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

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14
votes
0answers
219 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$: (Mathematica code) ...
11
votes
0answers
165 views

Functional equation $f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}$

Problem: Find all continuous real-valued functions $f$ such that $$f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}.\tag{1}$$ Here $f$ is allowed to be defined only on a subset of $\mathbb{R}$. The only ...
8
votes
0answers
256 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
8
votes
0answers
360 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 ...
8
votes
0answers
357 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
60 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
7
votes
0answers
166 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 ...
7
votes
0answers
334 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 ...
7
votes
0answers
287 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
312 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
84 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 ...
6
votes
0answers
56 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
50 views

What is the actual significance of the lambda calculus for the formalization of math?

The Simply Typed Lambda Calculus was proposed initially as a foundational system for the formalization of mathematics. As such, I would expect that soon there would be attempts to implement most of ...
6
votes
0answers
251 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 ...
6
votes
0answers
2k 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 ...
5
votes
0answers
135 views

Apartness of reals and algorithm exctraction

I am trying to wrap my head around the notion of apartness in constructive mathematics and it turns out I lack understanding miserably. I would like to use as elementary notions as possible, in the ...
5
votes
0answers
127 views

Calculate $\sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \ldots $

$$ \sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}+ \frac{1}{2}\sqrt{\frac{1}{2}}}} \times\ldots$$ I already know ...
5
votes
0answers
49 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
202 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 ...
5
votes
0answers
132 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 $ ...
4
votes
0answers
63 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
82 views

What function satisfy: $f(x)+f^{-1}(x)=2x$?

What function satisfy: $f(x)+f^{-1}(x)=2x$? I have tried to substitute $x=f(x)$ to get $f^{(2)}(x)+1=2f(x)$ and subsequently plug in values to try to find $f(x)$ but to no avail. Please help thank ...
4
votes
0answers
35 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
4
votes
0answers
51 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
38 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 ...
4
votes
0answers
53 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 ...
4
votes
0answers
235 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
120 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$ ...
4
votes
0answers
27 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
35 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
90 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
104 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 = ...
4
votes
0answers
66 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 ...
4
votes
0answers
151 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
57 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
174 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
156 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 ...
4
votes
0answers
128 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 $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$? ...
4
votes
0answers
143 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) = ...
4
votes
0answers
152 views

What is this function called (looks like a variant of the exponential function)

We set (as usual) $\displaystyle{x \choose k} := \frac{x \cdot (x-1) \cdots (x-k+1)}{k!}$ for $x\in \mathbb{C}$. Now we can define a function $\displaystyle f(x) := \sum\limits_{k=0}^\infty {x ...
4
votes
0answers
121 views

What is the limit of the quotient of Stieltjes Constants?

What is the limit : $\displaystyle\lim_{n\to \infty }\frac{\gamma_{n-1} }{\gamma_{n}} $ Here $\gamma_{n} $ is the $n$-th Stieltjes Constant.
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
213 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 ...
3
votes
0answers
64 views

Find two homeomorphic topological spaces and a bijective continuous map between them which is not homeomorphism.

I'm aware that it is duplicate, but I'd like to know whether my example is appropriate or not. Let our function $f$ be on the set $\mathbb{Q}\cap\mathbb{Z}$ induced by standard topology of a line. ...
3
votes
0answers
23 views

an integral equation in a function with two arguments

Say we are given $C(s,t)=\min(s,t)+\zeta st$. How can we solve $$g(s,t)=C(s,t)+\lambda \int_0^1g(s,u)C(u,t)du.$$ Looking into some text books on integral equations I see that most of the kernels, $C$, ...
3
votes
0answers
32 views

Technical name for an almost-monotonic function

I'm wondering if there’s a technical name or short phrase that describes a function that’s monotonic, subject to some uniformly bounded amount of backtracking. $\exists \epsilon \forall x , y : y \gt ...
3
votes
0answers
55 views

Looking for a bound on a function involving $\sinh$

Fix $T > 0$ and let $t \in (0,T)$ let $c > 1$ be a constant (which may be bigger than $T$). Consider the function $$f(c,t,T) = \frac{\sinh ((T-t)c)}{\sinh (Tc)}.$$ I am looking for a bound of ...
3
votes
0answers
69 views

Proving injectivity of a multivariable function

Let I denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$$ ...
3
votes
0answers
31 views

Composition of functions injective implies one of them is injective?

Suppose we have $f: X \to Y $ and $g:Y \to Z $ maps. Prove that: $g \circ f $ injective $\implies$ $f$ injective $g \circ f $ surjective $\implies $ $g$ is surjective Attempt: Pick ...