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I have been through school. I have had a treatment of (Pre)Calculus (I,II,III).

I still feel like I don't know enough Trigonometry, or enough of Exponentials, or other functions.

Even through my engineering undergrad; I always felt, I lacked enough knowledge on these very essential "topics". They have been recurrent throughout my studies and had me in total surprise.

How mainstream, contemporary Calculus books enmeshes them into the real analysis topics (to me) felt like one giant over-bloated mess.

Here's the question - I want some books, which explore - Circular functions and/or Hyperbolic functions and/or Essential Exponentials. Exclusively and exhaustively.

What I've done - I have read "Introduction to Real Analysis" by Bartle and Sherbert. It was satisfying to touch actual mathematics without the advertised applications and implications. (I guess it's late transcendentals). Moreover, I'll attach this excellent book as well. I read through some trigonometry books but I am apprehensive of any geometrical methods to explain these functions now.

Kartik Pandey
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  • I think it would help to explain more fully what it is you feel you want to learn about elementary functions, and at what mathematical maturity level. The background you gave, followed by saying you've read Bartle/Sherbert, is like saying you have a background in pre-algebra and have read a precalculus book (gap between these two subjects is about 3-4 calendar years of study), and you want to learn more about how to multiply and divide (by hand) positive integers (something one learns several years before pre-algebra). – Dave L. Renfro Dec 26 '23 at 12:39
  • To follow-up on my previous comment, what exactly do you want that isn't in any of the many hundreds (even if restrict to the last 40 years) of trigonometry and precalculus textbooks that have been published? For example, two of the many such books I've taught from are Analytic Trigonometry with Applications by Barnett/Ziegler/Byleen (2003, 8th edition) and Algebra and Trigonometry with Applications by Munem/Foulis (1986, 2nd edition). – Dave L. Renfro Dec 26 '23 at 12:52
  • @DaveL.Renfro If you could please don't mind my background, tell me a resource on elementary functions, which "you" (at your level) find great, specifically Circular Functions. To let you know more, I am not a math student. I have had nominal exposure to Mathematical Analysis, Linear Algebra, Stochastic processes. I have dabbled with Algebra, Sets, Categories.

    I wanted to be exploratory in this case. What's there about these functions that I could have not known?

     – Kartik Pandey Dec 26 '23 at 12:54
  • My puzzlement is due to wondering how you could reach all these advanced areas and have studied Calculus I, II, III and have an undergraduate degree in engineering, without having studied (or learned along the way, as would be necessary for calculus courses, and certainly for engineering courses) the topics you are asking about. Usually when someone wants something beyond the standard high school and beginning college level courses in trigonometry and precalculus (what the two books I previously cited are for) (continued) – Dave L. Renfro Dec 26 '23 at 13:19
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    it is for more difficult/tangential topics that might be for someone interested in math contests OR it is for a more mathematically rigorous "analysis treatment" (e.g. where the functions are defined by functional equations or by power series, from which the properties of the functions are proved). However, it sounds like you're not interested in the latter. For the former, in the case of trigonometry maybe one or more of the books mentioned here (continued) – Dave L. Renfro Dec 26 '23 at 13:19
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    and in the case of precalculus maybe one or more of the books mentioned here and here and here. – Dave L. Renfro Dec 26 '23 at 13:19
  • @DaveL.Renfro You have a gold mine sir! Thank you. I chime well with the asker here.

    I really wanted to "revisit" these subjects again with maybe a fresh perspective. I'll definitely sift through these recommendations.

     – Kartik Pandey Dec 26 '23 at 13:51
  • < it is for more difficult/tangential topics that might be for someone interested in math contests OR it is for a more mathematically rigorous "analysis treatment" > Yes, I crave this. I request you to send those recommendations as well. Much thanks. – Kartik Pandey Dec 26 '23 at 13:53
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    send those recommendations as well --- Two real analysis texts I know of that are especially thorough with regard to a rigorous treatment of "precalculus functions" are Elementary Real and Complex Analysis by Shilov (1973/1996; see Sections 5.4-5.7, pp. 145-172) and Methods of Real Analysis by Goldberg (1976, 2nd edition; see Chapter 8, pp. 224-251). Maybe also Hawkins/Hawkins and Yandl. – Dave L. Renfro Dec 26 '23 at 20:57

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I am answering my own question.

I stumbled upon (and felt obliged to) attach these excellent series of blog articles, which covers the elementary functions in extensive depth and with unprecedented enthusiasm. They did succeed to satisfy me. Moreover, the author happens to be a user on MSE - @ParamanandSingh.

Throughout the series, various approaches were explored, which were all very insightful.

Exponential - 1 2 3

Circular Functions - 1 2 3

As for a book, I got interested with G.Hardy's "A Course of Pure Mathematics", especially the last two chapters. I would be appending a review soon.

Kartik Pandey
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