Alternative to Mathematical Methods books

[Figaro]

I have a background in Calculus (Simmons and Lang), Linear Algebra (Lang), and Differential Equations (Simmons). I want to read a book in Mathematical Methods but I find that the treatment is just too superficial.

Examples are, Mathematical Methods by Boas, Hassani, and Hobson.

I want to know some suggestions on books about the topics in those Mathematical Methods books, mainly I would like to focus on two topics: Integral Transforms (Fourier Analysis, Laplace transforms, etc) and Complex Analysis. I know there many suggestions in other threads but the catch here is that my goal is not to study each topic with a depth as say with what a mathematician wants but the level should be about for example, I've learned Calculus through say, Lang but I don't want to study books like Spivak, but Lang is certainly enough to get me through my study of Physics and of course with a certain amount of rigor in the Math aspect so as it wouldn't be called "cookbook/engineering math". To summarize, I would like to read books with a balance between theoretical and computational aspect, also it shouldn't assume familiarity and it's better if it is not too thick (~300-400 pages is about right) since I don't have the luxury of time but if it will sacrifice quality then I'll just stick to the one which is better, I just prefer moderate amount of pages i.e Lang. So far I have gathered some resources that I think will meet what I have in mind but feel free to give your opinion in my findings.

Integral Transforms:
Fourier Series by Tolstov (Sadly this doesn't include Laplace transform)
Fourier Series and Boundary Value Problems by Brown and Churchill (Engineery?)

Complex Analysis:
Complex Variables and Applications by Brown and Churchill ( Somehow longer, engineery?)

Probability:
Introduction to Probability 2nd Edition by Bertsekas and Tsitsiklis (Longer but I cannot find any better treatment, I have read the first two chapters and I'm very satisfied with it so I'll stick with it, opinions are still welcome)

Calculus of Variations:
Any classical mechanics book I think will do? (I find that Goldstein's treatment is too focused on the topics of classical mechanics but I don't know any other topics I could have used CoV).
Mathematics of Classical and Quantum Mechanics by Byron and Fuller (The chapter on Calculus of Variations is spot on and I think for physicist's needs, it is enough?)

*Tensor Analysis and Group Theory are excluded since it is more specialized (in a sense, any undergraduate in physics can get away without it aside from little bits here and there unless you are taking GR, Graduate QM, particle physics, also I already have resources that I'm satisfied with).

 

[George Jones]

I am not sure if it has enough computational detail for you, but a book that I like very much is "Mathematics for Physics and Physicists" by Walter Appel,

http://press.princeton.edu/titles/8452.html

It covers many of the same topics as "traditional" mathematical physics texts, but with a little more rigour.

 

[Figaro]

I am not sure if it has enough computational detail for you, but a book that I like very much is "Mathematics for Physics and Physicists" by Walter Appel,

http://press.princeton.edu/titles/8452.html

It covers many of the same topics as "traditional" mathematical physics texts, but with a little more rigour.

George, have you used this text? If yes, how do you compare this with the more popular books by Boas, Arfken, Hassani, Hobson? Is it self-contained in the topics (doesn't assume any previous knowledge) that are presented? I find his words about green's function interesting, I haven't found a "good" treatment of it from either of the four texts above.

"Several precise physical problems are considered and solved in Chapter 15 by the
method of Green functions. This method is usually omitted from textbooks on
electromagnetism (where a solution is taken straight out of a magician’s hat) or
of field theory (where it is assumed that the method is known). I hope to fill
a gap for students by presenting the necessary (and fairly simple) computations
from beginning to end, using physicist’s notation."

 

[Dr Transport]

this isn't a bad text and covers all of the transforms you'd ever need...

https://www.amazon.com/dp/0029490413/?tag=pfamazon01-20

 

[deskswirl]

I am not a fan of the usual mathematical physics texts although I own several and have used others (Boas, Kreysig, Mathews and Walker, Arfken, Hassani, Margenau and Murphy). If you were looking for a tolerable one Jeffreys and Jeffreys is the best, but very traditional.

