2011-06-09, 09:50 AM
I haven't really looked at your work yet (since I've just taken two finals and have my physics final tomorrow), but I swear I will get around to looking at it soon. My questions for now are:
1. Does the branch cut always have to be of length 2π?
2. What if the function wraps around itself more than once within this 2π interval? I don't have an example, but I'm sure there exists complex functions that wrap around themselves more than once within [0,2π
.
3. Not really related to anything, but I'm looking over the complex argument function, which is really just arctan(v/u) (or is it u/v?). Does it have a winding number of 2π, because it satisfies the Cauchy-Riemann equations, or is it just because of the singularity? Does satisfying the Cauchy-Riemann equations denote that a function has a singularity?
4. Can I imagine the value of the singularity as some sort of height between two branches? (like the distance between two leaves of a helicoid)
5. For what function would I get a value that's not 2πi, and all the constants are 1? (for example, not 2/z)
1. Does the branch cut always have to be of length 2π?
2. What if the function wraps around itself more than once within this 2π interval? I don't have an example, but I'm sure there exists complex functions that wrap around themselves more than once within [0,2π
.3. Not really related to anything, but I'm looking over the complex argument function, which is really just arctan(v/u) (or is it u/v?). Does it have a winding number of 2π, because it satisfies the Cauchy-Riemann equations, or is it just because of the singularity? Does satisfying the Cauchy-Riemann equations denote that a function has a singularity?
4. Can I imagine the value of the singularity as some sort of height between two branches? (like the distance between two leaves of a helicoid)
5. For what function would I get a value that's not 2πi, and all the constants are 1? (for example, not 2/z)