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

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31
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0answers
552 views

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 ...
11
votes
0answers
396 views

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 ...
9
votes
0answers
249 views

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 ...
9
votes
0answers
361 views

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 ...
8
votes
0answers
380 views

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 ...
7
votes
0answers
95 views

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 ...
7
votes
0answers
294 views

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 ...
7
votes
0answers
3k views

Finding general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$.

I'm trying to find the most general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$. I write this polynomial as $u(x,y)$. I calculate $$ \frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad \frac{\...
7
votes
0answers
324 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
6
votes
0answers
56 views

Why was the zeta function introduced?

I know the 'Zeta Function' is very useful in Mathematics, and that it has relations with many other functions (such as the 'Gamma Function'). I also know the 'Zeta Function' $\zeta(s)$ is defined as: ...
6
votes
0answers
154 views

How to invert this expression involving $\tanh^{-1}$?

I've got the expression: $ x = \tanh^{-1}(p) - \sqrt{\frac{2}{3}} \tanh^{-1}\left( \sqrt{\frac{2}{3}} p\right) $ How can I invert this function so I have a function $p(x)$? I thought about using ...
6
votes
0answers
310 views

Show that $1^k+2^k+…+n^k$ is $O (n^{k+1})$.

Let $k$ be a positive integer. Show that $1^k+2^k+...+n^k$ is $O (n^{k+1})$. So according to the definition of big-$O$ notation we have: $$1^k+2^k...+n^k ≤ n(n^k) = n^{k+1}$$ whenever $n>1$ Is ...
6
votes
0answers
255 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
5
votes
0answers
110 views

Arithmetic implications of different ways to geometrically construct an Hilbert's curve

I have a question on the relation between the geometric and the arithmetic representation of the Hilbert's space-filling curve. Geometric representation: consider the Hilbert's curve $f_h:[0,1]\...
5
votes
0answers
47 views

Find the inverse of the following piecewise defined function

Find the inverse of $f$ if $f(x)=$ $$ \begin{cases} \sqrt{2-x}, &\text{for $x<0$}\\ 1-x^2, &\text{for $x \ge 0$} \\ \end{cases} $$ My effort For $y=\sqrt{2-x}$ ,we find \begin{array}{...
5
votes
0answers
87 views

Proving not equicontinuity in $\Bbb R$ but equicontinuity in any other closed subset of $\Bbb R$

Let $F = {f_{n} | n ∈\Bbb N }$ be an infinite collection of functions $f_{n}(x)=e^{−n(x−n)^2} , x ∈ \Bbb R$. Prove that $F$ is not equicontinuous on $\Bbb R$ but equicontinuous on $[−a, a]$ for any $a ...
5
votes
0answers
64 views

$H^{1/2}$ function but not better

I am looking for an example of a function $f: [0, 1] \longrightarrow \mathbb{R}$ that is in the Sobolev space of order $1/2$, $H^{1/2}([0, 1])$, but not in the Sobolev space of order $1/2 + \...
5
votes
0answers
167 views

Existence of smooth extension of a function defined on a closed interval

Suppose $ f: [0,1] \rightarrow \mathbb{R}$ is a function such that derivatives of all orders exist ( at the end points of the interval the appropriate one-sided derivatives exist) and are continuous $ ...
5
votes
0answers
197 views

How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
4
votes
0answers
61 views

Prove that $f$ is invertible

Did I show enough to prove $f$ is invertible? Alternatively is there a more efficient way to do so? Thanks in advance for any help. Let $f : X \rightarrow Y $a nd $g : Y \rightarrow X$ be ...
4
votes
0answers
84 views

Why is the unit circle definition of trig functions not rigorous enough?

