Re: Simplification of complex expression

matthar@xxxxxxx wrote:
> 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[].

???
Using that definition of Sqrt[], suppose we take z1 = i and z2 = -1.
Then Sqrt[1-z1/z2]/Sqrt[z2-z1] would be i,
while 1/Sqrt[z2] would instead be -i.

David
.

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