Re: Simplification of complex expression

In article <1124894530.885572.182520@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
akhmel@xxxxxxxxxxx writes:
>Suppose we have an expression,
>
>Sqrt[1-z1/z2]/Sqrt[z2-z1], where z1 and z2 are complex variables. Does
>there exist a formula for simplification of this expression? The result
>should be 1/Sqrt[z2] with plus or minus. So does there exist a formula
>to define this sign?
>..

To make sense of the question, one has to assume
that one of the two branches is favored above the
other, which I'll do for now defining that Sqrt[] is
the principal value such that the |arg(Sqrt[..])|<=pi/2.
To simplify matters write
Sqrt[1-z1/z2]*Sqrt[z2]=Sqrt[z2-z1]
by simple multiplication, up to the negative sign in question.
If the sum of the two arguments of the factors on the
left hand side is larger than pi/2 or less than -pi/2,
this would "push" the product into the wrong branch
and need an extra minus sign. This would mean
|arg(Sqrt[1-z1/z2])+arg(Sqrt[z2])|>=pi/2
The argument of the PV is half of the original one of
within the Sqrt[],
|arg(1-z1/z2)/2+arg(z2)/2|>=pi/2
|arg(1-z1/z2)+arg(z2)|>=pi
now the sum of the arguments is the argument of the product,
opposite way of using the first step,
|arg((1-z1/z2)*z2)|>=pi
|arg(z2-z1)|>=pi
so the additional sign is never there for this special
combination with this definition of the Sqrt[].
.

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