Is exercise 33 ii) an error?...maybe not.

Hi

When looking at exercises with several parts as in this case, and having done many of Needham's exercises like this, one finds that there is almost always a theme or connection between the parts. Sometimes one finds (as in this case) that even though the parts are meant to be connected some inconsistencies in notation occur and can cause confusion. By attempting one part of the exercise in isolation we can miss these inconsistences.

Looking at part (i) of this exercise (33 on page 119), we see that we start by looking at the multifunction $\cos^{-1}(z)$. In my opinion, it's almost certain that, given part (i), Needham intended to write the same equation in part (ii).

The starting point of part (ii), $w=\cos z$ is inconsistent with the other parts and so it seems like a reasonable thing to do to consider this an error since the change I suggest makes the exercise as a whole totally consistent.

However if we look at part (i) with its reference to figure 26 on page 88 we find that we also need to swap all occurrences of $w$ and $z$ in figure 26.
These changes seem to me to be the best way to make the exercise and its associated diagram 26 consistent.

I do not claim that it is the only rewrite possible of this exercise, but its the one I prefer, and the exercise certainly cannot be acceptable the way it is printed in the current versions of the book.

Vasco

PS Since Needham uses $\text{Log}$ with a capital "L" for the principal branch of the logarithm I think that here we should use $\log$, the multifunction logarithm.

Yes I agree that this is also a way of re-writing the exercise and it could be argued that this way is better than my initial proposal since it leaves the diagram unchanged.
So I think each person can do their own re-write. The main thing is to be aware that as it appears in the book it is inconsistent and requires a re-write.

Vasco

Interesting discussion.  I must admit that by the time I had got to the end of my post I had changed my mind...the jarring I felt when I did the question should have been a big enough clue that this was not as Needham intended.  Now on Chapter 6, I am impressed how every part of this book weaves together as a whole and agree with Vasco that the elegance evident everywhere was absent from the phrasing of this part.

Yes my Log should have been a log...carelessness.

As an aside, my justification for saying that there was no need to show the negative square root was that it was redundant.  The negative root, only differing  from the positive by and argument of pi, would be automatically covered on another branch of the log function (which repeats every 2*pi, which of course Log does not!).  Have I got the right approach?


telemeter

$\hspace{5em} log [w + \sqrt{w^2-1}] = ln|w + \sqrt{w^2-1}| + i \, arg (w + \sqrt{w^2-1})$.

The negative root would have to be considered in every branch of log, so the total number of values is potentially $2 \cdot \infty$. That is, a branch of log for every integer times two values for the square root, as in the second equation of Section VII The Logarithm Function (p. 98). This is essentially 2 values for every time that $w$ makes a complete traversal of the ellipse in Fig [26], p. 88. Now having thought about it a bit more, I can see how one can think of the two values of the square root as creating two branches of log, but then one must still describe the infinity of branches.

I'm not presenting this on a stone tablet. This is just the way I see it.

Gary

Gary

In your document you say that the square root function has two branches. I think it has one branch point at the origin of order 1, and an infinite number of branches.
The $\log$ function has one branch point at the origin of infinite order (usually called a logarithmic branch point) and an infinite number of branches.

Vasco