Books for Analysis & Physics: Advice Needed

[Enjolras1789]

Hi,
I am a first year physics grad student. I was curious to get some advice on some books. I only ever took real analysis undergraduate, and even at that sometimes I worry about how good my understanding is. I was curious to know what people think are good backs to learn real analysis and more importantly complex analysis (of which I know nothing). Also, I love Mary Boas's book on maths for physicists, but I am curious to know if there are any higher level books on math for physicists that one would recommend?
And are there any opinions on what the best book is to read if one wants to try to read about general relativity from no particular background in it beyond standard undergraduate physics education? Thanks.

 

[Kummer]

How well do you know undergraduate analysis? Do you know the following topics:
Completeness property
Limits of sequences and delta/epsilon, Cauchy sequences
Subsequences
Limsups and Liminfs
Continous functions, delta/epsilon definition, sequence definition
Properties of continuous functions
Uniformly continuity
Infinite series and how to test convergence of infinite series
Power series, radius of convergence of power series
Uniform convergence of series of functions, and power series
Differentiation and its properties, Taylor's theorem
Integration on a line, Riemann and Darboux definitions.

Do you know all of this?

 

[Enjolras1789]

I do not know the Darboux definition.

 

[rudinreader]

If you are really lost as to what math to study, I would recommend analysis, and for sake of completeness, I would recommend baby Rudin as a starting point. If you are already good at undergraduate analysis, then you will just finish the book all the more quickly. Of course, I don't mean reading the theorems, I mean solving the problems.

A handful of complex analysis books are written with no analysis prerequisites, such as Serge Lang's, but again it's for sake of completeness that I recommend baby Rudin as a starting point. For example, the residue calculus loses its charm if you haven't already scoped out the real variable situation.

 

[quasar987]

I do not know the Darboux definition.

Darboux's definition is just the characterizations of the riemann integral in terms of "riemann sums". As a physicist, it's probably the dfn you know best.

 

[Xevarion]

Elias Stein's "Princeton Lectures in Analysis" series is great for analysis after you've had an intro like baby Rudin or a beginning class. He teaches Fourier Analysis and PDEs in book 1, Complex Analysis and basic analytic NT in book 2, and measure theory and bits of functional analysis, geometric measure theory, and ergodic theory in book 3. There are lots of great problems.