# Questions tagged [functional-equations]

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### $f' = e^{f^{-1}}$, a third time

I am of the impression the differential equation $f' = e^{f^{-1}}$ was considered on mathoverflow for the first time here: How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$? It was found ...
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### Solving the integral equation $y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$

Could you please help me to solve the following integral equation? $$y(x)=e^{-x^2/2}+\lambda\int^{+\infty}_{-\infty}e^{-i(x-z)}y(z)dz$$ I tried to turn the exponentiential term into its trigonometric ...
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### 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 ...
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### 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 ...
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### A question about decompositions of rational functions

Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...
181 views

### Largest asymptotic growth for $2f(x)-f(2x)$

I am trying to find smooth functions $f : \mathbb{R}_+ \to \mathbb{R}_+$ such that the quantity $$\Delta_f(x) := 2f(x)-f(2x)$$ is positive for $x$ large enough and has the greatest asymptotic growth. ...
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### Solving a difference functional equation by using Laplace transform

Consider the operator $T:L^{2}(0,r+1)\longrightarrow L^{2}(0,r+1)$: \begin{equation*} Tu(x)=:u(x)+a\mathbf{1}_{(0,1)}(x)u(x+r)+b\mathbf{1}_{(1,1+r)}(x)u(x-1),% \text{ }x\in (0,r+1), \end{equation*} ...
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### Square root of a function on a finite set

Let $S$ be a finite set and $f \colon S \to S$ be an arbitrary function. How can we find all functions $g \colon S \to S$ with $f = g \circ g$? If $f$ and $g$ are both required to be invertible, the ...
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### Functional equations and normal distribution

Let $\alpha \neq 1.$ If $X,Y$ are two independent random variable such that $U=X+Y$ and $V=X+\alpha Y$ are independent, then $X$ and $Y$ are normally distributed. In term of characteristic functions ...
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### 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 ...
368 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 ...
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### Solve $(x-a)^{\alpha +1} - \lambda*(b-x)^{\alpha + 1} = C(\frac{a+b}2 - x)^{\alpha}$ over $\mathbb R$ [closed]

I have been having trouble solving the following equation. Any help would be appreciated. Let a,b,C,$\alpha$,$\lambda$ be real numbers with $C < 0$, $0 < \alpha < 1$, $\lambda > 1$. We ...
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### Functions that satisfy a reverse triangle inequality: do they have a name?

Let $f : \mathbb{R}^n \to \mathbb{R}_+$ satisfy $$f(a) - f(b) \le C f(a - b)$$ $\forall a, b \in \mathbb{R}^n$ for some $C \ge 0$. Is there a name for such functions? (I would be happy to have a name ...
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### book recommendation about iterative functional equations [closed]

I would like to learn about iterative functional equations. I read this book, but it doesn't include such functional equations. I tried this, but it was too general for my purpose. Finally I read ...
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Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...