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

Posted 9 years ago
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 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 ?
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Posted 9 years ago
 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
Posted 9 years ago
 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.
Posted 9 years ago
 Oh, nevermind.FullSimplify does it perfectly http://reference.wolfram.com/language/ref/FullSimplify.html