Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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14
votes
1answer
863 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 formal-power-...
13
votes
0answers
312 views

Find a function such that $f^{-1}=f'$

Let $f:\Bbb{R}^+\rightarrow\Bbb{R}^+$ be a differentiable bijection and let $f$ satisfy: $f'=f^{-1}$ (where $f^{-1}$ denotes the inverse of $f$). Find $f$. This comes from a facebook page "...
11
votes
3answers
553 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, \phi^...
10
votes
0answers
125 views

$f(f(x)) = 1 + x^2$, then what is f(1)?

I get $f(f(a)) = a^2 + 1 = f(f(-a))$, and so $f(a)^2 + 1 = f(a^2 + 1) = f(-a)^2 + 1$, so $f(a) = f(-a)$ or $f(a) = -f(-a)$, but then I donot know what to do next. Thanks for any help.
10
votes
0answers
234 views

Non-trivial solutions of a functional equation $(f \circ f \circ f)(x)=x$

I have come to this from the topic of self-inverse functions. Let's consider a more complicated case where: $$(f \circ f \circ f)(x)=x \tag{1}$$ Also assume that $f(x)$ is continous on some non-...
10
votes
2answers
218 views

Find all the functions $f: \mathbb{N} \to \mathbb{N} $ such that $(m+f(n))(n+f(m))$ is a perfect square for all $m,n$

Let $N$ be the set of natural number. Find all functions $f: \mathbb N \to \mathbb N$ , such that the number $(m+f(n))(n+f(m))$ is perfect square for all natural numbers $m$ and $n$. I was unable to ...
8
votes
0answers
175 views

Existence of function satisfying $f(f'(x))=x$ almost everywhere

My project is to Study the existence of a continuous function $f : \mathbb{R} \rightarrow \mathbb{R}$ differentiable almost everywhere satisfying $ f\circ f'(x)=x$ almost everywhere $x \in \mathbb{R}$...
8
votes
0answers
117 views

Uniqueness of solutions of a functional equation

Problem: Find all continuous and strictly increasing functions $f\colon(0,\infty)\to\Bbb R$ with $$f(3x)-f(2x)=f(2x)-f(x)$$ for all $x>0$. A class of solutions is given by $f(x)=ax+b$, where $a>...
8
votes
0answers
174 views

Is this $f(x) = x+1$ the only solution to this functional equation.

I am considering the problem of finding all functions $f:(0,\infty)\rightarrow(0,\infty)$ satisfying the functional equation: $$f\big(xf(y)+f(x)\big) = 2f(x)+xy.$$ I have been able to prove the ...
8
votes
1answer
199 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 $Q(x)$ ...
8
votes
0answers
229 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 ...
7
votes
0answers
154 views

Imaginary part of expression involving complete elliptic integral

Suppose an expression $$ \tag 1 G(z) = \frac{1}{\sqrt{(z-1)^{3}(z+3)}}\text{K}\left( \sqrt{\frac{16z}{(z+3)(z-1)^{3}}}\right), $$ where $$ K(x) \equiv \int \limits_{0}^{\frac{\pi}{2}}\frac{dy}{\sqrt{1 ...
7
votes
1answer
270 views

Continuous functional equation

Problem. Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that for all $x \in \mathbb{R}$, we have $$f(1-x) = 1 - f(f(x)).$$ I have obtained that $f(x) = f(f(f(f(x))))$ for all $x ...
7
votes
0answers
80 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 ...
6
votes
1answer
101 views

Differential Equation with inverse function $\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$

$$\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$$ I know $f(x) = ax+b$ is a solution. How can I find other solutions?
6
votes
1answer
127 views

Find all functions that satisfy $f(x+f(y))=f(x)-y$

here is the problem Here is my solution : $x=y=0$ gives $f(f(0))=f(0)$ $x=0; y=f(0)$ gives $f(f(f(0)=0=f(0)$ (because $f(f(0))=f(0) \iff f(f(f(0)))=f(f(0))=f(0)$) $x=0$ gives $f(f(y))=-y$ $x=0 ; ...
6
votes
1answer
149 views

Is it possible to find aproximation of conformal map from sequences of complex points?