It has recently come to my attention that the usual unit circle derivation of the elementary trigonometric functions isn't considered rigorous enough. Apparently, this has to do with problems ...
4
votes
0answers
44 views

Find how many such complex numbers exist

Let $f:\mathbb{C}\to\mathbb{C}$ be defined by $f(z)=z^2+iz+1$. How many complex numbers $z$ are there such that $\text{Im}(z)>0$ and both the real and the imaginary parts of $f(z)$ are integers ...
4
votes
0answers
79 views

Compact sets of compact-open topology

Let $X$ and $Y$ be topological spaces,$X$ not compact and $Y$ metric, denote with $C(X,Y)$ the set of continuous functions between $X$ and $Y$ and put on $C(X,Y)$ the compact-open topology. My ...
4
votes
0answers
48 views

Find $f(x)$, when $\left[f''(x)\right]^2\cdot f(x)=\left[f'(x)\right]^3,\ \forall x\in[0,1]$

Let $f:[0,1]\to\mathbb{R}$ be a twice differentiable function with $f''(x)>0,\ \forall x\in[0,1]$, $f(0)=f'(0)=1$ and $\left[f''(x)\right]^2\cdot f(x)=\left[f'(x)\right]^3,\ \forall x\in[0,1]$....
4
votes
0answers
90 views

Why is cos at $\pi/2$ not undefined?

If the $\cos$ function is based off of the ratio of the adjacent side of Euclidean, right triangle, with fixed hypotenuse length (such as the unit circle), then how does this correspond to a defined ...
4
votes
0answers
75 views

If $f$ and $g$ are analytic functions, does $f \circ g - g \circ f = g- f$ implies $f = g$?

The question is in the title. There are actually two variants, one for real analytic functions and the other one for complex analytic functions. It came up as an attempt to solve this question by ...
4
votes
0answers
37 views

a weak notion of flow in a metric space

I am seeing the definition of flow in a metric space : $f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$ Note that the condition is ...
4
votes
0answers
36 views

Given a function $\phi$ and the set of iterates $(\phi^t)$ under self-composition, can you construct $\phi^t $ for a real continuous parameter?

$\phi$ is given as any function with the following properties: 1. $\phi$ is strictly increasing 2. $\phi$ is continuous 3. $\phi$ is a mapping of the interval $[0,1]$ into itself. 4. $\phi^n\circ\phi^...
4
votes
0answers
87 views

Mathematical Olympiad Problem

Let $\Bbb{R}$ be the set of real numbers. Determine all functions $f:\Bbb{R}\longrightarrow \Bbb{R}$ satisfying the equation $$f(x+f(x+y))+f(xy) = x + f(x+y)+yf(x)$$ for all real numbers $x$ and $y$.
4
votes
0answers
72 views

Finding period of a function? Am I doing something wrong?

I need to find the period of the function $$\large{\frac{\sin (\sin {nx})}{\tan{\frac{x}{n}}}}$$. According to me, the period of $\sin (\sin (nx))$ should be $\large{\frac{2\pi}{n}}$ and the period ...
4
votes
0answers
63 views

Proving $f$ cannot be onto

If $f$ maps finite sets $A$ to $B$ and $n(A) < n(B)$, prove that $f$ cannot be onto. Proof by contradiction: If $f: A→B$ and $n(A) < n(B)$, $f$ is onto. Since, by definition of a function, $...
4
votes
0answers
322 views

Prove equation has only one root in a specific interval

Prove that the following equation has only one solution in the interval $[-\text{min}(a_i), +\infty]$: $f(x) = \left(\sum_{i=1}^n \frac{1}{a_i + x}\right)\times \left(\sum_{i=1}^n \frac{a_i b_i}{(a_i ...
4
votes
0answers
139 views

Show that f is periodic if $f(x+a)+f(x+b)=\frac{f(2x)}{2}$?

