📖 5.0% done with Complex Analysis by Elias M. Stein & Rami Shakarchi

📖 5.0% done with Complex Analysis by Elias M. Stein & Rami Shakarchi

A nice beginning overview of where they’re going and philosophy of the book. Makes the subject sound beautiful and wondrous, though they do use the word ‘miraculous’ which is overstepping a bit in almost any math book whose history is over a century old.

Their opening motivation for why complex instead of just real:

However, everything changes drastically if we make a natural, but misleadingly simple-looking assumption on f: that it is differentiable in the complex sense. This condition is called holomorphicity, and it shapes most of the theory discussed in this book.

We shall start our study with some general characteristic properties of holomorphic functions, which are subsumed by three rather miraculous facts:

  1. Contour integration: If f is holomorphic in \Omega , then for appropriate closed paths in \Omega

    \int\limits_\gamma f(z)\,\mathrm{d}z = 0.

  2. Regularity: If f is holomorphic, then f is indefinitely differentiable.
  3. Analytic continuation: If f and g are holomorphic functions in \Omega which are equal in an arbitrarily small disc in \Omega , then f = g everywhere in \Omega .

This far into both books, I think I’m enjoying the elegance of Stein/Shakarchi better than Ahlfors.

Syndicated copies to:

Leave a Reply

Your email address will not be published. Required fields are marked *

To respond on your own website, enter the URL of your response which should contain a link to this post's permalink URL. Your response will then appear (possibly after moderation) on this page. Want to update or remove your response? Update or delete your post and re-enter your post's URL again. (Learn More)