The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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8
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2answers
666 views

On sort-of-linear functions

Background A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies $$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$ $$ (2)\;\; f(\alpha x) = \alpha f(x) $$ for all $ x,y \in ...
0
votes
2answers
133 views

How to solve the following system?

I need to find the function c(k), knowing that $$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$ $$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$ $$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$ ...
1
vote
1answer
183 views

conversion of a powerseries $-3x+4x^2-5x^3+\ldots $ into $ -2+\frac 1 x - 0 - \frac 1 {x^3} + \ldots $

This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general. Q1: I considered the function $ ...
4
votes
1answer
414 views

Find a function such that $f(\log(x)) = x \cdot f(x) $

I recently read an article in which the author describes how to find some functions that obey to certain recursion relationships. If we want to find a function that satisfies, for example, $f(x^a) = ...
1
vote
1answer
1k views

Solve the functional equation $f (x+y)=f (x)+f (y)+xy (x+y)$, $f$ continuous at $0$

Find all functions $f:\mathbb{R} \to \mathbb{R}$, that are continuous at the point $x=0$ and satisfy: $$f (x+y)=f (x)+f (y)+xy (x+y) \ \ \forall x,y \in \mathbb{R} $$
4
votes
3answers
501 views

Find all continuous functions $f$ on $\mathbb{R}$ satisfying $f(x)=f(\sin x)$ for all $x$.

Find all continuous functions $f$ on $\mathbb{R}$ satisfying $f(x)=f(\sin x)$ for all $x$.
2
votes
0answers
197 views

How to find a function which satisfies such functional equation?

How to find a function which satisfies: $$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$ where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
1
vote
3answers
119 views

Solving Series of equations

I have the following series of equations (n+2 equations n+2 variables): $k_0q_0+\lambda q_0 + c_0 = 0$ $k_1q_1+\lambda q_1 + c_1 = 0$ $k_nq_n+\lambda q_n + c_n = 0$ $q_1+q_2+....+q_n = 1$ ...
0
votes
1answer
187 views

Solving this set of quadratic equations

I have a set of quadratic equations of the form.. $ 2S_0q_0(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_0 = 0$ $ 2S_1q_1(q_0S_0 + q_1S_1 + ...... + q_iS_n)+k_1 = 0$ . . . $ 2S_nq_n(q_0S_0 + q_1S_1 ...
5
votes
1answer
214 views

Generalization of cos: is this function known?

Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$). Consider a function $f_2$ ...
6
votes
1answer
305 views

Iterated polynomial problem

Polynomial $P$ satisfies $P(n)>n$ for all positive integers $n$. Every positive integer $m$ is a factor of some number of the form $P(1),P(P(1)),P(P(P(1))),\ldots $. Prove that $P(x)=x+1$.
3
votes
2answers
316 views

Implicit function $y = e^{(y-1)/x}$

I'd like to know if the function $ y = f(x) : [0,1] \rightarrow [0,1]$ defined implicitly by the transcendental equation $$\displaystyle y = e^{(y-1)/x}$$ is "well known" (name, properties) or is ...
2
votes
0answers
367 views

Solving $f(f(x))=g(x)$ equations [duplicate]

Possible Duplicate: Square root of a function (in the sense of composition) I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known ...
0
votes
1answer
159 views

Finding $\alpha$ such that $f(\alpha(x+y))=f(x)+f(y)$

Problem taken from the link: http://web.mit.edu/rwbarton/Public/func-eq.pdf I am stating the question here For which $\alpha$ does there exists a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ ...
14
votes
4answers
554 views

Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$

In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)) = f(x) + x$$ (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...
1
vote
2answers
537 views

Solving the functional Equation $f(f(x))=f(x)+x$

Let $f$ be continuous on $\mathbb{R}$. Then how to find all continuous functions satisfying $f(f(x))=f(x)+x$
5
votes
1answer
204 views

How to solve DE that relate values of derivatives at different points?

I try to solve for the specific function $f(x) = \frac{2-2a}{x-1} \int_0^{x-1} f(y) dy + af(x-1)$ It looks similar to the function used to find the Renyi's parking constant because it came out from a ...
7
votes
5answers
1k views

Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$

Find all polynomials $p(x)$ such that for all $x$, we have $$(x-16)p(2x)=16(x-1)p(x)$$ I tried working out with replacing $x$ by $\frac{x}{2},\frac{x}{4},\cdots$, to have $p(2x) \to p(0)$ but then ...
7
votes
1answer
600 views

Polynomials which satisfy $p^{2}(x)-1 = p(x^{2}+1)$

Can we find a polynomial $p(x) \in \mathbb{R}$ such that $\text{deg}\ p(x)>1$ and which satisfies $$p^{2}(x)-1=p(x^{2}+1)$$ for all $x \in \mathbb{R}$. This question can be very well identified with ...
14
votes
3answers
762 views

Continuous function satisfying $f^{k}(x)=f(x^k)$

How does one set out to find all continuous functions $f:\mathbb{R} \to \mathbb{R}$ which satisfy $f^{k}(x)=f(x^k)$ , where $k \in \mathbb{N}$? Motivation: Is $\sin(n^k) ≠ (\sin n)^k$ in general?
2
votes
3answers
1k views

Function Satisfying $f(x)=f(2x+1)$

If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and satisfies $f(x)=f(2x+1)$, then its not to hard to show that $f$ is a constant. My question is suppose $f$ is continuous and it satisfies ...
23
votes
3answers
3k views

Prove that this function is bounded

This is an exercise from Problems from the Book by Andreescu and Dospinescu. When it was posted on AoPS a year ago I spent several hours trying to solve it, but to no avail, so I am hoping someone ...
15
votes
6answers
722 views

Solution to $1-f(x) = f(-x)$

Can we find $f(x)$ given that $1-f(x) = f(-x)$ for all real $x$? I start by rearranging to: $f(-x) + f(x) = 1$. I can find an example such as $f(x) = |x|$ that works for some values of $x$, but not ...