Book(s) to gain practice with green's functions, spherical harmonics

[WannabeNewton]

Hi guys! I was wondering if anyone knew of a particularly nice book that taught one how to solve physics problems that need the use of green's functions and/or spherical harmonics. I can't seem to find a book that actually does this other than Jackson but I'd rather not tread there (I'm guessing this is usually done in a Jackson type class but I never went the Jackson route; instead I went straight to the classical field theory route which I love much more but where you don't really see these computational methods show up). I am talking, in particular, about physics books and not math methods books. As much as I find these computational techniques highly non elegant and annoying in comparison to pure math / proofs, I need the techniques under my belt, it would seem, as I have stumbled upon many problems that required the use of green's functions etc. but had absolutely no idea how to use them to solve the problem. Thanks for any and all suggestions.

 

[Lavabug]

Hi guys! I was wondering if anyone knew of a particularly nice book that taught one how to solve physics problems that need the use of green's functions and/or spherical harmonics. I can't seem to find a book that actually does this other than Jackson but I'd rather not tread there (I'm guessing this is usually done in a Jackson type class but I never went the Jackson route; instead I went straight to the classical field theory route which I love much more but where you don't really see these computational methods show up). I am talking, in particular, about physics books and not math methods books. As much as I find these computational techniques highly non elegant and annoying in comparison to pure math / proofs, I need the techniques under my belt, it would seem, as I have stumbled upon many problems that required the use of green's functions etc. but had absolutely no idea how to use them to solve the problem. Thanks for any and all suggestions.

I don't know if such a thing exists, but I would also be interested. Arfken (a math methods book) has sections on this but isn't really pedagogical.

Maybe look for some differential equations/PDE cookbook that covers Green's function solutions? Maybe some engineering books might have something relevant, in the context of circuits.

I'm personally considering getting this as a special functions reference, since I always found the Boas book I have a bit lacking:
https://www.amazon.com/gp/product/0486435210/?tag=pfamazon01-20

Also I think any sufficiently advanced modern optics book with problems on diffraction probably covers Green's functions methods too. I could recommend one but it's written in Spanish, so I don't know if it would be of much help: "Óptica Física", Carreño.

 

[WannabeNewton]

Thanks for the suggestions. I don't know any Spanish unfortunately. The rather recent EM text by Franklin also has a chapter devoted to green's functions but I do not know how good it is; it would be very fortunate if it is indeed good because I am exclusively interested in applying these methods to EM and gravitation and not much outside of that.

 

[marcusl]

Chapter 2 of R. Collin, Field Theory of Guided Waves (2nd ed.) presents an extensive (over 100 pages) treatment of Green's functions, both scalar and dyadic. He applies these techniques to solve physical transmission line and waveguide problems throughout the remainder of the book.

Warning: it's roughly comparable to Jackson in difficulty.

For a gentler introduction, maybe look to a text on PDE's?

 

[WannabeNewton]

Hi marcusl. I appreciate the help! To be honest I've never actually ever looked inside Jackson; I am scared to do it because of its hearsay notoriety in difficulty. I'll see if I can take a look at Collin. For the other part, do you have a particular book PDE - type book in mind that goes into this stuff e.g. solving poisson's equation and the likes?

 

[alissca123]

Hassani gives a really nice treatment of Green Functions in his book
Mathematical Physics: A Modern Introduction to its Foundations
https://www.amazon.com/dp/0387985794/?tag=pfamazon01-20

 

[WannabeNewton]

Hi alissca123! I have looked at Hassani before but to be honest I highly dislike the book; I hate most mathematical methods books (not to be confused by methods of mathematical physics books like the ones by Hilbert and Courant or Reed which are superb). The reason is because, like Hassani, they hand wave much of the proofs of theorems and lemmas and existences and employ techniques without actually justifying said techniques with proofs.

 

[micromass]

I know you don't want pure mathematics books, but I'll answer anyway. Two books are

"Fourier Analysis and Its Applications" by Folland
"Introduction to Partial Differential Equations" by Folland.

The Fourier analysis book is brilliant. Very nicely written and quite rigorous. Furthermore, it doesn't really require many prerequisite knowledge. It's one of my favorite books on basic Fourier Analysis.

I haven't had the chance to go through much of the PDE book, but I guess it's very good as well. Although the topics covered look somewhat advanced.

