The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...
5
votes
2answers
72 views
$f,g:[a,b]\to [a,b]$ are continuous, $f\circ g=g\circ f$ and $f$ is 1-1. Show that $\exists t\in [a,b]$ so that $f(t)=g(t)=t$
Using the Intermediate Value Theorem on $h(x)=f(x)-x$ and $r(x)=g(x)-x$ I can easily show that $\exists t_f,t_g\in[a,b]$ so that $f(t_f)=t_f$ and $g(t_g)=t_g$. It remains to show that $t_f=t_g$. Since ...
5
votes
2answers
178 views
Solving the functional equation $f(x) = f(x / \phi) f(x / \phi^2 - 1)$
I'm trying to find a function $f(x)$ such that the spacing between consecutive roots looks like the infinite Fibonacci word:
$$1, \phi^{-1}, 1, 1, \phi^{-1}, 1, \phi^{-1}, 1, 1, \phi^{-1}, 1, 1, ...
4
votes
2answers
155 views
I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)$. I know that there are other questions that are asking the same thing, but I'm trying to figure this out ...
0
votes
2answers
35 views
Some functional equations in two variables
I have two questions.
i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation
$$
|f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|}
$$
has a solution ...
0
votes
2answers
46 views
$f(\vec{x_1}+\vec{x_2})=f(\vec{x_1})f(\vec{x_2})$
What is the general solution to $f(\vec{x_1}+\vec{x_2})=f(\vec{x_1})f(\vec{x_2})$ where $\vec{x}$'s are in discrete vector space $x\in \{n_1\vec{e_1}+n_2\vec{e_2}+n_3\vec{e_3},n_1,n_2,n_3 \in Z\}$?
0
votes
2answers
70 views
General aggregation functions
Is there a way to find all or some functions which "aggregate" numbers and are non-isomorphic to addition. I mean functions which are commutative and associative:
$f(x,y)=f(y,x)$
...
-1
votes
2answers
57 views
Context problems of Number theory and functional equation
I can't solve the following problems, please help.
1) Find all primes $p$ and $q$ such that $p^q+q^p$ is a prime.
2) Solve $2^x+3^y=z^2$ in integers.
3) Find all $f: \mathbb{Q} \rightarrow ...
6
votes
1answer
527 views
Equation about generating functions and subfactorial
Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
4
votes
1answer
75 views
Uniqueness of solution of functional equation
I have a function $f(x_1,x_2) \colon \mathbb{R}^2_{+} \to \mathbb{R}_{+}$ positive homogenous: $$
f(\lambda x_1, \lambda x_2) = \lambda f(x_1,x_2), \; \forall \lambda > 0
$$
and such that ...
1
vote
1answer
114 views
Cauchy functional equation
Is there $U\subset \Bbb R^2$ with Lebesgue measure $0$ such that
$$f(x+y)=f(x)+f(y)$$
for all $(x, y)\in U$ implies $f(x+y)=f(x)+f(y)$ for all $(x, y)\in\Bbb R^2$ ?
1
vote
1answer
94 views
A functional equation related to the exponential function
Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow
\mathbb{R}$ are two functions and satisfies the following relation
\begin{equation}
f(xy)=f(y)^{g(x)}
\end{equation}
...
1
vote
1answer
152 views
Analytic solution of a certain functional equation
Here's my question. Let $b_2$, ..., $b_d \in \mathbb{C}$ ($d$ is an integer greater than 2), and consider the functional equation
$$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$
...
0
votes
1answer
42 views
inverse function , asymptotics ..
let be a function given by $ f^{-1} (x) = \sqrt x + G(x) $
with $ G(x)= \sum_{n=0}^{N}a_{n} \sin(nx+ \pi/4) $ finite fourier series with N big
my questio is , if for $ x \rightarrow \infty$ the ...
-2
votes
1answer
137 views
find all function f,g that satisfy
Find all function $f,g$ that satisfy: $$g(x)-g(y)=1/6 (x-y)(f(x)+f((x+y)/2)+f(y))$$
For $y=0$ we have an equation in $f$: $$4(x-y)(f(x/2)-f(x/2+y/2))=xf(y)-yf(x)$$ How can i do it?
0
votes
0answers
76 views
solution of d’Alembert’s equation.
i know that equation for d’Alembert’s equation. is looking so
$g(x+y)+g(x-y)=2*g(x)*g(y)$
so am trying to find actual solution for this equation,first i took $x=y=0$ and i got
$2*g(0)$=$2*g(0)^2$ ...
0
votes
0answers
105 views
Could anyone derive a formula for this?
Edited:
I want to get a sentiment score of various sentences and I've tried coming up with an equation that could satisfy the conditions that are inherent to each sentence (It's estimated mood as ...
0
votes
0answers
55 views
Additivity of averages
Let us suppose that we have a system $(A,B)$ consisting of two independent subsystems $A$ and $B$. Suppose that $A$ has $m$ states having energies $E_1^{(A)}, \dots, E_m^{(A)}$ with probabilities ...
0
votes
0answers
187 views
$ f(nx)=f(x), \qquad n \in \mathbb Z^+$
Let $f$ satisfy the following equation,
$$ f(nx)=f(x), \qquad n\text{ a positive integer}.$$
Then I know that the most general solution is
$$ f(x)= C_{+}x^{2\pi i m/\log n}+C_{-}x^{-2\pi i m/\log ...
