I am trying to integrate an equation that requires an angle term calculated using ArcTan[x,y]. A number of issues have been mentioned about ArcTan in some of the older questions (Link here) and I wanted the ArcTan function to give out 0 as an output when I give ArcTan[0,0] as an input.

I have defined my ArcTan function as:

myArcTan[x_,y_] := Which[x > 0 || y > 0 || x < 0 || y < 0 , ArcTan[x, y], 
x == 0 && y == 0, 0];

Question1: Is myArcTan correctly defined for what I am trying to do?.

When I use myArcTan as one of the terms in the indefinite integral given below, I get a different answer than when using ArcTan. Definite integral, however, gives the same result for both the cases:

Integral1 = Integrate[myArcTan[x, y], y, x]
Integral1 /. {x -> 1, y -> 0}
Out[2]= 1/4

Integral2 = Integrate[ArcTan[x, y], y, x] 
Integral2 /. {x -> 1, y -> 0}
Out[4]= 0

Integrate[myArcTan[x, y], {y, -1, 1}, {x, -1, 1}]
Out[5]= 0

Integrate[ArcTan[x, y], {y, -1, 1}, {x, -1, 1}]
Out[6]= 0

Question 2: One of the answers given in the older question (Link here) uses Arg[x + i y] but an indefinite integral using Arg does not evaluate. Only a numerical approximation using NIntegrate works on it. Why is this so?, can one do an indefinite integral using Arg?.

  • 2
    $\begingroup$ An indefinite integral is not a function. It’s an equivalence class of functions up to constant difference (the $+C$ from calculus). Mathematica gives one representative of that equivalence class. Perhaps that’s what you’re seeing. From the Help in Mathematica 11: “Different forms of the same integrand can give integrals that differ by constants of integration.” $\endgroup$ – Steve Kass Apr 11 '18 at 17:55
  • $\begingroup$ Right, I totally forgot about it. But, I'm still not sure about the problem of ArcTan[0,0] being undefined and my implementation of it (myArcTan). $\endgroup$ – dykes Apr 11 '18 at 18:40
  • $\begingroup$ Have you tried out some of the alternate expressions listed in Wikipedia? $\endgroup$ – J. M. is away Apr 11 '18 at 21:07
  • $\begingroup$ @J.M. As per Wikipedia myArcTan looks correct to me other than the ArcTan[0,0] value which is undefined in Wikipedia and 0 by my definition. $\endgroup$ – dykes Apr 11 '18 at 23:13
  • 1
    $\begingroup$ I see. I'll only note that it's more natural to use Piecewise[]: Piecewise[{{ArcTan[x, y], x != 0 || y != 0}}, 0] $\endgroup$ – J. M. is away Apr 12 '18 at 13:01

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