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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).

2
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2answers
739 views

Functional equation $f(x^2+y)=f(x)+f(y^2)$ from Olympiad

Let $f: \mathbb{R} \to \mathbb{R}$ $$f(x^2+y)=f(x)+f(y^2)$$ How do I find all functions that fulfill this equation? I tried to just write many equalities but it just doesnt help.
6
votes
3answers
2k views

Classifying Functions of the form $f(x+y)=f(x)f(y)$ [duplicate]

Possible Duplicate: Is there a name for such kind of function? The question is: is there a nice characterization of all nonnegative functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x+...
5
votes
0answers
382 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function $f:...
1
vote
2answers
1k views

Indeterminate equation and functional equation

I was wondering what differences and relations are between indeterminate equation and functional equation? Are they the same concept? Thanks and regards!
1
vote
1answer
59 views

Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance $\rho(...
0
votes
1answer
108 views

General solution to homogeneous difference equation

With a given example $$ a_{n-1} = ca_{n-2} $$ general solution: $$ a_{n} = c . c . a_{n-2} $$ $$ = c . c . a_{n-3} $$ $$ = c^n a_0 $$ Question: Find the general solution for the ...
4
votes
1answer
422 views

Solving (and proving) a combinatorial functional recursive equation

I have a sequence of functions $f_k(n)$ defined as follows: $f_1(n)=n^{n-2}$ $f_k(n)=\sum_{i=1}^{n-1}f_{k-1}(i)\cdot(n-i)^{n-i-2}\cdot{n-k \choose n-i-1}$ My goal is to find and prove a closed-form ...
0
votes
3answers
243 views

Recurrence relation - How to solve this recurrence relation

a person invests 1000 at a bank at 4 percent compound interest compounded annually and every year government and bank charges amounting to C are deducted and if An is the value of the investment at ...
0
votes
1answer
210 views

Solving general solution of recurrence relation by iteration

$$a_{n-1} = ca_{n-2} $$ Hence $$a_n = c \cdot c \cdot a_{n-2} $$ $$ = c \cdot c \cdot c \cdot a_{n-3} $$ ...... $$ = c^na_0 $$ Why is there a iteration on the constant $c$ ?
3
votes
2answers
268 views

Understanding difference equation

I was given an example $$R_n = R_{n-1} + R_{n-2} $$ This equation is given as an second-order equation. Why is it so?
11
votes
2answers
262 views

Can every real function be represented as two shifted even functions?

I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be ...
18
votes
1answer
2k views

Find continuous functions such that $f(x+y)+f(x-y)=2[f(x)+f(y)]$

Here is the problem: Find all continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x+y)+f(x-y)=2[f(x)+f(y)]\;\;\;(1).$$ Here is my attempt: Fix $\delta>0$ and let $C=\int_{0}^{\delta}...
4
votes
2answers
860 views

Functional derivative of $\int \left( \frac{df^2 }{d^2 x} \right)^2 dx$

According to page 7 of the PDF document $$ \frac{\delta}{\delta f} \int \left( \frac{df^2 }{d^2 x} \right)^2 dx = \int \frac{df^4}{d^4 x} dx $$ I would like help proving this statement. Although ...
3
votes
1answer
105 views

All functions with the property $ k \mid f(m+n) \iff k \mid f(m)+f(n)$

Let $\mathbb N$ be the set of all positive integers. How can one find all functions $f: \mathbb N \to \mathbb N$ such that $$ k \mid f(m+n) \iff k \mid f(m)+f(n)$$ For all positive integers $k$.
2
votes
3answers
105 views

Defining a function with certain properties

I'm a bit rusty in mathematics so I need your help please :) I need a function $y$ that satisfies: $$\begin{align*} y &= ax\\ y &= \left\{\begin{array}{ll} x &\text{if }x\geq 0;\\ 0 &\...
5
votes
3answers
150 views

Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$

I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas. PS. I've tried plotting the the initial values into "...
8
votes
3answers
337 views

Solving $\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$

I recently came across this equation : $$\forall x \in \mathbb{R}_+^*, f'(x) = f\left(\frac1{x}\right)$$where $f \in \mathcal{C}^1(\mathbb{R}, \mathbb{R})$. I've done the following, but I'm stuck at ...
5
votes
3answers
518 views

Does the functional equation $f(1/r) = rf(r)$ have any nontrivial solutions besides $f(r) = 1/\sqrt{r}$?

Repeating for the sake of TeX rendering: Does the functional equation $f(1/r) = rf(r)$ have any nontrivial solutions besides $f(r) = 1/\sqrt{r}$?
3
votes
1answer
257 views

A differential-functional equation: $f'(f^{-1}(x)) = 1/g(x)$

Problem: Given $g(x)$, solve the equation $f'(f^{-1}(x)) = \frac{1}{g(x)}$ for an invertible and differentiable function $f(x)$. So far I have tried setting $y = f^{-1}(x) \Leftrightarrow x = ...
2
votes
2answers
2k views

Converting polar equation to cartesian coordinate polar equation and back again?

