Why ArcTan[x,y]-ArcTan[x,-y] = 2 ArcTan[x,y] is not equivalent in the WL?

Ideas
1

I’m working with integrals that yield pairs of terms of the form ArcTan[x,y]-ArcTan[x,-y] or equivalent expressions involving Arg or complex Log. For Mathematica’s choice of the branch cut, this is equal to 2 ArcTan[x,y]. However, it appears that the only way to achieve this simplification, or even to get Mathematica to acknowledge that these are equivalent, is to use a custom transformation function like

negt[ArcTan[x_, y_]] := -ArcTan[x, -y]

in Simplify or FullSimplify.

Have I missed something?

2

In[2]:= ArcTan[x, y] - ArcTan[x, -y] - 2 ArcTan[x, y] /. {x → -1, y → 0}

Out[2]= -2 Pi
1 Like
3

In[1]:= $Assumptions = {x > 0}; Simplify[ArcTan[x, y] - ArcTan[x, -y] - 2 ArcTan[x, y]]
Out[1]= 0