Questions tagged [functional-equations]

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55 votes
14 answers
10k views

Does any research mathematics involve solving functional equations?

This is a somewhat frivolous question, so I won't mind if it gets closed. One of the categories of Olympiad-style problems (e.g. at the IMO) is solving various functional equations, such as those ...
Qiaochu Yuan's user avatar
51 votes
4 answers
17k views

Function satisfying $f^{-1} =f'$

How many functions are there which are differentiable on $(0,\infty)$ and that satisfy the relation $f^{-1}=f'$?
C.S.'s user avatar
  • 4,735
45 votes
10 answers
10k views

The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$ (so $c_0=0$ is imposed). First things that ...
Pietro Majer's user avatar
  • 56.5k
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 ...
Fan Zheng's user avatar
  • 5,119
29 votes
3 answers
2k views

Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$, ...
Alexandre Eremenko's user avatar
28 votes
4 answers
2k views

For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant?

Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true : $$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - ...
Aditya Guha Roy's user avatar
27 votes
1 answer
7k views

Are there any non-linear solutions of Cauchy's equation $f(x+y)=f(x)+f(y)$ without assuming the Axiom of Choice?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be s.t. $f(x+y) = f(x) + f(y), \ \forall x, y$ It is quite obvious that this implies $f(cx)=cx$ for all $c \in \mathbb{Z}$ and even further: $\forall c \in \...
Ignas's user avatar
  • 583
20 votes
9 answers
10k views

What is the indefinite sum of tan(x)?

What is the indefinite sum of the tangent function, that is, the function $T$ for which $\Delta_x T = T(x + 1) - T(x) = \tan(x)$ Of course, there are infinitely many answers, who all differ by a ...
Herman Tulleken's user avatar
18 votes
3 answers
4k views

Is there a general solution for the differential equation $f''(x) = f(f(x))$?

I'm currently an undergraduate studying differential equations and I've been fixated on the differential equation $f''(x) = f(f(x))$ for the past 2 days. I can't seem to crack it but it feels like it ...
Trevor Krumrine's user avatar
16 votes
3 answers
3k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
zeraoulia rafik's user avatar
16 votes
2 answers
2k views

Getting a differential equation for a function from a functional equation of its Mellin transform

If $f$ is a locally integrable function then its Mellin transform $\mathcal{M}[f]$ is defined by $$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$ This integral usually converges in a ...
Armin Straub's user avatar
  • 1,372
14 votes
2 answers
1k views

Modular equations for quasimodular forms

This problem is motivated by this question and by teaching modular polynomials for the classical modular invariant $j(\tau)$. The latter implies that if we consider the fields of modular functions $\...
Wadim Zudilin's user avatar
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 ...
David Corwin's user avatar
  • 15.1k
12 votes
4 answers
827 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{...
ocg's user avatar
  • 453
12 votes
2 answers
666 views

Series defined by a fixed-point functional equation

Description I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...
Samuele Giraudo's user avatar
11 votes
2 answers
829 views

What are the iterates of $x \mapsto 1 - \sqrt{1-x^2}$?

Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
Gro-Tsen's user avatar
  • 29.9k
11 votes
2 answers
1k views

Which trigonometric identities involve trigonometric functions?

Another question that's getting no answers on stackexchange: Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled ...
Michael Hardy's user avatar
11 votes
3 answers
241 views

How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?

Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...
Tim Campion's user avatar
  • 60.6k
10 votes
1 answer
718 views

A conjecture about certain values of the Fabius function

The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations $$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$ and $$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\...
Vladimir Reshetnikov's user avatar
10 votes
2 answers
597 views

Proving the simple form of a function from statistical mechanics

I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
LuckyJollyMoments's user avatar
10 votes
3 answers
839 views

Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses

The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...
Ivan Meir's user avatar
  • 4,782
10 votes
2 answers
471 views

A functional equation involving the inverse function

$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
Iosif Pinelis's user avatar
9 votes
4 answers
2k views

How may I find all continuous and bounded functions g with the following property?

Find all continuous and bounded functions $g$ with : $$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$ I have posted this question here, but received no answer.
Dattier's user avatar
  • 3,737
9 votes
1 answer
355 views

Some nice functional equations for $q$-continued fractions

Given $\large q=e^{2\pi i \tau}$. Define, $$\alpha(\tau) = \sqrt2\,q^{1/8}\prod_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$ $$\beta(\tau) = q^{1/5}\prod_{n=1}^\infty\frac{ ...
Tito Piezas III's user avatar
8 votes
2 answers
686 views

Solving functional equation $f(xy)=f(x+y)$ and Diophantine equations

It is well known that the only solution is $f$ a constant function. However, by putting some restrictions on the functional equation, we might get other solutions, with potential implications to ...
Vincent Granville's user avatar
8 votes
1 answer
640 views

Hahn-Banach theorem with real extended valued function

Hello to everyone, My problem is the following: I have this version of the Hahn-Banach theorem: Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any convex function. Let $W$ be a vector ...
alef87's user avatar
  • 83
8 votes
2 answers
381 views

Linearizing a power series by conjugation

Let $\mathfrak{I}:=\big\{ \, f:=\sum_{k=0}^\infty f_k z^k \in\mathbb{C}[[z]]\; : \text{s.t. }\; f_0=0 \;\text{ and }\; f_1=1\big\}$. A most basic result about linearization states that, for any $f\...
Pietro Majer's user avatar
  • 56.5k
8 votes
1 answer
454 views

Functional equation with Fourier transform and $\frac{1}{x} f(\frac{1}{x}) $

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation: $$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$ $\...
Bertrand's user avatar
  • 1,121
8 votes
1 answer
481 views

$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?

