[Maxima-bugs] [maxima:bugs] #4536 atan2(0, 0) undefined error while evaluating a limit

[Raymond T. <rt...@us...>]

I notice that in the first case, we have `atan2((tan(x)-1)/2,-(tan(x)-1)/2)`.    This is basically  `-%pi/4` for small x.

I suppose if we get a `atan2(0,0)` expression, we could look at the limit of `atan(y/x)` as x approaches 0


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**[bugs:#4536] atan2(0,0) undefined error while evaluating a limit**

**Status:** open
**Group:** None
**Created:** Sat Apr 05, 2025 07:17 PM UTC by Barton Willis
**Last Updated:** Wed Apr 09, 2025 12:05 PM UTC
**Owner:** nobody


I think this is correct:
~~~
(%i9)	xxx : integrate(log(cot(x)-1),x);
(%o9)	x*log(1/tan(x)-1)-(2*x*log((tan(x)^2-2*tan(x)+1)/2)+(%pi-2*atan2((tan(x)-1)/2,-((tan(x)-1)/2)))*
log(tan(x)^2+1)+2*%i*li[2](-((%i*((%i+1)*tan(x)+%i-1))/2))+2*%i*li[2](%i*tan(x)+1)-2*%i*li[2](1-%i*tan(x))-2*%i*li[2](-((%i*((%i-1)*tan(x)+%i+1))/2))-4*x*log(tan(x)))/4
~~~
But Maxima is unable to find  the following  limit of `xxx`:
~~~
(%i10)	limit(xxx,x,%pi/4,'minus);
atan2: atan2(0,0) is undefined.
~~~
 
 Locally setting the non-user level option `generate-atan2` allows the calculation to finish (but I haven't attempted to show that the result is correct):
 
~~~
(%i11)	xxx : block([?generate\-atan2 : false], integrate(log(cot(x)-1),x));
(xxx)	x*log(1/tan(x)-1)-(4*x*log((tan(x)^2-2*tan(x)+1)/2)+3*%pi*log(tan(x)^2+1)+4*%i*
li[2](-((%i*((%i+1)*tan(x)+%i-1))/2))+4*%i*li[2](%i*tan(x)+1)-4*%i*li[2](1-%i*tan(x))-4*%i*
li[2](-((%i*((%i-1)*tan(x)+%i+1))/2))-8*x*log(tan(x)))/8

(%i12)	limit(xxx,x,%pi/4,'minus);
(%o12)	-((%i*li[2](%i+1))/2)+(%i*li[2](1-%i))/2-(%pi*log(2)-2*%pi*log(-2))/8-(3*%pi*log(2))/8
~~~



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