Textbook Recommendations: Complex Analysis

[dm4b]

Hello,

I was interested in learning more about complex analysis. Also, very interested in analytic continuation. Can anyone recommend a good text that focuses on complex analysis.

Also, is there a good textbook on number theory that anyone recommends?

Thanks!

<mentor - edit thread title>>

 

[member 587159]

Hello,

I was interested in learning more about complex analysis. Also, very interested in analytic continuation. Can anyone recommend a good text that focuses on complex analysis.

Also, is there a good textbook on number theory that anyone recommends?

Thanks!

<mentor - edit thread title>>

Number theory: The theory of numbers by Niven

Complex analysis: I heard that Ahlfors is great.

 

[dm4b]

Number theory: The theory of numbers by Niven

Complex analysis: I heard that Ahlfors is great.

Thanks, I just ordered the Niven text, it was pretty cheap on Amazon. Other one looks pricey, but I may have to get it!

 

[MatthewNITX]

There is also
Lars, Complex analysis
Zil, First course in complex analysis.
But if your course is all about theories, and demonstrations, i suggest you using Complex Variables by the great author James Brown.P.S. i am taking this year the same course under the name called "Mathematical methods for physics, and i am using Complex variables and Applications by James brown.

Good Luck!

 

[alan2]

Ahlfors. Churchill and Brown if you are more applied oriented.

 

[dm4b]

Thanks again for the recommendations, much appreciated!

 

[alan2]

By the way, Ahlfors is available for $17 on Amazon in a paperback international edition.

 

[MidgetDwarf]

There is also Van Eyden for Number Theory. Very concise book. No wasted pages. Definition, lemma, theorem, corollary style. Get an old edition.

If this too difficult, try Number Theory: A friendly introduction... by Pomersheim.

 

[MidgetDwarf]

For complex Analysis. Try Real and Complex Analysis: Shilov. It is of the pure math book variety.

Zill is garbage in my opinion. It teaches methods only. It is good for Eletrical Engineers who only want to solve problems that have Complex Analysis elements. But I think a working engineer should be able to read a pure based undergraduate math book.

 

[Avogadro Number]

For a concise easy reading treatment, maybe "A friendly approach to complex analysis" by A. Sasane and S. Maad-Sasane.

 

[mathwonk]

the good news is that almost all books on complex analysis are good, probably because the subject itself is clean and beautiful.

The key tools are plane topology, path integration, and infinite series, which it helps to understand somewhat beforehand.

I still found it difficult to grasp for a long time, especially residue theory, or the computational technique of laurent series, until I ran across the book by frederick p. greenleaf (out of print of course), so of course i recommend that one. Although Ahlfors is a master, I do not like his book, which to me is unintuitive.

My favorite is probably the book by Cartan, but it is very theoretical and a bit terse, although masterfully eloquent. A book in the same vein that I also like is the one by Lang.

Some people I respect like Levinson and Redheffer. and of course if you want a more engineering approach, there is churchill, and the later rewrites by brown.

if you are a budding theoretical mathematician, there is a very fine but very theoretical treatment by george mackey, a harvard genius who just wrote out his class lecture notes and still produced an incredibly flawless treatment.

finally just as a throwaway for the extremely ambitious, or as a future project, there is riemann's thesis and his great followup on abelian functions. riemann essentially created the subject it seems.

 

[Infrared]

I like the book by Stein and Shakarchi. The problems are pretty good and it has some fun applications (prime number theorem, sums of four squares) in the later chapters.

 

[Ssnow]

Hi @dm4b I suggest: "Elementary Theory of Analytic Functions in One or Several Complex Variables'' of Henri Cartan and a second book "A second course in Complex Analysis" of W.A. Veech that has some elements of number theory (especially connected to complex analysis).
Ssnow

 

[kith]

There is also the magnificent "Visual Complex Analysis" by Tristran Needham who approaches the subject from a geometric perspective.

 

[MathematicalPhysicist]

I see that no one has suggested the books by markushevich and conway for single variable. From the little that i read, these books seem to be right on the spot for a mature maths student.

 

[member 587159]

I see that no one has suggested the books by markushevich and conway for single variable. From the little that i read, these books seem to be right on the spot for a mature maths student.

I second Conway's book. I have it at home and it is a quick read if your topological preliminaries are settled.

 

[FactChecker]

There is also the magnificent "Visual Complex Analysis" by Tristran Needham who approaches the subject from a geometric perspective.

I like the supplemental knowledge that this book gives with its geometric approach, but it takes a very long time (400-500 pages) to get to some fundamental subjects like integration and Cauchy's formula. I don't think that I would recommend it as the primary book to study from.

 

[kith]

I like the supplemental knowledge that this book gives with its geometric approach, but it takes a very long time (400-500 pages) to get to some fundamental subjects like integration and Cauchy's formula. I don't think that I would recommend it as the primary book to study from.

I mostly agree. I mentioned it because it is one of those rare books were a unique approach to a subject is executed masterfully and I wanted to bring this to the attention of the OP. I don't think that there's a danger that the OP might use it as a primary text for his studies if he is after a rigorous treatment. The book makes it explicitly and implicitly clear that it isn't a rigorous text.

 

[jasonRF]

I mostly agree. I mentioned it because it is one of those rare books were a unique approach to a subject is executed masterfully and I wanted to bring this to the attention of the OP. I don't think that there's a danger that the OP might use it as a primary text for his studies if he is after a rigorous treatment. The book makes it explicitly and implicitly clear that it isn't a rigorous text.

I'd place the 'danger' level at 0, since the OP hasn't been here for more than 18 months. In any case the recommendations can still be valid for other readers.

jasoon