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I'm learning complex variables. What is the Arg of $\sqrt{(t-1)(t-2)}$ at the point $t=0$? Suppose we have define the cut $[1,2]$. Can we define at the upper side of the cut Arg of $t-1$ to be $0$ and $t-2$ to be $3\pi$. Or the initial Arg if of these two must have some correlations and cannot be arbitrarily defined?

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What relation does $t$ have with $z$? What do you mean by $[1,2]$ as a cut for the argument function - shouldn't the cut extend infinitely outward from the origin? – anon May 27 '12 at 2:06
So you've come back to edit, but you haven't engaged with the comment or the answer. Is $t$ supposed to be $z$, or what? You won't get the kind of answer you want if you don't meet us halfway! – Gerry Myerson May 27 '12 at 13:15
@GerryMyerson I'm sorry I mistyped the question on mobile phone... – Strin May 28 '12 at 6:37
Do you really mean to ask whether the argument of $t-2$ can be $3\pi$? or do you mean to ask about the argument of $\sqrt{t-2}$? I don't see how it's possible for the argument of $t-2$ to be $3\pi$. – Gerry Myerson May 28 '12 at 7:08
@GerryMyerson so based on what assumptions of the branch cut can we uniquely define the value of the function at every point? I know how to define for the case $\sqrt{t-2}$ but I'm confused in that case. – Strin May 28 '12 at 11:00
up vote 4 down vote accepted

Assuming $t$ is supposed to be $z$, at $z=0$ we find that $(z-1)(z-2)$ is $2$, and the argument of $2$ is zero. If I've misunderstood the question, please clarify.

EDIT based on revised version of the question, and comments by OP:

It's probably best if you look at the text Raymond mentions. For what it's worth, you can take the branch cut to be the interval $[1,2]$, and then at $t=0$ you can, if you like, choose for your function to have the value $\sqrt2$ (and thus argument zero). Making a choice of the function value at $t=0$ then uniquely specifies the function value everywhere (except on the branch cut, where it's not defined).

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Short answer : the arguments must belong to the same branch and the branch (or 'sheet') depends of the cut chosen.

This new question (with the additional square root) is more interesting and subtle than it seems.

Let's illustrate it with a plot of the argument of $\sqrt{(z-1)(z-2)}$ :

out branch

From the complex branch point of view you may :

  • consider the cut $(-\infty,1)$ and $(2,\infty)$ : in this case, because of the continuity requirement at $(1,2)$, all the arguments must belong to only one of the sheets shown above (upper, lower or one of the other hidden parallel surfaces...). Since $z=0$ is on a branch cut this choice is clearly ambiguous (the upper value is often preferred and sometimes none of them...)

  • or the cut will be $(1,2)$ (with continuity at both sides) and your arguments will all belong to the surface below (or one of the parallel surfaces). The picture is a little misleading and the part at the right should be shifted up or down by an offset $\pi$ to assume continuity at the front or the back when $\Re(z) \in (1,2)$ (with an 'acceptable' jump of $\pm 2\pi$ at the other side).

middle cut

Both points of view are handled with care in Ablowitz and Fokas 'Complex Variables' (2.3).

Hoping this clarified more,

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