Suppose $a$ and $b$ are distinct real numbers and $f$ is a continuous real function such that $\frac{f(x)}{x^2}$ goes to 0 when $x$ goes to infinity or minus infinity. Suppose that$ f(x+a)+f(x+b)=\...
4
votes
0answers
33 views

Function in Lipschitz space

I'm looking for a function that is in $W^{1,1}(0,1)$ but only in the Lipschitz space $\mathrm{Lip} (\alpha, L_2(0,1))$ for $0<\alpha < 1$. $\mathrm{Lip}(\alpha, L_2(0,1))$ is defined as the set ...
4
votes
0answers
40 views

How to read an expression that is ambiguous?

(1) How should I parenthesize $\log n \log \log n$? Also: (2) What general rule/rationale is used to do this parenthesization? To elaborate; I see why $\log\log n$ is unambiguous, but $\log n \log .....
4
votes
0answers
409 views

Inverse logarithmic integral

If the expansion of the logarithmic interval is$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$ what is the inverse of the function?
4
votes
0answers
92 views

How to efficiently calculate $ax+b$ once I know $a$ and $b$?

What's the cheapest way to calculate $ax+b$ several times once I know the values for $a$ and $b$? For instance, the cheapest way to calculate $ab+x$ several times once I know the values for $a$ and $...
4
votes
0answers
138 views

Find generating function For sequences

Can anyone out here help? The exercise says: "Find the generating function for each of the sequences below (the general term is given)" Now, the question is how do you find one for those: a) $U_n = 3$...
4
votes
0answers
215 views

Decrease of $L_1$ norm of piecewise constant functions after some “averaging”

In the course of a project, I ended up looking at what happens to the $L_1$ of real-valued piecewise-constant functions after some particular king of smoothing — but am currently stuck at a particular ...
4
votes
0answers
68 views

How prove this $f=C$ if $4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$

Question: if $f:\mathbb{Z}^2\to \mathbb{R}$ is bounded ,and for any $x,y\in \mathbb{Z}$,we have $$4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$$ show that $$f\equiv C$$ where $C$ is constant. My try:...
4
votes
0answers
189 views

Lower semicontinuous and discontinuous everywhere real bounded function?

Does there exist an $f:\mathbb{R}\rightarrow\mathbb{R}$ that is bounded such that for any $a$ then $f^{-1}(a,+\infty)$ is open but $f$ is discontinuous everywhere? Such a function seems too likely to ...
4
votes
0answers
65 views

Decomposability in the tensor product sense of functions of two variables

Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$. Let $F$ be a function $$ F:S\times T\longrightarrow K $$ where $K$ is a topological field ...
4
votes
0answers
173 views

How can I Create an integral that can only be evaluated via complex contour integration?

In Richard Feynman's book, Surely You're Joking Mr. Feynman!, he says: One time I boasted, "I can do by other methods any integral anybody else needs contour integration to do." So Paul [Olum]...
4
votes
0answers
137 views

How do you call functions that fulfill $f(x)=\pm f(\pm 1/x)$?

A function $f(x)$ that fulfills $f(x)=\pm f(-x)$ is called (a)symmetric even/odd. How do you call functions that fulfill $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$? "$\color{red}{\text{...
4
votes
0answers
144 views

Finding a series of functions orthogonal to all $x^{2n}$ but one

We need to find a series of functions $f_m$ verifying the property $$\int_{-\infty}^{+\infty}\! x^{2n} f_m\!(x) \;{\text d}x=\delta_{nm}.$$ We already have found $$f_0(x) = \frac{e^{-\sqrt{|x|}}\sin(\...
4
votes
0answers
116 views

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 ...
4
votes
0answers
51 views

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 ...
4
votes
0answers
215 views

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....
3
votes
0answers
43 views

Function with $f(f(n))=f(n-1)f(n+1)-f(n)^2$

Let $\mathbb{N}$ denote the set of positive integers. Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$f(f(n))=f(n-1)f(n+1)-f(n)^2$$ for all $n\geq 2$? If $f$ is linear, ...
3
votes
0answers
32 views

What are some standard methods for solving functional equations?

I have searched the internet for methods on solving functional equations, unfortunately, most of them consist mainly of substituting values for the variables or on guessing solutions. I think those ...