1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ @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
2
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.