All Questions
Tagged with ca.classical-analysis-and-odes functional-equations
8 questions
31
votes
1
answer
2k
views
$f'=e^{f^{-1}}$, again
This question is a spin-off of this one, in which the OP asks whether there is a solution $f:\mathbb R\to\mathbb R$ of the functional equation (not exactly an ODE) $f'=e^{f^{-1}}$, where $f^{-1}$ is ...
- 5,169
13
votes
6
answers
4k
views
Finding f such that f(f(x))=g(x) given g
Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...
- 15.4k
12
votes
4
answers
831
views
Relating the roots of polynomials to the solution sets of certain functional equations
Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in \mathbb{R},\;\text{and}\;f^0=\text{...
- 453
7
votes
4
answers
827
views
$n$th root of $(a,b) \mapsto (\operatorname{gm}, \operatorname{am})$
Suppose $0 < a < b$, and let GM and AM be respectively the geometric and arithmetic means of $a$ and $b$. Does the mapping $(a,b) \mapsto (\operatorname{GM}, \operatorname{AM})$ have a well-...
5
votes
1
answer
761
views
Is it possible to express the functional square root of the sine as an infinite product?
Cross-post from MSE.
It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that ...
- 4,829
5
votes
2
answers
309
views
A functional equation for a family of functions indexed by the symmetric group $S_3$
$\newcommand{\C}{\mathbb C}$A question asked recently was as follows:
For the symmetric group $G:=S_3$, is it possible to construct functions $t_g\colon\C\to\C$ that satisfy the convolution identity
...
- 128k
5
votes
1
answer
667
views
solution of functional equation $f^{\circ k}(x) = x$
The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]:
...
- 183
2
votes
2
answers
416
views
What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?
What is the solution, $f(n)$, of the following functional equation:
$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$
where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...
- 153