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I have expressions:

$F(s)=\sqrt{\left[\left(a+b\right)s+c\right]\mp\sqrt{\left[\left(a-b\right)s+c\right]^{2}+d}}$

where $s$ is a complex number; and $a,b,c,d$ are reals such that

$a>0,\,b>0,\, a\neq b,\, c>0,\, d\geq0$. I am trying to figure out whether

it is possible to simplify these expressions when $d=0$,

but I am afraid I have forgotten how to deal with the

constructs of the type $\sqrt{z^{2}}$ for the complex $z$ arising here.

Can anybody help me please?

Leszek

Clarification 1:

I may be terribly uneducated, but what else can $\sqrt{z}$ mean, if not "the square root of $z$"?

I never assume that the expressions of which the square roots are taken are integers. Please read my description above.

I think my problem can be expressed in a simpler way: what is the result of calculating $\sqrt{z^{2}}$ when $z$ is complex, and how this result depends on the actual form of $z$, as is described?

When $z$ is real, then $\sqrt{z^{2}}$ is equal to the absolute value of $z$. Obviously I expect this is not the case when $z$ is complex.

Assuming a special case of real $s$ and $a > b$, we would obtain simplified expressions:

$F(s)=\sqrt{2bs}$ when there is "-" in the formula for $F(s)$,

and $F(s)=\sqrt{2as+2c}$ when there is "+" in the formula for $F(s)$

I need to derive the counterparts of such simplified expressions obtainable for complex $s$, both when $a > b$ and when $a < b$.

Leszek

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    $\begingroup$ What does $\sqrt z$ mean? (And, please, don't answer “the square root of $z$”.) $\endgroup$ Commented Oct 9, 2022 at 17:11
  • $\begingroup$ Do you need a general algebraic form, or is what you are stating simply a generalization of a specific numerical expression? Note that for integers $m$ and $n,$ it is not possible in general to express $\sqrt{m + \sqrt n}$ using integers and a single radical symbol, but for certain special cases it is possible. It might also be helpful to know what you need the simplification for, since it might actually be unnecessary for what you want to do. $\endgroup$ Commented Oct 9, 2022 at 19:01
  • $\begingroup$ I have added Clarification 1 $\endgroup$
    – Leszek
    Commented Oct 10, 2022 at 19:53
  • $\begingroup$ The complex square root isn't well-defined so whatever it is you mean by $\sqrt{z}$ need to be explained. It's not normally a meaningful expression. $\endgroup$ Commented Oct 10, 2022 at 20:50

1 Answer 1

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The function square root $\sqrt{z}$ is defined on the complex plane minus a curve connecting the points $z = 0$ and $z =\infty$. The curve is usually taken as the positive or negative half-axis of the real line. Therefore, the result $\sqrt{z^2}$ depends on the chosen branch. Moreovere, if you take a point on the curve, be careful which side of the curve cointains this point.

A simplification can be implemented, for instance, in Mathematica by a description of the fixed branch. Maybe the following example in Mathematica will clarify the role of the branch definition:

In[1]:= f[z_] = Sqrt[z^2 - 1] - Sqrt[z + 1] Sqrt[z - 1];
        f[I - 1.] // Simplify
Out[2]= 1.5723 - 2.54404 I
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