I am trying to understand Green's functions, specifically with respect to solving a differential equation $F=-\nabla^2x$. This involves the Green's function of the Laplacian operator.
My understanding is that the Green's function of a differential operator $L$ is $G(x,s)$ such that $LG(x,s)=\delta(x,s)$ where $\delta(x,s)$ is the Dirac delta function. I appreciate that I am being a bit loose in my terminology here, in that x and s may be scalars or vectors.
It seems that from this definition, a particular $L$ can have multiple Green's functions, both as obvious families and with seemingly different forms.
For example for the 2D Laplacian operator, the green's function listed on Wikipedia at https://en.wikipedia.org/wiki/Green%27s_function#Table_of_Green's_functions is
$G(x,s)=1/(2\pi)ln(\sqrt((x-s_x)^2+(y-s_y)^2))$
Which upon checking fits the definition above. We add any other function to this, providing the second derivative of this function is zero. For example
$G(x,s)=1/(2\pi)ln(\sqrt((x-s_x)^2+(y-s_y)^2)) +2x +2y$
Would appear to fit the definition also.
I have also seen the definition
$G(x,s)=1/(2\pi)/\sqrt((x-s_x)^2+(y-s_y)^2)$
This definition seem much more useful because it tends to zero as $x$ or $y$ become large. This seems very useful when using the Green's function to solve differential equations.
I wanted to check, am I correct in thinking that all these equations are Green's Functions of the 2D Laplacian operator? If so, then is there a reason why the first definition seems relatively common, despite the last version seeming to me to be more useful?