Of course there is also https://www.amazon.com/dp/0125850506/?tag=pfamazon01-20 book. It doesn't cover what you want, but it's still very nice!

 

[marcusl]

Hi marcusl. I appreciate the help! To be honest I've never actually ever looked inside Jackson; I am scared to do it because of its hearsay notoriety in difficulty. I'll see if I can take a look at Collin. For the other part, do you have a particular book PDE - type book in mind that goes into this stuff e.g. solving poisson's equation and the likes?

Well, Green's function techniques aren't exactly elementary, no matter what book you look in.

I learned PDE's from Berg and McGregor, which I recall was based around solving physics problems. I can't recall whether it had much on Green's functions, though.

EDIT: If you are comfortable with Hilbert and Courant, then you should look at Jackson. It's not that hard...

 

[WannabeNewton]

EDIT: If you are comfortable with Hilbert and Courant, then you should look at Jackson. It's not that hard...

I am measurably more comfortable with proofs than I am with computations, that is the issue and it seems like Jackson is a slew of computations (just ask Micromass about how bad I am at computations lol)

 

[WannabeNewton]

Of course there is also https://www.amazon.com/dp/0125850506/?tag=pfamazon01-20 book. It doesn't cover what you want, but it's still very nice!

>.> I knew you would link it! Anyways, you've shown me the Folland ones before and as much as I would love to do a pure math book (hooray for functional analysis!) on this subject I know I will fall into the same boat where I get so engrossed by the theory that I end up not learning the computational methods and get flung back to the current situation (remember how I wanted to do Hassani the summer after high school ended to learn about exterior calculus and you suggested Lee and the rest is history...)

 

[George Jones]

The Fourier analysis book is brilliant. Very nicely written and quite rigorous. Furthermore, it doesn't really require many prerequisite knowledge. It's one of my favorite books on basic Fourier Analysis.

I, too, think that this book is brilliant, nicely written, and rigorous. And its last chapter is on Green's functions.

Hi alissca123! I have looked at Hassani before but to be honest I highly dislike the book; I hate most mathematical methods books

Anyways, you've shown me the Folland ones before and as much as I would love to do a pure math book (hooray for functional analysis!) on this subject I know I will fall into the same boat where I get so engrossed by the theory that I end up not learning the computational methods and get flung back to the current situation (remember how I wanted to do Hassani the summer after high school ended to learn about exterior calculus and you suggested Lee and the rest is history...)

Isn't the complement of the union of these two sets of books (at which you don't want to look) empty?

 

[WannabeNewton]

Isn't the complement of the union of these two sets of books (at which you don't want to look) empty?

Yes, unfortunately I realize I am asking for something that is too ideal. I do apologize for that, I did not intentionally intend to ask for something this idealistic. I would love to go with a pure math perspective but I have to pick up these computational tools at some point or another. I will take a look at the PDE books mentioned by marcusl and Lavabug, mix and match them as needed, and if all else fails I will try to muster the courage to tackle Jackson . Thanks George!

 

[jasonRF]

Overall, Jackson isn't all that bad in my opinion. I do really like Collin's chapter on Green's functions in general and I learned a lot about them from his book, but if spherical harmonics are of special interest he only explicitly discusses them in text in the context of dyadic Green's functions which ads extra complications (he of course expects you to use them to solve scalar Poisson's equation in the problems ...). I actually think Jackson is more friendly for spherical harmonics, and Collin certainly is not any easier!

Other options for green's functions in electromagnetics (but not necessarily spherical harmonics) are Dudley's, "mathematical foundations for electromagnetic theory," and the very good (but out of print) "radiation and propagation of electromagnetic waves," by Tyras. Dudley has my favorite exposition on Green's functions, although much of his presentation is 1-D. Tyras has many excellent examples of Green's functions for the Helmholtz equation; his presentation of the radiation from a point source above a dielectric half-space is the clearest I have seen anywhere.

For some oldies that use lots of special functions including spherical harmonics there of course are always Stratton's "electromagnetic theory" and Smythe's "static and dynamic electricity." I don't think you will find modern treatments including delta functions in them, though.

Then of course you can look in applied math kinds of books such as those by Stakgold. I wouldn't have the patience to wade through them, but someone must like them!

jason

 

[WannabeNewton]

Overall, Jackson isn't all that bad in my opinion.

You over - estimate my intelligence and computational abilities my friend.

