# 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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### Solving $\sum\limits_{k=1}^n e(x-x_k) = h(x)$ for $e(x)$, where $x_k$ and $h(x)$ are given (updated)

I would like to find the function $e(x)$ which solves $\sum\limits_{k=1}^n e(x-x_k) = h(x)$, where $x_k$ and $h(x)$ are given. There are no restrictions on any of the $x_k$ or $h(x)$ except that $h(x)$...
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### Characteristic functional equation of a Theta Function

Define the following as a "simple" theta function $$\vartheta(q) = \sum_{n=0}^{\infty} q^{n^2} = 1 + q + q^4+q^9+ \ ...$$ Defined on the open unit circle on the complex plane. I'm trying to find ...
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### Prove that only quadratic functions $f$ solve the quadratic functional equation

Let $f$ be such that $f(x+y)+f(x-y)=2f(x)+2f(y)$, i.e. $f$ satisfies the quadratic functional equation. Then $f$ has to be such that $f(t)=\alpha t^2$. I am looking for an accessible proof of this. ...
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### Solving $f(x/2)^2=f(x)$

Does $\left[f(\frac{x}{2})\right]^2=f(x)$ imply $f(x)=\exp(Ax)$? How can I go about finding all the solutions to this equation?
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### Algebra problem that you have to assume certain criteria at the end.

I was trying to solve this problem: If $f(x)=\frac{ax+b}{cx+d}, abcd\neq0$ and $f(f(x))=x$ for all $x$ in the domain of $f$, what is the value of $a+d$? I start off by just plugging in and ...
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### Prove that $\sin^2(\pi x)$ is chaotic

My approach is based on the following from the book Chaos and Fractals: New Frontiers of Science, by Peitgen, Heinz-Otto, Jürgens, Hartmut, Saupe, Dietmar. To be more specific: "If $f$ is chaotic and ...
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### $f:\mathbb{R} \to \mathbb{R}$ we have $f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$ such function is polynomial of degree less than or equal to two. [duplicate]

Consider differential function $f:\mathbb{R} \to \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have $$f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$$ Then show that every such function is ...
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### How many polynomial functions exist such that $f(x^2) = (f(x))^2 = f(f(x))$ [closed]

How many polynomial functions $f$ of degree $\geq1$ satisfy $f(x^2) = (f(x))^2 = f(f(x))$ for all real $x$?
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### A Golden Ratio Functional Equation Sequence

I was looking at the equation $f^{-1}(x)=\int f(x)dx$ recently. One can note that it has an easy real-valued solution $f(x)=\phi^{\frac{\phi-1}{\phi}}x^{\phi-1}$ (by guessing for a solution of the ...
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### Find all $f$ that satisfies $f:\mathbb{R}\rightarrow\mathbb{R};f(x+y)+f(x)f(y)=(1+x)f(y)+(1+y)f(x)+f(xy)$

Find all $f$ that satisfies: $1, ~f:\mathbb{R}\rightarrow\mathbb{R};\\ 2,\forall x,y\in\mathbb{R},f(x+y)+f(x)f(y)=(1+x)f(y)+(1+y)f(x)+f(xy);$ Maybe we can prove it's derivable or it's a linear ...
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