi}) = – frac{1}{4 pi} int x^i x^j (big( 2 pi (E^* + 2S^*) psi^{-2} + frac{7}{8} (f_{il} f_{jm} hat{A}^{lm} hat{A}^{ij} ) psi^{-8} big) (alpha psi))$

$latex mathcal{M}(Delta^{-1} f_{alpha psi}) = + O() $

5.- $latex boxed{Delta Theta_beta = frac{3}{4}mathcal{D}^i (mathcal{D}_j(2alpha psi^{-6}hat{A}^{ij})) }$ con $latex Theta_beta := mathcal{D}_i beta^i$

Para esta ecuación, $latex f_{Theta_beta}:=frac{3}{4}mathcal{D}^i (mathcal{D}_j(2alpha psi^{-6}hat{A}^{ij}))$

$latex M(f_{Theta_beta}) = – frac{1}{4 pi} int frac{3}{4}mathcal{D}^i (mathcal{D}_j(2alpha psi^{-6}hat{A}^{ij}))$,

$latex D^i(f_{Theta_beta}) = – frac{1}{4 pi} int x^i frac{3}{4}mathcal{D}^i (mathcal{D}_j(2alpha psi^{-6}hat{A}^{ij}))$,

$latex Q^{ij}(f_{Theta_beta}) = – frac{1}{4 pi} int x^i x^j frac{3}{4}mathcal{D}^i (mathcal{D}_j(2alpha psi^{-6}hat{A}^{ij}))$

$latex mathcal{M}(Delta^{-1}f_{Theta_beta}) = + O()$

6.- $latex boxed{Delta beta^i = mathcal{D}_j(2alphapsi^{-6}hat{A}^{ij})-frac{1}{3}mathcal{D}^i Theta_beta}$

Finalmente, tenemos $latex f_{beta^i}:=mathcal{D}_j(2alphapsi^{-6}hat{A}^{ij})-frac{1}{3}mathcal{D}^i Theta_beta$

$latex M(f_{beta^i}) = – frac{1}{4 pi} int mathcal{D}_j(2alphapsi^{-6}hat{A}^{ij})-frac{1}{3}mathcal{D}^i Theta_beta$,

$latex D^i(f_{beta^i}) = – frac{1}{4 pi} int x^i (mathcal{D}_j(2alphapsi^{-6}hat{A}^{ij})-frac{1}{3}mathcal{D}^i Theta_beta)$,

$latex Q^{ij}(f_{beta^i}) = – frac{1}{4 pi} int x^i x^j (mathcal{D}_j(2alphapsi^{-6}hat{A}^{ij})-frac{1}{3}mathcal{D}^i Theta_beta)$

$latex mathcal{M}(Delta^{-1}f_{beta^i}) = + O()$

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A la hora de resolver las diferentes ecuaciones elípticas CFC tenemos dos posibilidades para fijar las condiciones en la frontera, cada una con sus mas y sus menos.

La primera consiste en hacer un desarrollo multipolar de los terminos fuente en armónicos esféricos, de manera que cuantos mas términos consideremos mas cerca podremos colocar la frontera.

La segunda consiste en compactificar el dominio, lo que nos permite reducir todo el universo a un cubo unidad y considerar Minkowski en su frontera, puesto que ésta corresponde a infinito.

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