Complex Analysis for Number Theory

I got hold of a book from Anatoly A. Karatsuba with the title Complex Analysis for Number Theory.

Chapter 1. The Complex Integration Method and Its Application in Number Theory.
Chapter 2. Riemann Zeta
Chapter 3. Dirichlet L-functions

Paragraph 1. Generating Functions in Number Theory. Section 1. Dirichlet Series. Can it be more to the point? This book is clearly written with the intention of efficiently transferring the required tools of complex analysis to number theory students. Chapter 2 introduces the Riemann Zeta function.

Why is this book priced at $179.95 at Amazon? Because it is hard-cover? It is a book from the 19-nineties. This is educational material. Transfer it to electronic format and price it reasonably.

Video lectures on multivariable calculus ( and more ) ...

I read a commentary on the future of universities.

Education and knowledge have become more accessible in the internet age. Students can do a lot of their work online or at home. But does that mean that universities will become obsolete? Of course not. - Books have to be written, exams have to be prepared and marked, research has to be done and published and students will always want some form of live interaction with a professor or tutor. Even if all lectures would be given in the form of video lectures then these lectures have to produced and as is the case with textbooks, redone every few years. Things have changed and will continue to change. Let's hope for the better.

Some video lectures, I recommended:
- [ NEW ] Multivariable calculus ( Berkeley )
- Video lectures number theory
- [News] - Video Lectures Algebraic Topology ( for Undergraduates )
- video lectures complex analysis

No set of video lectures is better than a well written text-book with lots of examples and exercises. Like this magnificent book on Complex Analysis for example ( I'll start M337 october next year ):

Touching complex analysis

Prof. Mattuck is in top form in lecture 6 when he talks about Euler and the beauty of complex numbers. Although it is a lecture in the DE series it can be watched as a stand-alone lecture. So if you are doing MST121, MS221 or M208 it is great fun to watch this lecture. He even touches the field of Complex Analysis when he explains differentiating $e^{i\theta}$. He notes ( jokes ) that time is always a real variable but he isn't so sure when the next Einstein comes around: he may very well decide we need complex time! Anyway, how would you integrate $\int{e^{x}}\cos{x}\ dx$ ? Prof. Mattuck says these integrals are easy if you switch to the complex domain. Finally he solves the beautiful equation $x^n+1=0$ ( as we have seen in MS221, M208 ).



Hopefully, after or during MST209, I will be able to analyze the synchronization of metronomes problem using differential equations some day.

Complex Analysis

What is shear impossible with linear algebra... transforming a rectangle to a circle is easy with the complex mapping z: -> Exp[z].

AMATH 401/501 Video Lectures

Because of my special interest in number theory I read ahead on the topic of complex analysis ( M337 ) which is a prerequisite for analytical number theory. I like to watch video lectures because it gives me an idea about how the topic is presented to students. Until a few days ago I wasn't able to find any lectures on the topic. Although the lectures aren't currently online at the site of the University of Washington they must have been in the past or they are available to enrolled students only. Anyway, some student has been so very kind to upload 38 lectures given in 2009 of +/- 50 min each in a total of 6GB compressed video data to the filesharing site Rapidshare. I do not know if the student violated copyright laws or anything. I don't think so. I think the files are offline because having the files online involves quite a lot of bandwidth for the university which is expensive. Mathematics video lectures have a global appeal and seem to be very popular.

Complex mapping

'Reading ahead' on Complex Analysis it suddenly daunted how I can easily draw graphs of complex mappings using Mathematica. - Where would I be without Mathematica?
Define f: C->C by f[z]=Tan[2z].
We investigate the image of the unit-circle in the complex plane under f.

As you can see this is a simple procedure with the Mathematica function ParametricPlot.

Moebius transformations video

( This video has been watched over a million times on YouTube. )
( Update: 10/4-'10 Added Complex Analysis tag. )