# Tagged Questions

Elementary questions about functions, notation, properties, and operations such as function composition.

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### What is that thing that keeps showing in papers on different fields?

A few months ago, when I was studying strategies for the evaluation of functional programs, I found that the optimal algorithm uses something called Interaction Combinators, a graph system based on a ...
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### Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
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### Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
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### Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2}$$ I would like to extend the idea for ...
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### Multiplicative Identity analog for absolute value

Is there a standard name for the function: $$f(x) = \begin{cases} x & \text{if |x|≤1;}\\ 1/x & \text{if |x|>1;}\\ \end{cases}$$ And is there a potential application of this ...
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### Can you add new functions to the set of elementary functions such that every function has an anti-derivative?

Its fairly well known that not every elementary function has an elementary anti-derivative. The common examples of this are $\exp(-x^2)$ and $\sin(x)/x$. The general workaround to this problem is to ...
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### To determine if$f^{-1}(x)$ is periodic function or not? $f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$

$$f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$$ $P(x)$ is polynomial with degree $n$. $m$ is an positive integer and $m>1$ What is the algoritm to determine $f^{-1}(x)$ is periodic function ...
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### Why is this change of variables true?

I have an equation $F_{xx}+yF_{yy}+{1\over 2}F_y=0$ defined on $y<0$. I found that the characteristics are $\alpha={2\over 3}(-y)^{3\over 2}-x,\,\,\,\,\,\beta={2\over 3}(-y)^{3\over 2}+x$ and that ...
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### Name of principal root function modified to return real values if possible

Is there a concise name for the function $f_n(x)\colon\mathbb{C}\to\mathbb{C}$ which returns the principal $n$th root of $x$, except in the case when $n$ is odd and $x$ is a negative real number, in ...
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### Is there some official name for this function?

$$\sqrt{1 - (-1 + x\bmod 2)^2}\cdot\operatorname{sign}(-2 + x\bmod 4)$$ Like half-circles connected to each other to look like waves: Its plot looks smooth, but the function is actually not smooth....