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 ...

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8
votes
2answers
171 views

What do logarithms distribute over?

I notice that division distributes over addition Root extraction distributes over multiplication What operator do logarithms distribute over: ie: what non-constant function $H \in C^2 \rightarrow C$ ...
6
votes
2answers
251 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, ...
5
votes
2answers
85 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 ...
2
votes
2answers
39 views

function equation with translation of independent variable

The following has come up in some work I'm doing: If $\frac{f(x+a)}{f(a)}=g(x)$ , where $g(x)$ is given and $a \geq 0$ is a constant, what is $f(x)$ ? We can assume that $g(x)>0 ~ \forall x$ . Of ...
1
vote
2answers
50 views

A scaling functional equation

I want to find a closed form of a function satisfying $$G(4z)=\frac{G(z)}{2z},$$ unfortunately I am not experienced with scaling problems, so I have no idea and would be thankful for any hints. ...
1
vote
2answers
98 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
0
votes
2answers
46 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
54 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
78 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
67 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 ...
-3
votes
2answers
248 views

find all function f,g that satisfy

Find all function $f,g$ that satisfy: $$g(x)-g(y)=\frac{1}{6} (x-y)(f(x)+f((x+y)/2)+f(y))$$ 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
1answer
25 views

Functional Equation and totally multiplicative functions

Find all $f:\mathbb{C}\rightarrow\mathbb{C}$ s.t. $\forall a, b\in \mathbb{C}, f(ab) = f(a)f(b)$. I could deduce a lot of things about what happens at the roots of unity, and 0, but I can't find out ...
0
votes
1answer
49 views

$f(x)=xf(1)$ Doubt

I just started learning Functional Equations and I was working on a problem that asked me to find all functions satisfying a certain condition, and at some point I got $f(x)=xf(1)$, is there a way to ...
0
votes
1answer
50 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
0
votes
1answer
32 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
0
votes
1answer
75 views

A functional equation for homogeneous functions of degree zero

Let a function $F: \textbf{R}^2_{++} \rightarrow \textbf{R}$ satisfy the relation: if $F(x,y)=F(x',y')$ then $F(x,y)=F(x+x',y+y')$. It is easy to prove that under the additional assumption of ...
0
votes
1answer
48 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
0
votes
1answer
112 views

More general reflection formula for Gamma function

It is know that $\Gamma(z)\, \Gamma{(1-z)}=\pi \csc( \pi z)$. Is there any formula for $\Gamma{(a+z)}\Gamma{(a-z)}$ where $a$ is a rational number, i. e., $a=p/k$ with $p, k$ integers and $z$ is ...
0
votes
1answer
51 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 ...