Comments on: Branch points and cuts http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts/ somewhere near the beginning. Mon, 01 Dec 2008 21:29:05 +0000 http://wordpress.org/?v=2.5 By: Gordon Worley http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts/#comment-230115 Gordon Worley Fri, 23 Nov 2007 15:02:49 +0000 http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts#comment-230115 To see why there's a branch point at infinity, I think it's most helpful to think about the projective plane, since you can "see" the complex infinity better there. Since there is only one infinity on the complex plane, you can think of it as 0 being at the south pole and infinity at the north pole. So we then can easily see that a branch point at 0 often will imply a branch point at infinity since the reason for the break is often that you get multiple values as you rotate around 0 (just expand the circle until it's around infinity instead of around 0). To see why there’s a branch point at infinity, I think it’s most helpful to think about the projective plane, since you can “see” the complex infinity better there. Since there is only one infinity on the complex plane, you can think of it as 0 being at the south pole and infinity at the north pole. So we then can easily see that a branch point at 0 often will imply a branch point at infinity since the reason for the break is often that you get multiple values as you rotate around 0 (just expand the circle until it’s around infinity instead of around 0).

]]> By: John Bebawy http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts/#comment-214521 John Bebawy Sun, 14 Oct 2007 02:10:19 +0000 http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts#comment-214521 I like the second explaination. and would like to elaborate more with a way to visualize it. if you walk along aa curve through a point, you have a certain gradient (slope) G1, and if you walk in the opposite direction you also have another G2. if G1 and G2 are crossing each other, then you have an abrupt change, then you have a discontinuity, then you have branch point. Also if you can determine one G and can't determine the other because it's at infininty, then it's also a branch point. PS: I advise you to verify this idea, but it sounds like a good generalization. I like the second explaination. and would like to elaborate more with a way to visualize it. if you walk along aa curve through a point, you have a certain gradient (slope) G1, and if you walk in the opposite direction you also have another G2. if G1 and G2 are crossing each other, then you have an abrupt change, then you have a discontinuity, then you have branch point. Also if you can determine one G and can’t determine the other because it’s at infininty, then it’s also a branch point. PS: I advise you to verify this idea, but it sounds like a good generalization.

]]> By: Adam Hughes http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts/#comment-72190 Adam Hughes Sun, 19 Nov 2006 22:04:10 +0000 http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts#comment-72190 A simple way that I keep it in mind (may not be what you are looking for), is that a branch point is "cut" in the graph. If we do not make the cut, the graph will repeat itself forever. For example, if I had a circle in a plane, represented by CosX. If I limit X to 0-2pi, that corresponds to one rotation. If I don't limit x, the circle keeps going around and around and around forever. 0-2pi is then my branch point, and outside of this domain, the function repeats and is not analytic. A simple way that I keep it in mind (may not be what you are looking for), is that a branch point is “cut” in the graph. If we do not make the cut, the graph will repeat itself forever. For example, if I had a circle in a plane, represented by CosX. If I limit X to 0-2pi, that corresponds to one rotation. If I don’t limit x, the circle keeps going around and around and around forever. 0-2pi is then my branch point, and outside of this domain, the function repeats and is not analytic.

]]> By: ObsessiveMathsFreak http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts/#comment-64262 ObsessiveMathsFreak Tue, 10 Oct 2006 10:30:49 +0000 http://www.tangentspace.net/cz/archives/2006/09/branch-points-and-cuts#comment-64262 I'm not overly clear on this either, but my current thinking is that the branch point is the end point of the branch cut, and so if you've wound around it, you've gone from one copy of the C plane to another. Once you're on the second copy of C, the answers you get will be the second set of answers to the problem, so when you return to you "original" point, you're really on another sheet, so the answer will be different. But there are multiple choice for a branch cut! But, I think they should all have the same branch point. Hence winding around the branch point in the domain plane will take you to a different point in the range plane. The above probably made no sense, and the following links will likely be of little help, but I'll post them anyway. <a href="http://www.wlym.com/antidummies/part60.html" rel="nofollow">These</a> <a href="http://www.wlym.com/antidummies/part61.html" rel="nofollow">pages</a> can at least be said to have pretty pictures, though exactly what this guys is doing escapes me. He's big on rhetoric, short on analysis, but the animations are good. I’m not overly clear on this either, but my current thinking is that the branch point is the end point of the branch cut, and so if you’ve wound around it, you’ve gone from one copy of the C plane to another. Once you’re on the second copy of C, the answers you get will be the second set of answers to the problem, so when you return to you “original” point, you’re really on another sheet, so the answer will be different. But there are multiple choice for a branch cut! But, I think they should all have the same branch point. Hence winding around the branch point in the domain plane will take you to a different point in the range plane.

The above probably made no sense, and the following links will likely be of little help, but I’ll post them anyway. These pages can at least be said to have pretty pictures, though exactly what this guys is doing escapes me. He’s big on rhetoric, short on analysis, but the animations are good.

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