I do really like Collin's chapter on Green's functions in general and I learned a lot about them from his book, but if spherical harmonics are of special interest he only explicitly discusses them in text in the context of dyadic Green's functions which ads extra complications (he of course expects you to use them to solve scalar Poisson's equation in the problems ...).

I will make it an objective to go find it in the library and see how tractable it will be for me. Both you and marcusl have recommended it. Thanks for the plethora of other suggestions as well. I will have to sit down sometime and sort out which ones I can use and cannot use. Thanks again!

 

[marcusl]

Other options for green's functions in electromagnetics (but not necessarily spherical harmonics) are Dudley's, "mathematical foundations for electromagnetic theory,"
...
For some oldies that use lots of special functions including spherical harmonics there of course are always Stratton's "electromagnetic theory" and Smythe's "static and dynamic electricity." I don't think you will find modern treatments including delta functions in them, though.

I couldn't relate to Dudley's book and got rid of it after it had sat unused on my shelf for some years. Probably that means that Wannabe will love it!

I didn't mention Stratton because it is too old to conform to modern usage. He doesn't use the notation G(x|x') and doesn't call it a Green function, for instance.

Smythe is a lovely book, and I forgot that he actually does use Green functions by name. He doesn't introduce them coherently in one place, however, but sprinkles them all through the book. Jackson and Collin are better places to learn them, IMO.

So long as we are mentioning awesome physics books, why not throw in Morse and Feshbach? They devote a hundred pages in vol.1 to solving a zillion physics problems with GF. Vol. 2 treats dyadic GF's. Since most physicists complain about M&F being too mathematical, it might appeal to Wannabe.

 

[WannabeNewton]

So long as we are mentioning awesome physics books, why not throw in Morse and Feshbach? They devote a hundred pages in vol.1 to solving a zillion physics problems with GF. Vol. 2 treats dyadic GF's. Since most physicists complain about M&F being too mathematical, it might appeal to Wannabe.

I've seen this one (Vol 1) in the library before actually but didn't open it up. I'll take a look. Thank you so much for the suggestions, I really appreciate it!

 

[jasonRF]

You over - estimate my intelligence and computational abilities my friend.

Thanks for calling that out - when I re-read my statement it sounds ridiculous. The problems are hard, of course. I was trying to imply that the way Jackson motivates Green's functions and provides a number of examples is not as bad as all that and I think it is worth a look.

I did just look on my shelf and realized that I actually have a book on engineering electromagnetics that has a final chapter on Green's functions that walks the reader through it in a simple way. It is "advanced engineering electromagnetics" by Balanis. Not mathy, so you may not be interested, but a much more systematic presentation than Jackson. More basic than Collin as well. In the context of EM, I can't imagine it could get much easier to follow. On the down side, I find the entire book mostly uninspiring.

If you want a mathematical approach then the suggestions that interest you from some of the other folks may be better suited to you, of course.

have fun!

jason

 

[jasonRF]

Just found this great, free book on green's functions in physics. Again, not highly mathematical, but you can instantly see if you like it;

http://freescience.info/go.php?pagename=books&id=1435

jason

 

[atyy]

Another free book you may like to try is Peter Olver's Intro to PDEs. Chapter 6 is about Green's functions and impulse responses.

http://www.math.umn.edu/~olver/pdn.html

 

[WannabeNewton]

Christ I did not realize there was so much out there on this. Anything PDE related scares the living hell out of me :[
I will try to sort out what to use and not to use as efficiently as possible. Thank you all for the continual suggestions, I really appreciate it!

 

[atyy]

Anything PDE related scares the living hell out of me :[

I thought you liked differential geometry

 

[WannabeNewton]

I thought you liked differential geometry

Yeah I love differential geometry and I have a lot of interest in DEs in the way of flows and their connection to lie derivatives but in my last post I meant things like methods of solving PDEs scare me

 

[atyy]

Yeah I love differential geometry and I have a lot of interest in DEs in the way of flows and their connection to lie derivatives but in my last post I meant things like methods of solving PDEs scare me

lApparently Lie derivatives have something to say about whether one has a nice solution.
http://books.google.com/books?id=nFSJn7dIYysC&dq=stephani++lie+derivative&source=gbs_navlinks_s
http://books.google.com/books/about/Equivalence_Invariants_and_Symmetry.html?id=YuTzf61HILAC

Anyway, Green's functions are only for linear equations (as far as I know), so they can't be that scary ultimately. They are basically what engineers call an impulse response for linear systems.
http://www.cds.caltech.edu/~murray/courses/cds101/fa02/faq/02-10-23_green.html

 

[WannabeNewton]

lApparently Lie derivatives have something to say about whether one has a nice solution.
http://books.google.com/books?id=nFSJn7dIYysC&dq=stephani++lie+derivative&source=gbs_navlinks_s

Oooh it's the same Stephani who co authored the GR text with Ohanian. Very nice thanks.

