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## Main Question or Discussion Point

I have lately been trying to understand branch points and branches used for complex, multivalued, functions.

From doing questions and reading online, however, I get an unclear view of what branches points are.

Take the function: [tex] f(z)_a = (z^2+1)^\frac{1}{2} [/tex] or [tex] f(z)_b = log(z-1) [/tex]

Based on my (limited) understanding of branch points, it seems that the points where the function is 0, f(1)_a and f(-1)_a, are branch points, or where it becomes undefined f(1)_b = log(0).

I do not understand why the function is 0, and sometimes undefined, at a branch point. I cannot find any text online that explains, clearly, exactly when you get a branch point.

Related to this of course is the notion of branches, which seem to be parts of the function, separated by a branch cut (a line between two branch points?).

Take for example, a more complicated example, [tex] f(z) = (z^2-i-1)^\frac{1}{4} [/tex]

What method would I then use to find the branch points and the branches of this function?

I assume: [tex] z^2-i-1 = 0 [/tex] Gives the two roots of z (two branch points), such that, generally, the nth root of z equals the branch points?

What about finding the branches, I assume that's related to the 1/4 exponential,but how do you explicitly go about doing it?

I have been trying to understand this for a few days, so any explanations would be really appreciated!

From doing questions and reading online, however, I get an unclear view of what branches points are.

Take the function: [tex] f(z)_a = (z^2+1)^\frac{1}{2} [/tex] or [tex] f(z)_b = log(z-1) [/tex]

Based on my (limited) understanding of branch points, it seems that the points where the function is 0, f(1)_a and f(-1)_a, are branch points, or where it becomes undefined f(1)_b = log(0).

I do not understand why the function is 0, and sometimes undefined, at a branch point. I cannot find any text online that explains, clearly, exactly when you get a branch point.

Related to this of course is the notion of branches, which seem to be parts of the function, separated by a branch cut (a line between two branch points?).

Take for example, a more complicated example, [tex] f(z) = (z^2-i-1)^\frac{1}{4} [/tex]

What method would I then use to find the branch points and the branches of this function?

I assume: [tex] z^2-i-1 = 0 [/tex] Gives the two roots of z (two branch points), such that, generally, the nth root of z equals the branch points?

What about finding the branches, I assume that's related to the 1/4 exponential,but how do you explicitly go about doing it?

I have been trying to understand this for a few days, so any explanations would be really appreciated!