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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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Richard Fateman

Posts: 1,402
Registered: 12/7/04
Re: Handling branch cuts in trig functions
Posted: Mar 26, 2013 1:17 AM
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this is still nonsense.

-3 is a square root of 9, whether the 9 was produced by squaring 3 or
squaring -3.

-x is a square root of x^2 whether the x^2 was produced by squaring x or -x.

It doesn't matter whether x is positive or negative.

On 3/25/2013 5:52 AM, G. A. Edgar wrote:
> In article <kimoma$hru$1@speranza.aioe.org>, Nasser M. Abbasi
> <nma@12000.org> wrote:
>

>>
>> But I am using Maple 17?
>>
>> -----------------------------------
>> ans:=simplify(sqrt(sec(x)^2)) assuming x::positive;
>>
>> 1
>> --------
>> |cos(x)|


this is wrong; see below
>>
>> simplify(abs(sec(x))- ans);
>> 0
>>

well, this should be zero.
>> -------------------------------------
>>
>> Unless x::positive implies x::real (since positive does
>> not apply to complex numbers). Is this what you meant?

>
> Yes, positive implies real. You will also get that result assuming x
> is negative, or assuming x is an integer, and so on. Not only on the
> reals, but also on any subset of the reals we have sqrt(x^2) = abs(x) .


If you visualize f(z)=sqrt(z^2) in the complex plane, you can specialize
it for real z and see if it corresponds to abs(z).
>
>>
>> So Maxima was wrong then:
>>
>> sqrt(sec(x)^2);
>> |sec(x)|
>>
>> No assumptions!


Yes, this is wrong. The issue, at its core, is that computer algebra
systems are not programmed to deal with multiple-valued object
in a satisfactory way.

>
>
> We cannot tell whether Maxima is wrong unless we know whether Maxima
> assumes x is real (when you do not tell it). Maple assumes x is
> complex, as was said. Perhaps the documentation for Maxima tells you
> about this?
>
> sec(1+i) is about .4983370306+.5910838417*i,
> and the square-root of the square of that is itself, not its absolute
> value. (Assuming principal branch.)
>





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