# Functions of functions

Is there such a thing as the study of the calculus of functions (I can think of no better term for it!)

eg:

if $f_0(x)=\sqrt{n}\log(n)$

then $f_1(x)=\sqrt{\sqrt{n}\log(n)}\log(\sqrt{n}\log(n))$

and $f_2(x)=\sqrt{\sqrt{\sqrt{n}\log(n)}\log(\sqrt{n}\log(n))}\log(\sqrt{\sqrt{n}\log(n)}\log(\sqrt{n}\log(n)))$; etc.

where each successive $n$ is replaced with $f_0$. What is the best way to write this?

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Google "iteration theory" and "function". For a nice survey of the early historical work in this area, see Daniel S. Alexander's book A History of Complex Dynamics: From Schröder to Fatou and Julia. – Dave L. Renfro Jun 13 '14 at 14:06
Thanks! Will take a look :) – martin Jun 13 '14 at 14:50
It occurred to me that if you change "function" to "real function" in the search I gave, then you'll get stuff involving real functions of a real variable, which might be of more immediate relevance to you. I tried this, and perhaps it's better than the search I originally gave (if you want real functions), but I still saw an awful lot of hits that look like iteration of complex-valued functions was involved. – Dave L. Renfro Jun 13 '14 at 14:58
Wikipedia article up in top position - will start there! Thanks for the tip :) – martin Jun 13 '14 at 15:06
The best beginning text I know of for this topic is A First Course In Chaotic Dynamical Systems by Robert L. Devaney. This essentially requires only a first semester calculus course (differential calculus), although the preface says "first year of calculus". This book was used every Spring for a course offered at a math/science high school (in Louisiana, USA) that I taught at during 1996-1999 (I didn't teach this particular course, however). Note that Devaney has other books, one with a similar title, that are more advanced. – Dave L. Renfro Jun 13 '14 at 16:03

Let $f_{0}(n)=\sqrt{n}\log(n)$ and $f_{m}(n)=f_{m-1}(f_{0}(n))$ for $m\in\mathbb{N}$.
I wonder whether there is a way to write this for $m\in\mathbb{R}$? – martin Jun 12 '14 at 20:25
I'm not sure what you mean by $m\in\mathbb{R}$? Perhaps I misunderstand but you provided a sequence of functions in your question which is indexed by the natural numbers and $0$. I don't think the index set involves $\mathbb{R}$. – user71352 Jun 12 '14 at 20:28