A differential geometry perspective.
Sooo... I'm not entirely sure if this will help, but I think that the answer to the "constant speed question" is not necesarily. Idealy we could make the circle be traced out at constant speed.
Here is how i think of it.
C:[0,1] --> S1
S1 will be the circle...
By D infinite do you mean an unbounded domain? I'm referring to subsets of the complex plane.
So yes if f is analytic, then f' is analytic, and if f' is bounded on all of C then by louiville's theorem f' is constant so then the rest of the derivatives are zero and hence bounded.
Sorry, I...
Ok my last post was trivial, but it led to this question
Assume f is unbounded and analytic in some domain D, and f' is bounded in D
does there exist a function for which the above holds and f'',f''',... are all unbounded in D?
Yeah another counter example is sin(z)+z
I'm going to post a better thread
let f be analytic in some domain not the entire plane, and let f' be bounded
can f'' or any other derivative for that matter be unbounded?
A general question I came up with and it might be trivial, but I'm not entirely sure what the answer is.
Does there exist a function analytic in the upperhalf plane that is unbounded and all of its derivatives are bounded but not identically zero?
or equivilently
does
d^n/(dz)^n(f)<M for...
It depends on your context.
According to Rudin (an analysis textbook author) there is a notion of a onesided derivative but in most cases the derivatives at the endpoints are underfined
Is it asking find the cirlce that is mapped to the line under the given transformation?
And the answer to 1. is similar to the map of the inverse transformation, but now the angle has changed.
It might be helpful to think of it as a composition of two operations, inversion, and then squaring...
Chiro
Thanks for the little exercise, I see how it works in euclidean space, but I'm worried now that it won't generalize to the poincare disk. My metric is pretty obscure and involves logarithms of ratios of euclidean norms, and it seems like a stretch that an innerproduct induced from it will...
Does it happen that the degree of f is j or greater than j?
What I would do is appeal to the fact that if the integral around a boundary of a simple jordan curve is 0 then the function must be analytic (holomorphic), and then since you have an analytic function times a conjugate function that...
No problem Zero
AlephZero,
If the equation of the plane is not well defined, would that imply that the points must all be colinear? Assuming they were all finite and none were infinite or anything strange like that.
I'm not an expert, but first let me see if I get the gist of your problem
You are given a set of points, and asked to show that the lie in the same plane
So you took the points, and constructed a plane that went through all of them
and Tah Dah they must be coplanar.
This sounds like it works...
Firstly thanks for your response.
I actually haven't ever heard of it as an optimization of an innerproduct before.
Im not sure what I could use for my "x-axis" though. I'm working in 2D in the poincare disk with the usual metric in place. Could I just use the typical complex plane...
I am trying to find the convex hull of a finite set in a hyperbolic space, particularly the Poincare disk, but the Upper Half plane works as well.
I know the following equivalent definitions of the Convex Hull:
1) It is the smallest convex set containing the points
2) If the set is...