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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:

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?

user5402
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    maybe you could give some examples of transcendental functions and what abou them is interesting. also see https://en.wikipedia.org/wiki/Special_functions – Will Jagy Jun 25 '15 at 19:12
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    At the risk of displaying my own ignorance: What do you mean by "transcendental function"? Any function that's not algebraic? That would include all sorts of wildness. Of not that then what do you mean? – David C. Ullrich Jun 25 '15 at 19:14
  • @DavidC.Ullrich https://en.wikipedia.org/wiki/Transcendental_function – user5402 Jun 25 '15 at 19:15
  • @WillJagy For example let $f(x)=\sum\limits_{n\geq 0}a_nx^n$. Give a necessary and sufficient condition on the sequence $(a_n)$ so that $f$ is a transcendental function. I mean such type of questions. – user5402 Jun 25 '15 at 19:17
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    "an analytic function that does not satisfy a polynomial equation", fine. There are many many references on analytic functions, typically in texts on complex analysis. Most of them are transcendental. – David C. Ullrich Jun 25 '15 at 19:18
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    https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms – Will Jagy Jun 25 '15 at 19:29

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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.

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Noah Schweber
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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.

Robert Israel
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  • I know some examples of transcendental functions but do you mean by "classes of transcendental functions"? – user5402 Jun 25 '15 at 19:18
  • For example, the class of functions you can get starting with the constant functions, the identity function, and $e^x$, and closing under multiplication, quotient, addition, composition. If we didn't include $e^x$, we'd just get the rational functions; when we do include $e^x$, we get a class of functions which are "rational relative to $e^x$," whatever that means. They include lots of transcendental functions (like $e^x$), but still have some very nice properties. – Noah Schweber Jun 25 '15 at 19:28
  • @NoahSchweber I got your point but that's too restricted. It's not what I'm looking for. – user5402 Jun 25 '15 at 19:58
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    @metacompactness Well, what are you looking for? – Noah Schweber Jun 25 '15 at 20:14
  • @NoahSchweber Some example: let $f(x)=\sum\limits_{n\geq 0}a_nx^n$. Give a necessary and sufficient condition on the sequence $(a_n)$ so that $f$ is a transcendental function. What are the transcendental functions whose derivative is algebraic? Let $f$ be an algebraic function and $\alpha\in\Bbb{R}$, what condition should we impose on $\alpha$ so that $\frac{d^{\alpha}}{dx^{\alpha}}f(x)$ is transcendental... I have a feeling that every question about transcendental functions is an open question. – user5402 Jun 25 '15 at 20:41