Tagged Questions

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 ...

136 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really stuck ...
134 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
72 views

What are the densities of branches of the euclidean tree?

The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$. (or ...
101 views

Find the least possible value of $n$ such that there exist $P(x), Q(x) \in \mathbb{Z}[x]$

Find the least possible value of $n, n \geq 2015$ such that there exists polynomial $P(x)$ with degree $n$, integer coefficients, the coefficient of the term $x^n$ is positive and polynomial $Q(x)$ ...
132 views

199 views

59 views

Well, similar questions have already been asked. But they are not identical and the solution methods offered there are not the same one as here. Anyway, I want to solve functional equation $f(m + f(n))... 0answers 52 views What can be said about a function with rotational symmetry of order other than 2? It is well known that an odd function is a function whose graph has rotational symmetry of order$2$(about the origin). Suppose the graph of$f:U \to \Bbb{R}$has rotational symmetry of some higher ... 0answers 396 views What special role plays the function$\pi^{\frac x\pi}$in analysis? I have tried to redefine some special functions in the most "natural" way, that is the way which allows to simplify the relations the most. I would call these functions "parelementary". The ... 0answers 61 views Is the product rule for logarithms an if-and-only-if statement? If a function$f(x)$is proportional to$\ln x$, then we know $$f(xy) = f(x) + f(y).$$ My question is, Is the converse true? If we know that, for an unknown function f, $$f(xy) = f(x) + f(y),$$ ... 0answers 116 views How to solve this finite-difference equation? How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say$f(x) = f_0(x)$for$x \in [0, \sqrt{2})$is a given function. ... 0answers 37 views Convergence of sum of antiderivative and derivative This question is inspired by this question: Solutions for$ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The question:... 0answers 84 views Is this a field of study? Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ... 0answers 75 views Solving functional equation Problem:find all continuous functions$f:[0,+\infty)\to [0,+\infty)$such that $$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$$ 0answers 56 views Could we compute$P(t^2)$? Let$P$be an operator such that$P(kx)=kP(x)$,$k \in \mathbb{C}$,$x$is a variable,$P(xy)=P(xP(y))+P(P(x)y)-P(x)P(y)$,$x, y$are variables. All variables commute. Let$P(t)=t$. Then$P(t^2)=2P(...
Define an extended algebraic function $f(a)$ as a function on $a$ that utilizes any combination of recursive extensions and inverses of sequentiation. Example: $a + 1$ , sequentiation. $a + a$, ...