Elementary questions about functions, notation, properties, and operations such as function composition.
14
votes
0answers
235 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
176 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 ...
9
votes
0answers
363 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
88 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 ...
8
votes
0answers
263 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
364 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
175 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
339 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
290 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
315 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
67 views
Problem based on $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$
Let
$$f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$$
for all real $x$ and $y$. Also, $f'(0) = -1$ and $f(0)=1$. What is then the value of $f(2)$?
I got it as $-1$ by using some algebraic ...
6
votes
0answers
85 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
74 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
$$
\frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad ...
5
votes
0answers
142 views
Calculate $\sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \ldots $
$$
\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$$
I already know ...
5
votes
0answers
50 views
$H^{1/2}$ function but not better
I am looking for an example of a function $f: [0, 1] \longrightarrow \mathbb{R}$ that is in the Sobolev space of order $1/2$, $H^{1/2}([0, 1])$, but not in the Sobolev space of order $1/2 + ...
5
votes
0answers
202 views
Decrease of $L_1$ norm of piecewise constant functions after some “averaging”
In the course of a project, I ended up looking at what happens to the $L_1$ of real-valued piecewise-constant functions after some particular king of smoothing — but am currently stuck at a particular ...
5
votes
0answers
134 views
Existence of smooth extension of a function defined on a closed interval
Suppose $ f: [0,1] \rightarrow \mathbb{R}$ is a function such that derivatives of all orders exist ( at the end points of the interval the appropriate one-sided derivatives exist) and are continuous $ ...
4
votes
0answers
69 views
Mathematical Olympiad Problem
Let $\Bbb{R}$ be the set of real numbers. Determine all functions $f:\Bbb{R}\longrightarrow \Bbb{R}$ satisfying the equation $$f(x+f(x+y))+f(xy) = x + f(x+y)+yf(x)$$ for all real numbers $x$ and $y$.
4
votes
0answers
84 views
What function satisfy: $f(x)+f^{-1}(x)=2x$?
What function satisfy: $f(x)+f^{-1}(x)=2x$?
I have tried to substitute $x=f(x)$ to get $f^{(2)}(x)+1=2f(x)$ and subsequently plug in values to try to find $f(x)$ but to no avail.
Please help thank ...
4
votes
0answers
55 views
Finding period of a function? Am I doing something wrong?
I need to find the period of the function $$\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
45 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
58 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
256 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
125 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
28 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
36 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
111 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$$
where $C$ is constant.
My ...
4
votes
0answers
156 views
Lower semicontinuous and discontinuous everywhere real bounded function?
Does there exist an $f:\mathbb{R}\rightarrow\mathbb{R}$ that is bounded such that for any $a$ then $f^{-1}(a,+\infty)$ is open but $f$ is discontinuous everywhere?
Such a function seems too likely to ...
4
votes
0answers
59 views
Decomposability in the tensor product sense of functions of two variables
Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$.
Let $F$ be a function
$$
F:S\times T\longrightarrow K
$$
where $K$ is a topological field ...
4
votes
0answers
175 views
How do zeros on the complex plane affect the real number line?
Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
4
votes
0answers
158 views
How can I Create an integral that can only be evaluated via complex contour integration?
In Richard Feynman's book, Surely You're Joking Mr. Feynman!, he says:
One time I boasted, "I can do by other methods any integral anybody else needs contour integration to do."
So Paul ...
4
votes
0answers
133 views
How do you call functions that fulfill $f(x)=\pm f(\pm 1/x)$?
A function $f(x)$ that fulfills $f(x)=\pm f(-x)$ is called (a)symmetric even/odd.
How do you call functions that fulfill $f(x)=\color{blue}\pm f(\color{red}\pm 1/x)$?
...
4
votes
0answers
143 views
Finding a series of functions orthogonal to all $x^{2n}$ but one
We need to find a series of functions $f_m$ verifying the property
$$\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
122 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
214 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
35 views
$f$ is having maxima at $\frac12$ show that $f\circ f$ is having minima at $\frac12$.
$f:[0,1]\to [0,1]$
$f(0)=0=f(1)$
$f$ is having local maxima at $x=\frac12$.
Show that $f\circ f(x)$ is having local minima at $x=\frac12$.
Using chain rule I was only able to find that $(f\circ ...
3
votes
0answers
18 views
Domain values of inverse funtion
If I'm plotting $$y=3e^{{x\over3}+1}$$ from $x=0$ to $x=1$ and on the same axes I want to plot its inverse $$y=3\ln\left({x\over3}\right)-3$$ but only for the domain values of $x$ given by the range ...
3
votes
0answers
35 views
Is there a name for this property of multiplication (and other functions)?
Suppose $x,y \in \mathbb{R_+}, x<y$, and $ 0 < \varepsilon \leq (y-x)/2$. It seems to me that $xy < (x+\varepsilon)(y-\varepsilon)$ and equivalently that $(x+\varepsilon)(y-\varepsilon)$ is ...
3
votes
0answers
24 views
Characterize in terms of fibre
I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you.
For a function $f:X\to Y$, and y an element of ...
3
votes
0answers
75 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
24 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
72 views
Real Analysis: Given $A_1$, $A_2$, $A_3$, which set is open, which is closed?
a. $A_1 = \{x: \sin(1/x) =0\}$
Clearly $\sin(1/x) = 0$ for $x = \frac{1}{n\pi}$, what can we say about this set?
b. $A_2 = \{x: x\sin(1/x)= 0 \}$
Again $\sin(1/x) = 0$ for $x = \frac{1}{n\pi}$, is ...