While talking about tetration with my friend the following idea (re)occured. $$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$ or variations of it like the weaker $$f(f(f(f(z)))) = z ,\...
mick's user avatar
  • 703
8 votes
0 answers
178 views

Zappa-Szép products of the group of integers with itself

Since my previous question didn't get much attention and I couldn't make any relevant progress on it, I thought it would be a good idea to "simplify" it by replacing monoids by groups. That is: ...
HeinrichD's user avatar
  • 5,402
7 votes
2 answers
800 views

On the consistency of the definition of the conductor for automorphic forms

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners: By its ...
Desiderius Severus's user avatar
7 votes
4 answers
819 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-...
Michael Hardy's user avatar
7 votes
2 answers
185 views

Characterizing when matrices are 'dissipative'

An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...
L. T. P. L.'s user avatar
7 votes
1 answer
149 views

On the functional equation $f(xf(y))=\frac{f(f(x))}y$ on arbitrary groups

In this answer, it was shown that there is no function $f\colon\mathbb Q_{+}^{*}\to\mathbb Q_{+}^{*}$ such that \begin{equation} f(xf(y))=\frac{f(f(x))}y \label{1}\tag{1} \end{equation} for all $x$...
Iosif Pinelis's user avatar
7 votes
0 answers
376 views

General solution of the Multiplicative symmetry equation $f(xf(y))=f(f(x)y)$ in nonabelian groups

As we know, the functional equation $f(xf(y))=f(f(x)y)$ was completely solved in abelian groups (by J. G. Dhombres, Solution... $f(x\ast f(y))=f(y\ast f(x))$, Aequationes Math. 15 (1977), 173--193, ...
M.H.Hooshmand's user avatar
6 votes
1 answer
565 views

Does it follow that $F^{(1)}(z)=F^{(2)}(z)$ for all $z \in \mathbb H$?

Let $\mathbb H= \{z \in \mathbb C\,:\, \textrm{Re}\,(z)\geq 0\}$ and for $j=1,2$ suppose that $F^{(j)}:\mathbb H\to \mathbb H$ is defined via $$ F^{(j)}(z) = \sum_{k=1}^{\infty} \frac{a_k^{(j)}}{z+\...
Ali's user avatar
  • 4,067
6 votes
1 answer
155 views

A second-order recursion (functional equation)

In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion, $$L(s+1)=L(s)+2s(2s+1)L(s-1).$$ Is it familiar to someone ? Is ...
Denis Serre's user avatar
  • 51.5k
6 votes
1 answer
495 views

What is known about the functional square root of the Riemann zeta function?

Let us consider the Riemann zeta function $\zeta(s)$, where $s$ can take on values on the domain $\mathbb{R}_{>1}$: $$\zeta(s) := \sum_{n=1}^{\infty} \frac{1}{n^{s}} .$$ I wonder what is known ...
Max Muller's user avatar
  • 4,485
6 votes
1 answer
723 views

On a generalization of the classical Cauchy's functional equation

I start with some known preliminaries on the problem: Classical result. The one-dimensional Cauchy functional equation $$ \forall x,y \in \mathbb{R}, \,\,\,f(x+y)=f(x)+f(y) $$ with $f:\mathbb{R}\to \...
Paolo Leonetti's user avatar
6 votes
1 answer
3k views

Techniques to solve equations involving a definite integral [closed]

Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, $\alpha\in\...
pbs's user avatar
  • 243
6 votes
1 answer
693 views

Is this method of "fractional sums" using a Fourier series viable?

Hi. I have this idea about developing what I call a "continuum sum", that is, a method to "add up a non-integer number of terms", i.e. to see if there is a "natural" way to assign a meaning to the ...
The_Sympathizer's user avatar
6 votes
1 answer
497 views

"inversion" of a convolution

I have the following relation: $$ \sum_{d|n} (1+1/x)^{d-1} F_{n/d}(x^d)=L_n(x) $$ where the right hand side is (for every $n$) a polynomial in $x$, which I have an expression for, but it's not ...
Martin Rubey's user avatar
  • 5,533
6 votes
0 answers
167 views

Boolean functional equations

My current approach to investigating reversible quantum gates requires the solution of Boolean functional equations. For example, $$f(x,y,z) = f(x,y \oplus f(x,y,z), z \oplus f(x, y, z)),$$ where $f\...
Helmut Bez's user avatar
6 votes
0 answers
136 views

Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...
Slava Rychkov's user avatar
5 votes
5 answers
1k views

Are there functions satisfying the following integral condition?

Can we find two functions $f$ and $g$ that are reasonably defined nontrivial(not everywhere zero, $f\neq g$, not linear polynomials) functions such that the following condition is satisfied? $$ f( \...
Chulumba's user avatar
  • 789
5 votes
3 answers
758 views

Does there exist another form of the derivative for polynomials?

Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that $$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$ for all $P, Q \in \mathbb{R}[X]...
Dattier's user avatar
  • 3,737
5 votes
1 answer
332 views

A variant of Cauchy-type functional equation conjecture

Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that $$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$ Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$ The answer is ...
math110's user avatar
  • 4,230
5 votes
2 answers
296 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 ...
Iosif Pinelis's user avatar
5 votes
1 answer
213 views

A functional equation in two complex variables

Let $X$ be a compact metric space, or just $X=\mathbb T$, the unit circle, if it helps. We consider only continuous, complex-valued functions on $X$. Let $\varepsilon >0$. Is there $\delta > ...
Tomasz Kania's user avatar
  • 11.3k
5 votes
2 answers
413 views

Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$ I think the following system of equations ...
TikoM's user avatar
  • 53

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