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 Difficulties in chapter 5 
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Joined: 15 Aug 2008, 10:21
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Post Difficulties in chapter 5
Hello everybody

I have some difficulties with the chapter 5 p. 94. I don't understand how we get
z = log r + it
So if there's someone who can explain this to me, I would be grateful.

Mysong


15 Aug 2008, 10:27
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Joined: 07 Jun 2008, 08:21
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Post Re: Difficulties in chapter 5
Hi
It's because on page 94, since Penrose defines w by writing w=e^z, which can be written in polar form as re^{i\theta}, then:
z=logw=log(re^{i\theta})=logr+log(e^{i\theta})=logr+i\theta

I hope this explains it in a way you can follow. If not ask again.


15 Aug 2008, 13:22
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Joined: 25 Feb 2008, 13:32
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Post Re: Difficulties in chapter 5
Mysong: Welcome to the forum. It's best to read vasco's equation from right to left, the \text{log} is the natural logarithm (the base of the logarithm is chosen as e ), and the properties of the arithmetic of logarithms have been invoked. "Obvious", but enlightening.

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15 Aug 2008, 14:15

Joined: 15 Aug 2008, 10:21
Posts: 2
Post Re: Difficulties in chapter 5
Ok
Thank you for your explanation. Now I understand.


15 Aug 2008, 19:43

Joined: 13 Aug 2009, 00:08
Posts: 13
Post Re: Difficulties in chapter 5
Just bought RTR and just discovered this forum, so apologies for delay!

Had to comment on this topic: vasco is correct, but to me the point is that when Penrose gives this expression for z (ie logw) the expression cannot be directly derived based on what he has said so far in the book. In fact, vasco's explanation/derivation is given by Penrose in a different form on the next page (p95 in my copy) after the para starting with "We can best understand...".

The closest Penrose gives to a justification for this expression is that when we multiply complex numbers we multiply the moduli and add the arguments. In fact in note [5.4] he admits that, from what has been given to us at this point in the book, an expression such as

z = log r + 5i\theta

also has the required properties for log{w}but the real factor of 5 I have used here would be wrong -- and he says that to prove that (ie that the imaginary part should not be a real multiple of i\theta you need calculus.

To summarize: the expression near the top of p94 is correct, and I know it is correct based on the mathematics I did at school, but it seems to me that it cannot be directly derived from what Penrose has said by that point in the book. Do I have that right?


13 Aug 2009, 00:24
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Joined: 07 Jun 2008, 08:21
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Post Re: Difficulties in chapter 5
In my opinion you are right. In order to prove what Penrose is saying about z=\log w you need Euler's formula:

e^{i\theta}=\cos\theta+i\sin\theta

which he has not proved or mentioned up to this point. He is just wanting to show all the interesting results and consequences of the complex logarithm without getting the reader bogged down in proofs which require knowledge of topics not yet covered.


15 Aug 2009, 07:26

Joined: 26 Mar 2010, 00:51
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Post Re: Difficulties in chapter 5
Hi I'm reading the book and find this site when meet difficulties :-)

I don't agree with vasco :-(

From my opinion, w=Image is infered from "z = log r + Image", so you can't said "z = log r + Image" can be infered from "w=Image"
similiarly, Euler's formula is also infered from the expression "z = log r + Image"

So I agree with Shaun Culver, but he didn't give answer to why "z = log r + Image"

I don't understand what flashbang said. :-(, Could you give me some more detail explanation about why "z = log r + Image"?

Thanks


26 Mar 2010, 01:18

Joined: 26 Mar 2010, 00:51
Posts: 2
Post Re: Difficulties in chapter 5
Sorry, I get it.

In fact, the Euler's formula can be infered from another method infinite series.

because it is dinner time, I will post the method later :-)


26 Mar 2010, 11:13
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