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Topic: Handling branch cuts in trig functions
Replies: 9   Last Post: Mar 26, 2013 4:54 PM

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Nasser Abbasi

Posts: 5,705
Registered: 2/7/05
Handling branch cuts in trig functions
Posted: Mar 24, 2013 4:35 AM
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I tried to simplify sqrt( sec(x)^2 ) but Mathematica will
only do this by assuming x is inside one branch, say
x>-Pi/2 && x<Pi/2 but Maple and maxima simplified it
but they gave the answer is terms of |sec(x)| to take
care of the sign which depends on the branch.

Here is plot of sec(x)

http://mathworld.wolfram.com/Secant.html

-----------------------------
In[37]:= Assuming[x>-Pi/2&&x<Pi/2,Simplify[Sqrt[Sec[x]^2]]]
Out[37]= Sec[x]

In[39]:= Assuming[x > Pi/2 && x < Pi, Simplify[Sqrt[Sec[x]^2]]]
Out[39]= -Sec[x]
------------------------------

If I just tell M that x>0, it will not simplify it.

------------------------------
In[38]:= Assuming[x>0,Simplify[Sqrt[Sec[x]^2]]]
Out[38]= Sqrt[Sec[x]^2]
-------------------------------

but Maple did it only with the x>0 assumption:

----------------------
restart;
simplify(sqrt(sec(x)^2)) assuming x::positive;

1
--------
|cos(x)|
restart;
simplify(sqrt(sec(x)^2));
/ 1 \
csgn|------|
\cos(x)/
------------
cos(x)
---------------------------------

On maxima 12.04.0

sqrt(sec(x)^2);
|sec(x)|

I think now that answer to sqrt(sec(x)^2) should be
|sec(x)| without need to give the branch. Since the only
different is the sign. Or is there something else here?

--Nasser



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