How to check if the Laplace of a function is 0?

Questions
,
1

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 ?

2

Oh, nevermind.

FullSimplify does it perfectly FullSimplify: Returns the simplest form of an expression—Wolfram Documentation

3

Ok, but FullSimplify does not work in this case:

In[98]:= Koncna[x_, y_] := A*Log[((x - a)^2 + y^2)/((x + a)^2 + y^2)]

In[99]:= Elpolje = -Grad[Koncna[x, y], {x, y}]

Out[99]= {-((
  A ((a + x)^2 + 
     y^2) (-((2 (a + x) ((-a + x)^2 + y^2))/((a + x)^2 + y^2)^2) + (
     2 (-a + x))/((a + x)^2 + y^2)))/((-a + x)^2 + y^2)), -((
  A ((a + x)^2 + 
     y^2) (-((2 y ((-a + x)^2 + y^2))/((a + x)^2 + y^2)^2) + (
     2 y)/((a + x)^2 + y^2)))/((-a + x)^2 + y^2))}

In[103]:= Simplify[-Grad[Koncna[x, y], {x, y}], {x, y}]

Out[103]= {(4 a^3 A)/(
 a^4 + 4 True^4), -((8 a A x y)/(a^4 + 4 True^4))}

What True? I don’t need True. I need expression. O.o

4

The second argument to Simplify has a meaning that might be very different from what you expect (it is forming assumptions). If you remove it you will get a result that probably is more to your expectations.