I think that 0 is the only fixed point of the sin function in the complex unit disc.

In other words sinz=z. I'm not sure how to prove it.

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- Jan 24th 2011, 05:45 AMrobeulerfixed points of sin in the complex unit disk
I think that 0 is the only fixed point of the sin function in the complex unit disc.

In other words sinz=z. I'm not sure how to prove it. - Jan 24th 2011, 06:50 AMroninpro
Maybe you can use Rouche's theorem? It seems possible that

If this is true, then you can conclude that and have the same number of roots, which would be 1.

Update: This turns out to be false, for . - Jan 24th 2011, 08:03 AMrobeuler
- Jan 24th 2011, 09:22 AMroninpro
You could also try to use the argument principle to measure the number of roots in the disk, but the function might be a little nasty to deal with. Putting ,

gives the number of roots in the disk. If you can evaluate this, then you are done.

Maybe somebody has a cleaner way of doing this.