I want to find equation of conformal map (= Fatou function $\Psi : z \to u$ ) which: maps some region of complex plane ( attracting petal) to right half of complex plane in u coordinate $Re(u) > ...
6
votes
0answers
85 views

Functional division $\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$

As the title suggests, the problem here is: Find all functions $f:\mathbb{Z}\to\mathbb{N}$ such that, for every $x,y\in\mathbb{Z}$, we have $$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$$ I ...
6
votes
0answers
382 views

Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for $$ \lim_{n\to-\infty}\sum_{i=n}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)} $$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + \frac{...
6
votes
1answer
175 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 $f(x_1,...
5
votes
0answers
50 views

Does there exist a function $f_{\Box,\Box}(\Box)$ making the formula $a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$ true?

Let $a$ and $b$ denote the resistances of two resistors. If they're put in series, the total resistance is $a+b$. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{...
5
votes
0answers
157 views

Solution of advanced functional differential equation

Statement Consider an advanced functional differential equation $$ Lf(x) = f(2x+\pi)+f(2x-\pi),\quad L\equiv\frac{d^2}{dx^2}+1. \tag{1} $$ Let's construct a solution of Eq. $(1)$ with finite ...
5
votes
2answers
76 views

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
5
votes
0answers
84 views

Does there exist a function $f$ such that $f(x)$ is an integer for only finitely many values of $x$?

Define $f : \mathbb{R} \to \mathbb{R}$ such that $f(x)\leq f(y)$ whenever $x\leq y$ and $f^{2018} (z) \in \mathbb{Z}$ $\forall z \in \mathbb{R}$. Does there exist a function $f$ such that $f(x)$ is an ...
5
votes
0answers
149 views

A nontrivial solution for $ f(f(x)) = \exp(\exp(x)) $

Consider the equation $$ f(f(x)) = \exp(\exp(x)) $$ Valid for all real $x$, $f(x) \neq \exp(x)$ (not identically equal everywhere), $f(x)$ is analytic and $$ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 +...
5
votes
1answer
97 views

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$. What I did was first plug in $x = 1$ to get $2f(1) = 1 \implies f(1) = \frac{1}{...
5
votes
0answers
110 views

Increasing function $f(x)$ such that $f(\gcd(x,y))=\gcd(f(x),f(y))$

This problem was largely inspired by this problem here. There were many counterexamples given to the problem, such as a multiplicative function that maps primes to a permutation thereof. However, if ...
5
votes
0answers
193 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $...
5
votes
0answers
879 views

Width of the Eiffel Tower as a function of height?

In the preface of Advanced Engineering Mathematics, 2nd Ed. by Zill and Cullen, it is claimed that the function relating the width of the Eiffel Tower as to the distance from its top, $x \mapsto f(x)$,...
5
votes
0answers
389 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function $f:...
4
votes
0answers
85 views

If $f: \mathbb R^n \to \mathbb R^n$ satisfies $\|f(y)\| = \|f(x + y) - f(x)\|$, is $f$ additive?

Question Main question: Let $\| \cdot \|$ be a norm on a finite-dimensional real vector space $V$. If $f : V \to V$ is a function satisfying $$ \| f(y)\| = \|f(x + y) - f(x)\| $$ for all $x, y ...
4
votes
1answer
170 views

Approximation of $\sum_{n=0}^\infty 2^{-n} \tanh(3^n x)$ in $x = 0$

I want to (asymptotic function) approximate the series $f(x) = \sum_{n=0}^\infty 2^{-n} \tanh(3^n x)$ around $x = 0$. I found the equation for $f(x)$: $$f(x) = \dfrac{1}{2}\ f(3x) + \tanh(x)$$ for $...
4
votes
1answer
76 views

For fixed $n \in \mathbb{R}$, what are all the continuous functions that satisfy $nf(x) = f(nx)$?

