Just in case my geek status was in any doubt, I’ve written a review of the textbook I read while vacationing in Jasper, Alberta.
Title: An Introduction to Laplace Transforms and Fourier Series
Author: P.P.G. Dyke
Original Publication Date: First printing 2001
Cover Price: None; Amazon.com lists the MSRP as $49.95 US.
Past textbook reviews can be found here.
Not surprisingly, this is an introduction to Laplace Transforms and Fourier Series. It also covers Fourier Transforms and the respective inverse transforms in more rigour than I’ve seen from other, application heavy textbooks on the subject.
The book states that “only a knowledge of elementary calculus and trigonometry is assumed.” While this is true, the linear algebra and complex analysis overviews are pretty intense for someone not already familiar with the topics. I recommend also coming in with a linear algebra background strong enough to be comfortable with inner products. As for the complex analysis, well, the final chapter is the only one that requires it, and this is used only to establish the form of the inverse Laplace transform. If you want to be completely comfortable with these final topics, come in with enough complex numbers background to calculate contour integrals with residues.
What I Was Looking For, And What I Found
As some of you know, my formal education background is mainly in physics (followed by education, my current day job. Well, morning and day job. Morning, day and evening job. Aw, heck, it’s the paying job.) In that degree, I took all of the formal math options I could manage, as I learn math best when it’s formally and carefully assembled from first principles. Sadly, I also took the engineering versions of the courses dealing with ordinary and partial differential equations. As a result, I was taught to use algorithms and jump through specific hoops to solve certain problems, with no motivating reason to be using those particular hoops to ensure that the correct results were coming out. I find it very difficult to learn that way, so I’ve spent much of the last decade looking for a textbook or two that would fill in the formal side and answer the many questions I had on the topic. This text answered most of those questions (as far as Laplace and Fourier Transforms are concerned), and pointed me to the subject area textbooks that should answer the rest.
Some of the specific questions I was looking for, as well as the answers given, are listed here to give the reader a feel for the level of detail present within the text.
- What possessed Laplace and Fourier to try these particular transforms, anyway?: Every presentation I’ve seen, including this one (unfortunately), starts out with the arbitrary definitions of said transforms with no indication of why one would even attempt those particular forms. I see the usefulness of them, and understand why, once discovered, they have become so popular for certain types of problems, but I still don’t know what those two were investigating when they stumbled across these particular transforms in the first place. Fortunately, this does, at least, indicate to me that the answers to those questions may well be found in a good book on Integral Transforms, which will soon be added to my shopping list.
- What is the connection, if any, between Fourier Series representations and Fourier Transforms?: This, thankfully, is answered with clarity and rigor. A large missing piece for me was the interpretation of the Fourier transform of a function. This piece is here. In short, the Fourier series representation of a function expresses a periodic function as the linear combination of (infinitely many) sine and cosine functions with frequencies that must be harmonic frequencies within a single period of the original function. The coefficients of the Fourier series indicate the relative amplitudes of each of these harmonics. The Fourier transform is similar, but it is not restricted to the harmonics natural to any particular length of the x-axis, as the domain runs over all real values of x. The resultant function is, in effect, a graph of the amplitudes of all possible frequencies present within the original signal function.
- Why do the inverse transformations take those specific forms?: In both cases, the courses I took merely quoted the inverse transformations and showed that they work for some known examples. This text derives the inverse transformations for both the Laplace and Fourier transforms in detail. This is particularly enlightening in the case of the Laplace transformation, which takes a real function, multiplies it by another real function, and then integrates said product to arrive at a third real function, and yet the inverse transformation ventures out of the reals and into the complex plane.
- What connection, if any, exists between the Laplace and Fourier transforms?: The similarities are made clear when the Fourier transform is represented in the complex plane.
I still have numerous other questions regarding the formalism of differential equations, particularly as they relate to solving ODEs with matrix methods, but this particular text doesn’t claim to touch that territory, so that’s not a problem with this at all. I only bring it up in case anyone reading this can suggest a nice, formal treatment of differential equations in general.
Answering the second question above. That’s been bothering me since November of 1997.
I still don’t know the answer to the first question above, and that’s the one that’s bothered me the most for the past decade.
The clarity of the text is good. As the author writes in Appendix C, “… it is often the case that an over formal treatment can obscure rather than enlighten.” Dyke includes the formalities in most cases, but always explains why those formalities are the appropriate ones. When the formalities are omitted, Dyke explains why that is so, and includes references to texts that will provide the missing details. It’s a good balance between the formal and the conversational. I give it 5 out of 6.
The structure of the text is well done. (Frankly, if you’re using LaTeX to typeset, it’s hard to mess this up given any sort of preplanning.) The index is effective, and the layout of the text is clear and consistent. When sections depend on previous sections, or will contribute to future sections, the relevant and appropriate comments are made. I give it 5 out of 6.
The examples are well chosen and abundant. There is a good mix of examples that are simple examples of the math and those that are more complicated due to their connection to real world applications. All are worked out in relevant detail. I give it 5 out of 6.
As with all texts in the Springer Undergraduate Mathematics Series of textbooks, there are numerous exercises at the end of each chapter, and complete solutions are provided in an appendix. The variety and depth of the exercises make this an excellent book for self study. (I suspect they would, unfortunately, deter some professors from assigning the text as a course book, as they’d need to farm out homework assignments from other sources. As the scope of this text doesn’t neatly fit any single course I’ve seen at the college level, that may not be a terribly big deal.) I give it 6 out of 6.
The completeness of the text is good. I, personally, would have preferred to have more of the formal details included, but this still provides more than any other text I’ve seen on the subject. Even so, I’ve never felt lost when reading, as Dyke always gives pointers to textbooks that fill in the missing details any time they have been omitted. I give it 5 out of 6.
The editing was good. I believe I found a few typos in some of the early, simple examples, where such errors are obvious and easily corrected by the reader. The later, more difficult examples, are free of any problems I can find. I give it 5 out of 6.
Overall, this text covers its own mandate well, even if it doesn’t fit any specific course I’m aware of. I’d recommend it to anyone looking for a formal treatment of these topics which are really treated in any formal manner. I give it 5 out of 6.
In total, An Introduction to Laplace Transforms and Fourier Series receives 36 out of 42.