Date: Mar 26, 2013 1:17 AM
Author: Richard Fateman
Subject: Re: Handling branch cuts in trig functions
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.)

>