For fixed $n \in \mathbb{R}$, what are all the continuous functions that satisfy $nf(x) = f(nx)$? I thought it would just be functions of the form $f(x) = kx$ but, for example, in the $n=2$ case we ...
4
votes
1answer
45 views

Find all functions such that if $I$ open bounded interval then $f(I)$ is also open of same length

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for all $I$ open bounded interval it follows $f(I)$ is also an open bounded interval of same length as $I$. It's easy to see $...
4
votes
2answers
176 views

Functional equation : $ f(1)^3 + f(2)^3 + \ldots + f(n)^3 = (f(1) + f(2) + \ldots + f(n))^2$

Find all function $f:\mathbb{N}\rightarrow\mathbb{N}\cup\{0\}$ satisfying $$ f(1)^3 + f(2)^3 + \ldots + f(n)^3 = (f(1) + f(2) + \ldots + f(n))^2$$ $\forall n \in \mathbb{N}$. Thank you, ...
4
votes
0answers
62 views

Functional equations in the form $f(x)-f(x-c)=P_n(x)$

Consider $f:\mathbb R\to\mathbb R$, and let $P_n$ be a polynomial with degree $n\ge 1$. Now, if $$f(x)-f(x-c)=P_n(x)$$ for some fixed constant $c$, then what assumptions on $f$ could let us ...
4
votes
0answers
202 views

Solving the functional equation $f(xf(x)+yf(y))=xy$ over positive reals

Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for all positive real numbers $x,y$ the following is true : $$f(xf(x)+yf(y))=xy$$
4
votes
1answer
127 views

Tricky composite function

Let $f: A\to A$ where $A\in (-1,\infty )$ and $f(x + f(y) + xf(y)) = y+f(x)+yf(x)$ for all $x,y \in A$ and $\frac{f(x)}{x} $ is strictly increasing for all $x \in (-1,0) \cup (0,\infty)$. Then find $...
4
votes
0answers
123 views

Solve this integral equation that results in a linear function

I need to find the family of real-valued single variable functions $F:\, (0,1) \to [0,1]$ that satisfy the following integral equation: $$\int_{\theta = 0}^{\pi} F\big(~ x\, \sin\theta ~\big)\text{d} \...
4
votes
0answers
132 views

Function with $f(f(n))=f(n-1)f(n+1)-f(n)^2$

Let $\mathbb{N}$ denote the set of positive integers. Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$f(f(n))=f(n-1)f(n+1)-f(n)^2$$ for all $n\geq 2$? If $f$ is linear, ...
4
votes
0answers
106 views

Functional equation: $f(x,y)=f(x+y,y)=f(x,x+y)$

Is there a nonconstant continuous function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ satisfying the functional equations $f(x,y)=f(x+y,y)=f(x,x+y)$? If the answer is yes, can we characterize all ...
4
votes
1answer
91 views

Functional equation with only two solutions?

Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation: $$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$ Unfortunately, I was able to show only ...
4
votes
0answers
67 views

Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...
4
votes
0answers
105 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, $$ f(xy) = f(x) + f(y), $$ ...
4
votes
1answer
100 views

Evaluation of $\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d\!}x$ for $\mu>0$

I'm trying to evaluate the integral $$ \Psi(\mu,\nu)=\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d}\!x\qquad(\text{for}\; \mu>0)\tag 1 $$ where $\nu\in\Bbb R$ and $f$ ...
4
votes
0answers
513 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
4
votes
0answers
174 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. ...
4
votes
0answers
297 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} \psi(\alpha)-\...
4
votes
0answers
63 views

Wronskian different from zero and solutions of ODE.

Let $a_0, \ldots , a_{n-1}$ continuous functions in an interval $I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let $\phi_1, \phi_2, \ldots,\phi_n$ $n$ are ...
4
votes
1answer
114 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 try:...