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I want to apply Cauchy rule (attached) to the following. I tried but in vain to code it using Mathematica. Any help to define this rule is highly appreciated.

sum1=Sum[E^(I*((2*p*Pi*x)/\[CapitalLambda]x + (2*Pi*q*y)/\[CapitalLambda]y +(2*Pi*r*z)/\[CapitalLambda]z))*a[p, q, r], 
   {p, -Infinity, Infinity}, {q, -Infinity, Infinity}, {r, -Infinity, Infinity}];

sum2 = Sum[Sx[p, q, r]/E^(I*(x*kx[p] + y*ky[p] + z*kz[p])), {p, -Infinity, Infinity}, 
{q, -Infinity, Infinity}, {r, -Infinity, Infinity}];

enter image description here

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