I am having some very trivial issues getting proper branch cuts of
elementary functions in the complex plane.
In [1]: from sympy import *
Exmpl. 1-------------------sqrt()---------------------------
1st numerically
In [2]:
N(sqrt(-1.+I*1e-10))
Out[2]: 5.0e-11 + 1.0*I
In [3]:
N(sqrt(-1.-I*1e-10))
Out[3]: 5.0e-11 - 1.0*I
Ok. sqrt-branch seems to lie on neg. Re-axis and is selected properly
2nd same thing symbolically
In [5]:
eps=symbols('eps')
In [6]:
limit(sqrt(-1+I*eps),eps,0,'+')
Out[6]: I
In [7]:
limit(sqrt(-1+I*eps),eps,0,'-')
Out[7]: I
In [8]:
limit(sqrt(-1-I*eps),eps,0,'+')
Out[8]: I
In [9]:
limit(sqrt(-1-I*eps),eps,0,'-')
Out[9]: I
Oops? Symbolic sqrt() does not seem to handle the branch cut properly(?)
Exmpl. 2----------------------------------------------
1st numerically
In [10]:
log(-1-I*1e-10)
Out[10]: 5.0e-21 - 3.14159265348979*I
In [11]:
log(-1+I*1e-10)
Out[11]: 5.0e-21 + 3.14159265348979*I
Ok. log-branch seems to lie on neg. Re-axis and is selected properly
2nd same thing symbolically
In [12]: limit(log(-1+I*eps),eps,0,'+')
Out[12]: I*pi
In [13]: limit(log(-1+I*eps),eps,0,'-')
Out[13]: I*pi
In [14]: limit(log(-1-I*eps),eps,0,'+')
Out[14]: I*pi
In [15]: limit(log(-1-I*eps),eps,0,'-')
Out[15]: I*pi
Oops? Symbolic log() does not seem to handle the branch cut properly(?)
I could probably 'find' more of that behavior with other elementary functions.
So, what am I doing wrong?
Xaver
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