If I attempt to simplify the sum ArcTan[1/x] + ArcTan[x]
with
ArcTan[ Simplify[ TrigExpand[ Tan[ ArcTan[1/x] + ArcTan[x]]]]]
Mathematica returns
During evaluation of In[103]:= Power::infy: Infinite expression 1/0 encountered. >> During evaluation of In[103]:= Power::infy: Infinite expression 1/0 encountered. >> During evaluation of In[103]:= Infinity::indet: Indeterminate expression ComplexInfinity + ComplexInfinity encountered. >> Indeterminate
even though the result is $\pm\pi/2$. The error messages occur during the call to TrigExpand
, and I can't think of a way to avoid them. Also, I tried in vain the assumption x > 0
, so that the result is not ambiguous.
The full expression to simplify is
(1/(16 a))Q^2 (-16 ArcTan[r1/a] + 1/(2 r1^2)(16 π r1^2 + 32 r1 a
+ π a^2 + 8 π r1^2 Log[r1] + 4 ArcTan[a/r1] (7 a^2 - 8 r1^2 Log[r1])
- 2 ArcTan[r1/(2 a) - a/(2 r1)] (a^2 + 8 r1^2 Log[r1])))
I know that the Log
s vanish after simplification and I was trying to prove this with Mathematica. It is possible if I split the expression into several parts, then simplify with the transformation rule for the addition of ArcTan
, because none of the other methods work in this case, but the transformation rule aborts when it meets an infinite expression.
Tan
, which yields1/0
in both cases. $\endgroup$ – Michael E2 Jul 15 '14 at 22:50x>0
, still the same. $\endgroup$ – auxsvr Jul 15 '14 at 22:51