I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe Shilov briefly touches on this in his "Elementary Real and Complex Analysis", but I'd like a more broad and in-depth treatment - book that, in general, deals with mostly functions (and maybe continuity/their limits).

Meant for a freshman mathematics major. Books that are entirely dedicated to this topic are preferred over others wherein this topic is only treated in passing.

  • $\begingroup$ @mvw I don't quite understand the source of your confusion. My question basically comes down to this: most books just give you a trigonometric function and tell you to work with it. Most books just give you transcendental functions, like logarithmic functions, and make you do exercises. I'm looking for a book that explains the anatomy of these functions; that explains what's really going on "under the hood" of a log function. One that constructs the trig functions analytically. In general; a book that gives a somewhat rigorous treatment of major elementary functions, and functions in general. $\endgroup$ – Ius Klesar Jun 7 '16 at 1:32
  • $\begingroup$ FYI, I believe Felix Klein is responsible for originally advocating that the elementary transcendental functions be defined by certain definite integrals. I made some comments about this here. See also The logarithm as a direct function by J. W. Bradshaw (1903). $\endgroup$ – Dave L. Renfro Jun 7 '16 at 14:28

I believe the book Real Numbers and Real Analysis by Ethan D. Bloch has what you are looking for.

  • $\begingroup$ I see - it does feature a chapter on the construction of these functions! I will check it out. Do you know of any other texts - perhaps more affordable texts? $\endgroup$ – Ius Klesar Jun 7 '16 at 12:40
  • $\begingroup$ It's the only one I know of. There was a similar question asked here and a few other replies: math.stackexchange.com/questions/1789692/… $\endgroup$ – user333870 Jun 7 '16 at 14:03

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