In the end it is just an artificial problem that comes from an incorrect choice of coordinates. La delta de Dirac o función delta de Dirac es una distribución o función generalizada introducida por primera vez por el físico británico Paul Dirac y, como distribución, define un funcional en forma de integral sobre un cierto espacio de funciones. Or you simply use symmetry to make $\vec a$ not point in the $z$-direction. Diagrama esquemático de la función delta de Dirac. For this you can just choose different spherical coordinates with respect to some other axis than $z$. I saw that the answer is related to Dirac delta function as: ( r) q 3 ( r) where 3 ( r) is the 3-dimensional Dirac Delta function. You have to make sure that your cover $\vec a$ with your domain. 1 I just got introduced to the Dirac delta function and one of the questions was to express volume charge density ( r) of a point charge q at origin. On the order hand, the delta function of a vector can be decomposed into the product of several delta functions, of the vector's components,į(\vec a)=\int f(\vec x)\prod_i\delta^3(x_i-a_i)\mathrm d x_i,\tag(\vec x - \vec a)), Where $f(\vec x)$ is an arbitrary function on the space, and $|J(\vec x)|$ is the Jacobian determinant. The Dirac notation allows a very compact and powerful way of writing equations that describe a function expansion into a basis, both discrete (e.g. It satisfiesį(\vec a)=\int \delta^3(\vec x-\vec a)f(\vec x)|J(\vec x)|\mathrm d^3\vec x, An alternative derivation, motivated by Hobsons derivative expression for solid spherical harmonics and utilizing Gauss Divergence Theorem, is presented. Let $\delta^3(\vec x-\vec a)$ represent a point density at $\vec a$. The nature of the Dirac delta-function singularity in the expansion theorem for an irregular solid spherical harmonic about another centre is discussed for the casel2.
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