Dealing with a Discontinuous ArcTan[] Function in Mathematica

[guerom00]

Hello everyone,

I copy-paste here a little bit of code:

In[4417]:= x = (1.5` + 0.` I);
y = (0.` + 0.` I);

(* 1 *) ArcTan[x, y]
(* 2 *) ArcTan[x // Chop, y // Chop]
(* 3 *) -I Log[(x + I y)/Sqrt[x^2 + y^2]]
{y == 0, Re[y] == 0, Im[y] == 0, Im[x] == 0}

Out[4419]= -1.5708 + 0. I

Out[4420]= 0

Out[4421]= 0. + 0. I

Out[4422]= {True, True, True, True}

The quantities x and y are calculated elsewhere.
The expression 1 for the arctangent is wrong since it gives me -pi/2
In expression 2, I chop the arguments and the result is right.
Expression 3 is used when x or y is complex and gives the right answer as well.
Finally, some tests…

Any thoughts ?
TIA

 

[phyzguy]

I agree it looks wrong, but I think ArcTan[x,y] is assuming x and y are both real. It must have a problem when thy are complex.

 

[guerom00]

The help page of ArcTan[x,y] indicates that it can handle complex arguments. It then uses the expression which involves the Log[] I gave.
Problem is: my quantities x and y are not complex according to the tests I make… Which is strange because I thought that the number 0.` is different from 0 (== wise)…

 

[phyzguy]

ArcTan[x] should handle complex arguments, but what does ArcTan[x,y] even mean when the arguments are complex?

 

[guerom00]

Like I said : http://reference.wolfram.com/mathematica/ref/ArcTan.html
Click the “More Information” and have a look at the last bullet :)

ArcTan[x,y] is simply ArcTan[y/x]. If x or y are complex, y/x is a complex number… No problem here.

 

[Hurkyl]

(for reference: http://reference.wolfram.com/mathematica/ref/ArcTan.html)

Also interesting:
http://www.wolframalpha.com/input/?i=ArcTan[1.5+++0.+I,+0.]



But really, let this be a lesson to you about numerical computation. You always have to be careful about calculations with numbers that are approximately some special value.

 

[guerom00]

Also interesting:
http://www.wolframalpha.com/input/?i=ArcTan[1.5+++0.+I,+0.]

Result : 0
Alternative form : -1.5708


WTF ?! This is clearly broken, no ?

Edit: Haha
Compare http://www.wolframalpha.com/input/?i=ArcTan[1.+0+I,+0.] and http://www.wolframalpha.com/input/?i=ArcTan[1.+0.+I,+0.]
Difference is subtle…

 

[Hurkyl]

WTF ?! This is clearly broken, no ?

No and yes.

No, in the sense that you should expect numerical algorithms to misbehave at special values. Yes, in the sense that it's a little surprising these values are special.


What's most likely happening internally is that there is an algebraic identity being used and your point is right on a branch cut of the terms involved. And because your numbers are approximate, different sides of the branch cut wind up being used in different terms.

 

[Hurkyl]

Difference is subtle…

That one's easy to explain -- the complex version of the function isn't being used in one of them.


http://www.wolframalpha.com/input/?i=1.0+++0+I+in+Reals
http://www.wolframalpha.com/input/?i=1.0+++0.0+I+in+Reals

 

[guerom00]

I see now… :)
Well, that doesn't solve my problem: I now have a function of this angle (I use this ArcTan[] to calculate an angle, obviously) which is discontinuous at one point… I'll have to make a specific test to avoid this very point.

Thanks :)