Properties of elementary functions book?

It's possible to define cos,sin as the unique solutions to a certain DE, ln(x) as a certain integral, etc.

Does anyone know of a nice analysis text that derives all of the properties of the elementary functions like this in a nice, thorough and comprehensive way? I have a couple of books with bits and pieces of the derivations, but I'd like a nice reference book where I know I'll find it all. I can't think of one myself, off the top of my head.

Reply 1

ODES_PDES

18

Original post by atsruser

It's possible to define cos,sin as the unique solutions to a certain DE, ln(x) as a certain integral, etc.

Does anyone know of a nice analysis text that derives all of the properties of the elementary functions like this in a nice, thorough and comprehensive way? I have a couple of books with bits and pieces of the derivations, but I'd like a nice reference book where I know I'll find it all. I can't think of one myself, off the top of my head.

If you write one I will buy it!
:tongue:

Reply 2

davros

16

Original post by atsruser

It's possible to define cos,sin as the unique solutions to a certain DE, ln(x) as a certain integral, etc.

Does anyone know of a nice analysis text that derives all of the properties of the elementary functions like this in a nice, thorough and comprehensive way? I have a couple of books with bits and pieces of the derivations, but I'd like a nice reference book where I know I'll find it all. I can't think of one myself, off the top of my head.

Tbh, most of the so-called "elementary functions of analysis" are defined as power series, from which their other properties are derived, because once you've defined what a power series is, and introduced the radius of convergence, you get all sorts of "nice" properties as a consequence - differentiability, integrability, etc - so you can just apply these results straightaway to your desired function.

ln x is generally introduced as the inverse of exp(x), although it is certainly possible to define it via an integral and show that this gives an equivalent definition - a number of elementary analysis books I have seen adopt this approach.

Definition of functions via an integral is relatively unusual, the most common example being the Gamma function which extends the factorial function.

Reply 3

atsruser

OP

11

Original post by davros

Tbh, most of the so-called "elementary functions of analysis" are defined as power series, from which their other properties are derived, because once you've defined what a power series is, and introduced the radius of convergence, you get all sorts of "nice" properties as a consequence - differentiability, integrability, etc - so you can just apply these results straightaway to your desired function.

ln x is generally introduced as the inverse of exp(x), although it is certainly possible to define it via an integral and show that this gives an equivalent definition - a number of elementary analysis books I have seen adopt this approach.

Definition of functions via an integral is relatively unusual, the most common example being the Gamma function which extends the factorial function.

I don't think that I was clear enough. I'm looking for a book that comprehensively treats (and not just asks the reader to do so in exercises) all of the various definitions of the elementary functions, and shows their equivalence e.g. we can define sin/cos via "solutions to a DE" approach, power series, or inverse of an integral approach (maybe more?). I want a reference book that shows all of them, in detail, that I can look up when the fancy takes me.

I've got some of that in various books, but I want the whole lot in one place.

Reply 4

Gregorius

14

Original post by atsruser

I don't think that I was clear enough. I'm looking for a book that comprehensively treats (and not just asks the reader to do so in exercises) all of the various definitions of the elementary functions, and shows their equivalence e.g. we can define sin/cos via "solutions to a DE" approach, power series, or inverse of an integral approach (maybe more?). I want a reference book that shows all of them, in detail, that I can look up when the fancy takes me.

I've got some of that in various books, but I want the whole lot in one place.

This perfectly reasonable request appears to be quite hard to answer! The properties of such elementary functions are listed in standard sources such as Abramowitz and Stegun (which has its modern online form here, with elementary functions here). But proofs are left to the reader!

The best I can suggest at the moment are books on elementary complex analysis; they usually contain reasonably thorough treatments.

Reply 5

atsruser

OP

11

Original post by Gregorius

This perfectly reasonable request appears to be quite hard to answer!

That's depressing. I would have thought that this would all be in some standard work somewhere.

The best I can suggest at the moment are books on elementary complex analysis; they usually contain reasonably thorough treatments.

I've never seen a comprehensive treatment such as I described - I guess it's too tempting a source of problems for an author to want to lay it all out in full.