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