Complex Analysis - Branch Definition

[Pyroadept]

Homework Statement

Hi everyone,

This is more of a definition clarification than a question. I'm just wondering if a branch is the same thing as a branch line/branch cut? I've come across a question set that is asking me to find branches, but I can only find stuff on branch lines/cuts and branch points in the textbooks.

Thanks!

Homework Equations

The Attempt at a Solution

 

[jackmell]

Homework Statement

Hi everyone,

This is more of a definition clarification than a question. I'm just wondering if a branch is the same thing as a branch line/branch cut? I've come across a question set that is asking me to find branches, but I can only find stuff on branch lines/cuts and branch points in the textbooks.

Thanks!

No. Suppose I take a long sheet of paper, grab it at the top and bottom and twist it around several times in the shape of a helix. It's got multiple levels now does't it. Now place that helix on top of the complex z-plane centered at the origin. Pick a point say z=1+i that is underneath the helix. Now above that point, there are multiple surfaces of that helix corresponding to the various "sheets" above it. Could you now identify a "section" of that helix so that it does not overlap? Sure, just cut out a slightly less than 2pi section of it, throw the rest away. That section now is a single-sheet above the complex plane.

Well multi-functions are also twisted sheets like that helix with multiple surfaces over each point in the complex plane but all the fundamental principles of Complex Analysis rely on functions being single-valued (single-sheeted). So in order to apply those theorems to multi-valued functions like $\sqrt{z}$, we likewise "cut out" a single-valued section of it, call that section a "branch" and where we cut it, we call the cuts "branch cuts".

Also, keep in mind that multi-valued functions are not all like the simple twisted helix I described above. They have many, many different forms but in general, they "twist" around in similar albeit contorted shapes like the helix and often in Complex Analysis we are concerned with picking out or "cutting out" a single-valued section of the function and calling that section a (single-valued) "branch" of the function.

 

[Pyroadept]

Thanks, that helps a lot!