In need of new direction/book recommendation; self studying math

Question

I self study mathematics in hope of one day getting a degree in Applied Mathematics. I will provide my mathematical background first as I think it will be necessary to you in giving advice.

(I am a year 11 high school student) I have made my way from linear algebra (Elementary linear Algebra; Anton) $\Rightarrow$ calculus 1-calculus 3 (Calculus Early Transcendentals; Anton, Bivens) $\Rightarrow$ introductory real analysis (Understanding Analysis; Abbott).

The degree to which I have completed these books vary.

Linear algebra book was my first course in "higher mathematics", I would say I did a rather poor job in that I avoided all proofs and skipped many questions (I did finish the book). But coming back to it a while ago in a proof involving inner products, the book seemed very clear and I am confident It will be a easy book to me now.

The calculus book had a huge reservoir of questions. There were hard questions in each chapter which I had to search up solutions for or abandon. I would say my competency in the book is a C to a B.

The real analysis book was immensely frustrating at first. I made sure that I had solutions and slowly worked through it. I did every single question in each chapter (seldom exceptions). Though I relied heavily on solutions, I made sure I understood every proof's method, derivation of certain results gradually became easier. At the end of the book, I felt comfortable in my understanding of the book and my proving abilities, especially after grinding out hard chapters like integral form of the remainder, computing the harmonic series, deriving the gamma function and Gauss product. I would rate my understanding $B^+$.

From the advice I received talking to a physicist, for me, developing a thorough understanding of series expansions and power series is more practical and useful than something fancy like a PDE book. I want to develop my abilities in power series representation, manipulation, resolving into and understanding certain special functions and so on. I searched for these books but most were aimed at graduate students. My question is: is there a comprehensive, thorough, well written, well motivated and long book covering the aforementioned topics. (I am prepared to do a complex analysis book first.)

Thank you for your valuable time.

EDIT: (a lot of replies have been about DE books, I know they are necessary but I feel they are too application oriented for me right now. I would say I am looking for something more "analysis" like.)

I am also self-studying analysis(Abbott) and probability th. with the hope of getting a degree in Applied Mathematics. — Quasar, May 01 '21 at 17:55
You don't mention some essential parts of analysis, some of them at the heart of applied mathematics, in particular Fourier Analysis (called "harmonic analysis) with Fourier series and Fourier Transforms, [and its cousin, application oriented, Laplace Transform], and orthogonal polynomials. You mention complex function theory. I recommand you warmly to begin its study in parallel with real Calculus: it's fantastic! Now, which book ? "Infinitesimal Calculus" By J. Dieudonné 1971 (english translation of the french original) — Jean Marie, May 01 '21 at 18:35
I think you should go ahead and study differential equations using a book such as Boyce and DiPrima. It will lead to power series solutions and special functions and Fourier series. By the way, I got a lot out of reading Strang's books such as Introduction to Applied Math and Linear Algebra and Its Applications. — littleO, May 02 '21 at 02:06
Three suggestions for an introduction to your specific desire that fit with your background: An Introduction to Real Analysis by Burton Swank Randol (some comments about Randol's book are included in this answer), A Text-Book of Convergence by William Leonard Ferrar, Advanced Calculus by Taylor/Mann (2nd or 3rd edition, last few chapters beginning with infinite series). — Dave L. Renfro, May 02 '21 at 13:35
Incidentally, some of these "interesting applications" of power series expansions might be of interest. — Dave L. Renfro, May 02 '21 at 13:40
Does this answer your question? How do I self-learn undergraduate math? — user1147844, May 30 '23 at 07:22

score 2 · Answer 1 · answered May 01 '21 at 18:14

Differential equations has a chapter on using power series as a method of solving. If you have not learned differential equations yet, you would do well to learn the basics, up to that chapter in any good book such as Edwards and Penney. (I am not sure what "Calculus III" is because it is very different at different schools.)

Power series and Laurent series are covered in a complex calculus course. I used Saff and Snyder, but you might want something more challenging.

Aside from a few applications here and there, those are the lower level undergraduate subjects that I know involve power series.

In need of new direction/book recommendation; self studying math

1 Answers1