I have a function:
G[x_, y_] := (A/(4*Pi))*Log[((x - a)^2 + y^2)/((x + a)^2 + y^2)]
And Laplace of this function should be zero, I just want to check if that is true. How do I do that?
I tried
In[62]:= Laplacian[G[x, y], {x, y}]
Out[62]= -((
50 ((20 + x)^2 + y^2) ((
8 (20 + x)^2 ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^3 - (
8 (-20 + x) (20 + x))/((20 + x)^2 + y^2)^2 - (
2 ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^2 +
2/((20 + x)^2 + y^2)))/(((-20 + x)^2 + y^2) Log[39])) - (
50 ((20 + x)^2 + y^2) ((
8 y^2 ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^3 - (
8 y^2)/((20 + x)^2 + y^2)^2 - (
2 ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^2 +
2/((20 + x)^2 + y^2)))/(((-20 + x)^2 + y^2) Log[39]) - (
100 (20 +
x) (-((2 (20 + x) ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^2) + (
2 (-20 + x))/((20 + x)^2 + y^2)))/(((-20 + x)^2 + y^2) Log[
39]) + (100 (-20 + x) ((20 + x)^2 +
y^2) (-((2 (20 + x) ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^2) + (
2 (-20 + x))/((20 + x)^2 + y^2)))/(((-20 + x)^2 + y^2)^2 Log[
39]) - (100 y (-((
2 y ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^2) + (
2 y)/((20 + x)^2 + y^2)))/(((-20 + x)^2 + y^2) Log[39]) + (
100 y ((20 + x)^2 +
y^2) (-((2 y ((-20 + x)^2 + y^2))/((20 + x)^2 + y^2)^2) + (
2 y)/((20 + x)^2 + y^2)))/(((-20 + x)^2 + y^2)^2 Log[39])
But as you can see I only get a horrible expression.
So, how do I check if the sum of partial derivations is zero for all x,y ?