Is there a theory of transcendental functions? Lately I've been interested in transcendental functions but as I tried to search for books or articles on the theory of transcendental functions, I only obtained irrelevant results (like calculus books or special functions). On the other hand, there's many books and articles on algebraic functions like:


*

*Algebraic Function Fields and Codes

*Topics in the Theory of Algebraic Function Fields

*Introduction to Algebraic and Abelian Functions
Are there any references for the theory of transcendental functions? Did anyone studied rigorously such functions or is this field of mathematics outside the reach of contemporary mathematics?
 A: The class of all functions is just too wild to study in general, so usually we focus on studying large collections of functions that still have certain nice properties. For example: algebraic, continuous, differentiable, Borel, measurable, . . .
"Transcendental" just means "anything not algebraic," so that's too broad. But there are many subclasses of transcendental functions which are nice: most continuous functions, for example, are transcendental, and we might say that calculus is the study of continuous functions.
But that's sort of dodging the point. One question we could ask is: do transcendental functions have any nice algebraic properties? That is, if what we care about is abstract algebra, are the algebraic functions really the only ones we can talk about? The answer is a resounding no, although things rapidly get hard, and I don't know much here. I do know that some classes of transcendental numbers have rich algebraic structure theory - see http://alpha.math.uga.edu/~pete/galois.pdf, or http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TranscendencePeriods.pdf.
A: A transcendental function is any function that is not algebraic.  That's about all you can say about them in general.  Of course there are specific classes of transcendental functions that do have interesting theories.
