Wrong atan2 of complex arguments

[rwst]

sage: atan2(CC(I), CC(I+1))
0.615479708670387
sage: atan2(I, I+1)
arctan2(I, I + 1)
sage: atan2(CBF(I), CBF(I+1))
...
/home/ralf/sage/src/sage/libs/pynac/pynac.pyx in sage.libs.pynac.pynac.py_atan2 (build/cythonized/sage/libs/pynac/pynac.cpp:21132)()
   1798         P = RR
   1799     if y != 0:
-> 1800         if x > 0:
   1801             res = py_atan(abs(y/x))
   1802         elif x < 0:
...
TypeError: No order is defined for ComplexBalls.

But both SymPy and Wolfram Alpha use the form atan2(y,x) = -I*log((x + I*y)/sqrt(x^2 + y^2)) for complex x,y:

In [18]: atan2(I,I+1)
Out[18]: 
      ⎛        ⅈ         ⎞
-ⅈ⋅log⎜──────────────────⎟
      ⎜   _______________⎟
      ⎜  ╱             2 ⎟
      ⎝╲╱  -1 + (1 + ⅈ)  ⎠

In [19]: _.n()
Out[19]: 0.553574358897045 + 0.402359478108525⋅ⅈ

which is arctan(1/2*I + 1/2).
(Wolfram reverses the argument and so one must input arctan(I+1,I) there)

Component: calculus

Author: Ralf Stephan

Branch/Commit: f5c35ff

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/22877

[rwst]

Branch: u/rws/atan2_of_complex_arguments_inconsistencies

[rwst]

New commits:

88e55d1

22877: Wrong atan2 of complex arguments

[rwst]

Commit: 88e55d1

[rwst]

Author: Ralf Stephan

[tscrim]

comment:6

LGTM.

[tscrim]

Reviewer: Travis Scrimshaw

[rwst]

comment:7

Thanks.

[vbraun]

comment:8

[sagelib-8.0.beta5]     if P in (float, RR, RBF):
[sagelib-8.0.beta5]                           ^
[sagelib-8.0.beta5] ------------------------------------------------------------
[sagelib-8.0.beta5] 
[sagelib-8.0.beta5] sage/libs/pynac/pynac.pyx:1807:27: undeclared name not builtin: RBF

[sagetrac-git]

Changed commit from 88e55d1 to f5c35ff

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

401237e	Merge branch 'develop' into t/22877/atan2_of_complex_arguments_inconsistencies
f5c35ff	22877: fix reality check

[vbraun]

Changed branch from u/rws/atan2_of_complex_arguments_inconsistencies to f5c35ff