# Questions tagged [functional-equations]

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### Is it possible to express the functional square root of the sine as an infinite product?

Cross-post from MSE. It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that ...
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### The uniqueness of a solution of a differential equation on a unit circle

Trying to solve one problem in the geometry of 2-dimensional Banach spaces, I arrive to the problem of uniqueness of the following differential equation on a function $r:\mathbb T\to(0,1]$ defined on ...
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### Criteria for $f(f(x))=g(x)$

I 'm searching about the solvability of the functional equation $f(f(x))=g(x)$. I have three questions about it: Let's be $g$ an arbitrary function and the functional equation $f(f(x))=g(x)$. Are ...
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### 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 ...
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Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$. Then is it true that $f(x+y)=f(x)... 2answers 157 views ### How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures? Consider the symmetric sequence$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$given by ... 1answer 124 views ### Solvability of a certain functional equation in simple$C^*$algebras For which simple unital$C^*$algebras does the following functional equation have a solution: $$d^2=0,\;{(d+d^*)}^2=1$$ The Calkin algebra and$M_{2n}(\mathbb{C})$are some examples. It is not ... 1answer 89 views ### 1D functional equation: solve for function with given expected value w.r.t normal density Given scalars$c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz)$\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation $$\begin{split} \mathbb ... 1answer 74 views ### Equation for a geometrical half surface from folding a flat curved in one direction surface in half [closed] I cannot not formulate the problem as i do not know how to model this idea. This is an open question not an mathematical exercise so if anybody has a good proposition i'm happy to use it. Sorry in ... 2answers 81 views ### Find the general solution to the Forsyth/Abel functional equation Let R(m,n) be defined on all the integers such that R(m,0)=m, R(0,n)=n, R(m,n)=R(n,m) and R(R(m,n),p)=R(m,R(n,p)) for all integers p. Thus R satisfies the Abel associativity equation. Let ... 0answers 67 views ### Characterisation of functions for which the Fourier transform commutes with a particular operator Defining the operator \phi by: \phi(f(x))=\frac{1}{|x|} f(\frac{1}{x}), and noting \mathcal{F} the Fourier transform on the real line, can we characterize all the functions (with real variable ... 2answers 274 views ### Seeking proof to an asymptotics of a recursion or functional equation My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion$$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\... 0answers 76 views ### Functional equation involving integrals and exponential Can we find on$\mathbb{R}^+$a real positive function$f(x)$(in$C^{\infty}$) such that: $$\int_0^{\infty} f(x) e^{\lambda \int_1^{x} f(t)^2 dt} dx=0$$ where$\lambda$is a complex number (with$0&...
I have the following problem. I have a function $v(x, \theta)$ that can be expressed in two ways, for all $x, \theta \in \Re$: $v(x, \theta) = u(x - \theta)$, where $u$ is strictly concave and ...