Customer Reviews

Integral Equations (Dover Books on Mathematics)

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Integral Equations. Some professors consider this a "minor topic" that can easily be covered as a few week lecture series amid others in an applied math course (that typically covers a variety of topics from PDEs to calculus of variations) or within some advanced differential equations course. Others think it's a fairly hefty subject that requires at least a bit more than one typical semester.

I'm not sure what this author thinks, but the book appears to fit the needs of the former. Or rather, this book is perfect for you if you've 1) already taken/studied a course on integral equations, 2) are currently only doing things involving LINEAR integral equations, and 3) just need a real "cookbook" to remind you how to solve various linear integral equations.

Sure, nonlinear equations are talked about, but it's fairly brief.

I won't pick this book apart too much, but I would like to note first a few problems I had with this book.

First off, in Chapter 1, a VERY rough introduction to fourier integrals is thrown out, and the book makes a very horrible claim that a fourier integral can be shown to have particular characteristic solutions with respect to +/-1 eigenvalues "with little difficulty." The problem is what I put in quotations: the whole process involves taking two integrals, one is of no real effort which is just a generic application of integration by parts, but the other is significantly more complicated. One is forced to use contour integration in complex variables (which, for this particular problem, may not have been handled in a typical undergraduate first course in complex calculus) that leads to using advanced calculus techniques (which could be assumed that the student knows, but if so, this approach is needlessly complicated and requires a GREAT deal of care) or to use the Bounded Convergence Theorem (which leads to a VERY simple result, but the BCT isn't exactly taught at the undergraduate level, which this book is aimed for). And so, no, I strongly disagree with this example being of little difficulty at all. You either have to be excellent at advanced calculus techniques and take great care, or you need to already be pretty well beyond the scope of the book.

This leads to the second problem of the book: roughly 70% of everything in it is "hand-waved," and the phrase "little difficulty" pops up all over the place erroneously. Not only this, but motivation is hardly, if ever, explained. This is particularly a nuisance in Chapters 2 and 4, where the whole POINT of everything in this book is just ignored, and the reader is left with just random derivations that end without going back to the purpose of said derivations. Then the next paragraphs lack any juxtaposition at all, so the "fresh" student is left scratching his head wondering why the book bothered.

That is, of course, providing the fresh student understood half of what was going on, again going back to the hand-waving.

Now, let's talk about the good parts of this book. The problems and cookbook methods themselves. The examples, though EXTREMELY few, are very descriptive. In many applied math textbooks, heavy with number-crunching, one often is forced to go through trivial examples (where, say, everything is a 1, so if you don't fully understand something, you're left wondering where exactly the values change in the methods if you change the numbers in the problem), but not here. Nontrivial examples are excellently presented, and stepping through each piece carefully shows everything that ever needed to be done. ...Here, because the calculations are pretty algebraic and require fairly rudimentary calculus, hand-waving is "okay" as far as I'm concerned. Someone not comfortable with calculus has no real reason to dive into this subject anyway.

Also, the problems themselves are very strong and bring a lot of the sections in each chapter together, ultimately giving one not only a sense of fulfillment, but a good bit of confidence in that one should, after doing the problems, be very acquainted with all the different ideas.

The problems are few, though. For 10 chapters, roughly 60 problems isn't much, but again, quality over quantity, and the difficulties curve properly. I would only complain about Chapters 5 and 8 having a few problems that just seem either "I'm Mr. Smarty Pants lemme show you what I know even if you won't use it" or "hurr waste of time."

Final remarks: a real average book, here. Again, this is best suited for someone who needs a (satisfyingly inexpensive) reminder of how to tackle linear integral equations and wants to focus primarily on applied integral equations with theory serving primarily to allow one to expand his ideas and move on to more challenging texts with a fair amount of confidence, or at the least the prerequisites necessary.