Anyway, Green's functions are only for linear equations (as far as I know), so they can't be that scary ultimately. They are basically what engineers call an impulse response for linear systems.

I'm not very good at these kinds of computations unfortunately and this is where my fear of the subject matter comes from. Thanks for the video!

 

[alissca123]

Hi alissca123! I have looked at Hassani before but to be honest I highly dislike the book; I hate most mathematical methods books (not to be confused by methods of mathematical physics books like the ones by Hilbert and Courant or Reed which are superb). The reason is because, like Hassani, they hand wave much of the proofs of theorems and lemmas and existences and employ techniques without actually justifying said techniques with proofs.

I'm confused haha.. you dislike hand-wavy books, but you are not looking for a pure math book...

You should also note that Hassani wrote two books, Math Methods For Students of Physics and Related Fields (maybe the one you looked at), and the one I mentioned earlier, which is a graduate level book.

I like the PDE book by Folland but I think that it's not the book you're looking for (not much computational stuff). Haberman and Myint-U are some nice books on applied PDE's. Or if you want to see these topics applied directly to E&M then I recommend that you take a look at Schwinger's Classical Electrodynamics :)

 

[WannabeNewton]

I'm confused haha.. you dislike hand-wavy books, but you are not looking for a pure math book...

I love pure math books but the problem is if I start doing one I'll get distracted and never get to my goal of learning the techniques lol. I'm not looking to get advanced; I mainly need to do things like evaluating the green's function solution, to poisson's, for a thin slowly rotating spherical shell and things like that. The sources I've seen use delta functions for thickness and all sorts of stuff that I haven't seen before so it confuses me. Thanks!

 

[Jorriss]

WBN, don't be ridiculous.

You are like 1,000,000,000x's more qualified for Jackson than necessary.

Just do Jackson.

 

[WannabeNewton]

WBN, don't be ridiculous.
...

Just do Jackson.

You're just saying that because you love me >.>. I mean I'll give it a shot but no guarantees!

 

[Jorriss]

No, use Jackson. The book is clear and pedagogical. For example, his introduction of delta functions as a tool for physicists is crystal clear.

 

[lurflurf]

I have Green's Functions with Applications by Dean G. Duffy. It is pretty good and covers some more than other books. It is not really about electromagnetism, with all the Jackson talk if that is what you want Dyadic Green Functions in Electromagnetic Theory by Chen-To Tai is a great book. Both as are many specialized books are priced high most places so look at the library. I learned spherical harmonics from The Theory of Spherical and Ellipsoidal Harmonics by E. W. Hobson, it is a classic with flavor, but these days people probably do not want to know that much about them.

 

[Meir Achuz]

Thanks for the suggestions. I don't know any Spanish unfortunately. The rather recent EM text by Franklin also has a chapter devoted to green's functions but I do not know how good it is; it would be very fortunate if it is indeed good because I am exclusively interested in applying these methods to EM and gravitation and not much outside of that.

Try it, I think you'll like it. It can give you a good background to go on with the more mathematical books.

 

[WannabeNewton]

Oh boy I had totally forgotten about this thread! Thanks lurf for the suggestions. Seems like it might be hard to get a hand on those books but I'll see what I can do. The Hobson book looks quite interesting; tis' quite an old book though eh :)?

And thanks clem, I had forgotten about Franklin but it seems sometime after this thread died out someone put up a rather cheap used version of the text. I agree, it would be nice to have a good solid base before moving on to the higher things. Basically what I'm trying to do is form a strong foundation for PDEs so that I can go on to study PDEs such as the full Maxwell's and Einstein's with distributions as sources because I'm trying to learn more about the "self-force" solutions. I just want to make sure I have down the basics of PDEs in electromagnetic theory before moving on to more advanced things. Up till now I have unfortunately not had much of an exposure to rigorous PDE theory.