Mathematics (or something like that)

This blog on PF is a copy of the posts on my blogspot.com blog. There are pages and content not compatible with PF blogs, so please check out the original! The URL is http://letepsilonbegreaterthanzero.blogspot.com

I will very quickly summarize the main points of the blog here

For the summer of 2012 I am reading through two books, Fourier Analysis by Stein and Shakarchi, and The Road to Reality by Penrose. I am blogging about thoughts I have from these books. I think it would be pretty cool to have feedback here =)

In fall 2012, I will be starting a PhD program with plans of continuing the blog over my course work. If I pass quals in two years (fingers crossed) this blog may change focus.

Anyways, every time I have a new post I will post a link here. I won't have any of the pages or other info on this blog, but at least the bulk of the post will be the same.
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# road to reality: chapters 1-5

Posted Jun3-12 at 10:37 PM by theorem4.5.9

View full entry at my blogspot.com blog: road to reality: chapters 1-5

Upon my reading of the preface, I am apprehensive to Penrose's claims. I do not have faith that this book is suitable to a lay audience, even one with prior experience to calculus. I do not believe that his explanation of the rational number system sheds much light to readers without logical backgrounds, and I can only extrapolate that his prose will be similar to this. Maybe this book isn't for the lay reader, but that's a plus to me. It's better suited to me!

I did not enjoy the first three chapters. Here the book mainly focuses on philosophy. I hold my own opinions on the subject, but I am glad see it included for those readers who haven't thought about these topics.

Chapter 3 and 4 were interesting and even enlightening at some points. I now see how a slide rule works is actually very simple! Though the most important thing I learned (which I'm a little embarrassed that I didn't learn this earlier) is that the exponential operation, say $z^\lambda$ is multivalued for arbitrary complex entries. I spent quite some time playing around with this and seeing where all the solutions lived in the complex plane (these spirals are quite pretty).

Here is something that I have been thinking about as well: Let clog denote what I will call the complex logarithm, and log the standard logarithm. The difference is how I interpret the input, as a real number or a complex number without an imaginary part. Recall then that
$$\mathrm{clog}(z) =\mathrm{log}(|z|) + i\theta + i2\pi k$$
and thus we immediately see that $\mathrm{clog}(1)=i2\pi k$. This isn't difficult, but it is unintuitive*. A second example of interest is $$i^i = e^{i \mathrm{clog}(i)}=e^{i\cdot (i\pi /2 + i2\pi k)} = e^{-\pi/2}e^{-2\pi k}$$
If I ignore how I constructed these solutions (i.e. forget which $k$ corresponds to the principal logarithm), I do not see which solution should be the "intuitive" solution, as they are all real! Penrose promises that the multivaluedness of these complex functions will become apparent later. Apart from winding, I'm not sure how.

* Apparently unintuitive is not a word. Even Merriam-Webster was not able to help me find the correct wording!
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