I am familiar with Churchill's Complex Variables book from my undergrad days which is okay, but better is the book by Carrier, Krook and Pearson:

https://www.amazon.com/dp/B001VF5C4M/?tag=pfamazon01-20

which is the best I have found without requiring advanced analysis or being encyclopedic (Markushevich). Pay close attention to the chapters on asymptotics and special functions in the complex plane. A excellent problem book which has many worked computations is by Speigel:

https://www.amazon.com/dp/0071615695/?tag=pfamazon01-20

After you are strongly grounded in complex variables I would suggest LePage:

https://www.amazon.com/dp/0486639266/?tag=pfamazon01-20

which covers Complex Variables, Fourier, Laplace, impulse response and Z transforms in that order. This is the book and approach I wish I had learned these topics from years ago. Two problems books which cover Fourier Analysis and Laplace are again by Speigel:

https://www.amazon.com/dp/0070602190/?tag=pfamazon01-20

https://www.amazon.com/dp/007060231X/?tag=pfamazon01-20

I am not familiar which Churchill's Fourier Series book. I thought his books on complex variables and operational mathematics to be haphazard.

A great book on green's functions with computational details is Duffy:

https://www.amazon.com/dp/1482251027/?tag=pfamazon01-20

For calculus of variations a good starting book is the one by van Brunt:

https://www.amazon.com/dp/0387402470/?tag=pfamazon01-20

if you have time a great introduction to tensors is by Grinfield:

https://www.amazon.com/dp/1461478669/?tag=pfamazon01-20

If you are on a budget I'd say get LePage to read along Churchill and Brown which if I remember correctly has a very light covering of Laplace Transforms. Most of these book should be avaliable in a local university library except maybe the last three but you can get through inter library loan.

 

[jasonRF]

Here are a couple more suggestions to look for in your library.

For complex analysis Churchill is fine. But I like the book by Saff and Snider better. I am familiar with the second edition which is good, and also includes sections on integral transforms to get you started.
https://www.amazon.com/dp/0133274616/?tag=pfamazon01-20
Deskswirl recommends Carrier, Krook and Pearson, which is truly excellent (I love my copy) but the theory part is terse and many of the problems are so difficult that I don't think it is the easiest book for most of us to learn the material from for the first time. The integral transform chapter is really outstanding, though. If your library has it you should look at it, and perhaps read as a supplement to another text like Churchill or Saff and Snider.

Green's functions: the book by Greenberg is a nice introduction.
https://www.amazon.com/dp/0486797961/?tag=pfamazon01-20
I think all of his books are pretty well written.

jason

 

[Figaro]

Here are a couple more suggestions to look for in your library.

For complex analysis Churchill is fine. But I like the book by Saff and Snider better. I am familiar with the second edition which is good, and also includes sections on integral transforms to get you started.
https://www.amazon.com/dp/0133274616/?tag=pfamazon01-20
Deskswirl recommends Carrier, Krook and Pearson, which is truly excellent (I love my copy) but the theory part is terse and many of the problems are so difficult that I don't think it is the easiest book for most of us to learn the material from for the first time. The integral transform chapter is really outstanding, though. If your library has it you should look at it, and perhaps read as a supplement to another text like Churchill or Saff and Snider.

Green's functions: the book by Greenberg is a nice introduction.
https://www.amazon.com/dp/0486797961/?tag=pfamazon01-20
I think all of his books are pretty well written.

jason

So the book by Carrier, Krook and Pearson is not suitable for an introduction, how about books by Gamelin? Bak and Newman? Any opinion? I've seen those two books being mentioned a lot, aside from Ahlfors, Stein and Shakarchi, etc which I think are geared towards hardcore pure mathematicians.

 

[jasonRF]

I have never looked at Gamelin, Bak and Newman, or Stein and Shakarchi very carefully, so I have no useful opinion. Ahlfors would be rough for a first exposure for most people.