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# Holomorphic Branches

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1. Let C denote spiral given by r= 2e^θ (in polars) . There is a unique holomorphic branch of log on ℂ\C s.t log(1)= 0.

For this branch determine:

log(i), log(3), log(-1), log(1000), log(-1000), log(2000)

I'm struggling to understand the intuition here. I know about how holomorphic branches of L(z) can can be found by restricting to say ℂ\[-inf, 0] etc, and we do this so the value of L(z) can be single valued. However, I'm unsure how to define L(z) in the case of a spiral. I have an idea but its just a guess really:

Say we define L(z) = log(|z|) + i*arg(z) + 2n*pi, where n is the number of times the spiral has wrapped around the origin by the time we get to the complex number we need (say for 1000, the spiral has wrapped around 0 times, but for 2000 its wrapped around once) (I'm sort of thinking of this in a 3d plane and 'moving up a level')

However, I've never seen an example with spirals (or infact any other curve that isn't a line in the complex plane) so I'm not sure if this is anywhere near. Any help appreciated
2. (Original post by Gome44)
Let C denote spiral given by r= 2e^θ (in polars) . There is a unique holomorphic branch of log on ℂ\C s.t log(1)= 0.

For this branch determine:

log(i), log(3), log(-1), log(1000), log(-1000), log(2000)

I'm struggling to understand the intuition here. I know about how holomorphic branches of L(z) can can be found by restricting to say ℂ\[-inf, 0] etc, and we do this so the value of L(z) can be single valued. However, I'm unsure how to define L(z) in the case of a spiral. I have an idea but its just a guess really:

Say we define L(z) = log(|z|) + i*arg(z) + 2n*pi, where n is the number of times the spiral has wrapped around the origin by the time we get to the complex number we need (say for 1000, the spiral has wrapped around 0 times, but for 2000 its wrapped around once) (I'm sort of thinking of this in a 3d plane)

However, I've never seen an example with spirals (or infact any other curve that isn't a line in the complex plane) so I'm not sure if this is anywhere near. Any help appreciated
Still at A level and have only watched videos regarding this topic and wiki...

However,

Draw the spiral by sampling at least every , between two arms of the spiral there is a curved strip of the complex plane. Following by continuity (starting from you get a branch of on

Hope this helps
3. (Original post by Naruke)
Still at A level and have only watched videos regarding this topic and wiki...

However,

Draw the spiral by sampling at least every , between two arms of the spiral there is a curved strip of the complex plane. Following by continuity (starting from you get a branch of on

Hope this helps
Sorry I don't really get what you mean by the bolded bits, is my idea correct?
4. (Original post by Gome44)
Say we define L(z) = log(|z|) + i*arg(z) + 2n*pi, where n is the number of times the spiral has wrapped around the origin by the time we get to the complex number we need (say for 1000, the spiral has wrapped around 0 times, but for 2000 its wrapped around once) (I'm sort of thinking of this in a 3d plane and 'moving up a level')
You have the right idea.

log(1)=0. And this branch will agree with the standard lnx in the range

exp(-2pi) < x < exp(2pi) of the real axis

To the right of exp(2pi) and as far as exp(4pi) the function is lnx + 2 pi i as we are one level up.

Across the curve C there will always be a 2 pi i discontinuity.
5. (Original post by RichE)
You have the right idea.

log(1)=0. And this branch will agree with the standard lnx in the range

exp(-2pi) < x < exp(2pi) of the real axis

To the right of exp(2pi) and as far as exp(4pi) the function is lnx + 2 pi i as we are one level up.

Across the curve C there will always be a 2 pi i discontinuity.
Thanks

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Nice question btw

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