OK, so I have the following polar equation: $r = Θ/20$ And I would like to translate this a little to the right, and down from the polar origin. Now, I figure since I know cartesian coordinate ...
0
votes
2answers
50 views

Reversing bijections defined via conditional expressions

Let's say that I have a variable $j$ defined by the following formula: $$j=\frac{n(n+2) + m}{2}$$ where $n$ and $m$ are two parameters, both integers, that satisfy the following conditions: $n\in \...
2
votes
1answer
489 views

Equation with a definite integral - can I differentiate it?

I have an equation like this: $$te^{t} = \int\nolimits_0^t e^\tau u(\tau)d\tau$$ I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how ...
23
votes
5answers
6k views

Is there a name for function with the exponential property $f(x+y)=f(x) \cdot f(y)$?

I was wondering if there is a name for a function that satisfies the conditions $f:\mathbb{R} \to \mathbb{R}$ and $f(x+y)=f(x) \cdot f(y)$? Thanks and regards!
1
vote
2answers
173 views

Finding 2^2^2^2^2 … n times

f(1) = 2 f(2) = 2^2 f(3) = 2^(2^2) f(4) = 2^(2^(2^2)) etc how can I find a closed form solution to this? Thanks.
2
votes
1answer
434 views

How to solve algebraically the equation $x = \frac{1}{2}\cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$

How to solve this trigonometric equation $x = \frac 1 2 \cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ ? The iterative solution seems to be 1.417. Can anybody suggest an algebraic ...
1
vote
1answer
215 views

Functional Equation Analysis

Given: $$F(F(n)) = n$$ $$F(F(n + 2) + 2) = n$$ $$F(0) = 1$$ where n is a non-negative integer. $$F(129) = ?$$ How can we solve such kind of functional equations? Is there any simpler approach ...
3
votes
1answer
228 views

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

Find all continuous $f:[0,1] \rightarrow [0,1]$ such that $f(1-f(x))=f(x)$.
4
votes
1answer
502 views

$f(x+f(y))=f(x-f(y))+4xf(y)$

Find all functions $f:R\rightarrow R$ which satisfy $f(x+f(y))=f(x-f(y))+4xf(y)$ $\forall x,y \in R$. I strongly suspect $0$ and $x^2+C$ to be the only solutions but, as is almost the case with ...
15
votes
3answers
853 views

The Notorious Triangle Problem

I was told this question by a friend, who said that their friend had thought about it on and off for six months without any luck. I have then had it for a while without any luck either. It is in the ...
0
votes
1answer
122 views

$x^3[f(x+1)-f(x)]=1$

Possible Duplicate: $x^3[f(x+1)-f(x-1)]=1$ Given that f is continuous and $x^3[f(x+1)-f(x)]=1$, determine $\lim_{x\rightarrow \infty}f(x)$ explicitly.
0
votes
1answer
130 views

$x^3[f(x+1)-f(x-1)]=1$

Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.
2
votes
3answers
257 views

Solving the functional equation $x[f(x+1)-f(x-1)]=1$ [duplicate]

Possible Duplicate: Solving the functional equation $f(x+1) - f(x-1) = g(x)$ How do I approach this problem $x[f(x+1)-f(x-1)]=1$.
7
votes
4answers
707 views

Solving the functional equation $f(x+1) - f(x-1) = g(x)$

Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies $$ f(x+1) - f(x-1) = g(x) $$
3
votes
0answers
86 views

Function Shape Reference

I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague. I would like to have a ...
12
votes
3answers
537 views

Evaluating $f(x) f(x/2) f(x/4) f(x/8) \cdots$

Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) \...
7
votes
5answers
2k views

$f(x) + f(1-x) = f(1)$

I was discussing with a friend of mine about her research and I came across this problem. The problem essentially boils down to this. $f(x)$ is a function defined in $[0,1]$ such that $f(x) + f(1-x) ...
12
votes
2answers
2k 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 \...
1
vote
2answers
147 views

How to solve the following system?

I need to find the function c(k), knowing that $$\sum_{k=0}^{\infty} \frac{c(k)}{k!}=1$$ $$\sum_{k=0}^{\infty} \frac{c(2k)}{(2k)!}=0$$ $$\sum_{k=0}^{\infty} \frac{c(2k+1)}{(2k+1)!}=1$$ $$\sum_{...
1
vote
1answer
195 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 $ f(...
4
votes
1answer
505 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) = ...
3
votes
1answer
2k 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
667 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 $x$.
0
votes
1answer
206 views

Solving this set of quadratic equations

I have a set of quadratic equations of the form: \begin{equation*} 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 \\ \vdots \\ 2S_nq_n(...
5
votes
1answer
239 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
481 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
349 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
426 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 function. ...
0
votes
1answer
204 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
1k 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
2k views

Solving the functional Equation $f(f(x))=f(x)+x$ [duplicate]

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