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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1answer
98 views

Are all functions on vectors in $GF(2^n)$ representable as polynomial functions?

In $GF(2)$, any function from an n-dimensional vector to a number is equal to a polynomial function of n variables. (See proof below.) Question: Is this true for other $GF$, especially $GF(2^8)$? ...
3
votes
3answers
114 views

Solving functional equation $f(x)f(y) = f(x+y)$

I'm having some trouble solving the following equation for $f: A \rightarrow B$ where $A \subseteq \mathbb{R}$ and $B \subseteq \mathbb{C}$ such as: $$f(x)f(y) = f(x+y) \quad \forall x,y \in A$$ ...
4
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0answers
76 views

Improvement IMO 1988 $f(f(n))=n+1987$

The following problem was given at IMO 1987. Prove that there is no function f from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$. So I tried to ...
0
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0answers
69 views
+100

Solutions to functional equation $ \gamma(s,t)=f(t \cdot g(s))+h(t) $

Let $$ \gamma(s,t)=f(t \cdot g(s))+h(t) $$ where $\gamma$ is a known function of $s \in \mathbb{R}$ and $t \in \mathbb{R}$ while $f$, $g$, and $h$ are unknown functions. Assume ...
7
votes
2answers
102 views

For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

A group $G$ and a function $f:G\longrightarrow G$ are given and for all $x,y\in G$ satisfying $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?
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1answer
14 views

Calculation of Sigma (Multiple Sigmas)

When I was studying Game Theory, I came across this equation: $$F_i(q_1, \ldots, q_n) = \sum_{s_1 \in S_1} \: \ldots \: \sum_{s_n \in S_N} \big\{ \prod_{j=1}^n q_j(s_j) \big\} \: \: f_i(s_1, ...
12
votes
2answers
138 views

Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$

Find all functions $f$ such that $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Let us plug in $n=1$ $f(f(1))+f(2)=3$ Since the function is from $\mathbb{N}$ to $\mathbb{N}$, ...
1
vote
1answer
26 views

A functional equation over integers

I was working in a problem in number theory and I blocked over the problem : Given functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$, $g:\mathbb{Z^2}\rightarrow \mathbb{Z}$ and ...
0
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1answer
21 views

Fixed Point Iteration $x = g(x)$ method for $y_1 = e ^{-x}$ and $y_2= \cos x$

The question reads as follows: Find the x and y coordinates of the intersection points by means of the $x = g(x)$ method. ( I believe they are referring to the Fixed Point Iteration method) The ...
18
votes
2answers
385 views
6
votes
4answers
517 views

Find a solution for $f\left(\frac{1}{x}\right)+f(x+1)=x$

Title says all. If $f$ is an analytic function on the real line, and $f\left(\dfrac{1}{x}\right)+f(x+1)=x$, what, if any, is a possible solution for $f(x)$? Additionally, what are any solutions for ...
4
votes
2answers
99 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get ...
5
votes
1answer
32 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
0
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0answers
7 views

find the distribution of 100 heart transplant patients at a low volume and high volume hospital using boxplot graph 0-40 mortality

Using a boxplot graph find the distribution show mortality rates within one year of 100 patients having heart transplants at various hospitals. The low volume hospital perform between 5 and 9 ...
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2answers
75 views

Functional equation $f(ax)=bf(x)$

What are all the solutions to the functional equation $f(ax)=bf(x)$, where $a,b>0$, and $f$ is continuous, strictly monotone and increasing, and $x$ ranges over the reals? references? proof? ...
2
votes
3answers
258 views

Solve the following functional equation

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ so that: a) $[f(x)+f(y)][f(x+2y)+f(y)]=[f(x+y)]^2+f(2y)f(y)$ b) for every real $a>b\ge 0$ we have $f(a)>f(b)$ As much as I know: ...
10
votes
4answers
118 views

Polynomial equation $f(x)f(2x^2)=f(2x^3+x)$

Find all polynomials $f(x)$, for which $f(x)f(2x^2)=f(2x^3+x)$. I have no idea how to do it.
0
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0answers
17 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
2
votes
1answer
36 views

Functional equation extended solution

The question is Find all functions $f:R \to R$ such that $$f(x+y)f(x-y)=(f(x)+f(y))^2-4x^2f(y)$$ Taking $x=y=0$, we get $f(0)^2=4f(0)^2 \implies f(0)=0$. Now take $x=y$ which immediately gives ...
8
votes
1answer
632 views

Equation about generating functions and subfactorial

Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following explanation) of variable $w$, try to solve out $G_n(w)$ for all $n\ge0$ from the ...
4
votes
0answers
95 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
0
votes
1answer
30 views

Functional equation, probably involving discrete differentiating?

I want to know how to solve this problem on functions. Question: Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying $$f(x+1)-f(x)=nf(x)$$ where $\mathbb{R}$ is set of real ...
15
votes
4answers
265 views

Find $f''(x)$ if $f\circ f'(x) = 4x^2 + 3$

Can you tell me the solution of this question? If: $f\circ f'(x)=4 x^2 +3$ then what is $f''(x)$? This was a question in math test which I just took yesterday. One function satisfying the ...
0
votes
0answers
16 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
2
votes
0answers
42 views

Is the product rule for logarithms an if-and-only-if statement?

If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$ My question is, Is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ ...
8
votes
1answer
109 views

Functional Equation $f(f'(x))=-f(x)$

Assume $f:\mathbb{R}_{>0}\to\mathbb{R}$ is differentiable and satisfies $\forall x>0:f(f'(x))=-f(x)$. What is $f(x)$? I know that $f(x)=\ln x$ is a solution, but I don't know if there is ...
3
votes
2answers
106 views

Solutions of the functional equation $f(2x) = \frac{f(x)+x}{2}$

How can I solve the following functional equation? $$f(2x) = \frac{f(x)+x}{2},$$ for $x \in \mathbb{R}$ with $f$ being a continuous function.
0
votes
1answer
48 views

Solutions of the functional equation $f(x) + f(qx) = 0$

How can I find the solutions of $$f(x) + f(qx) = 0,$$ where $q \in \mathbb{Q}, q\neq1, x \in \mathbb{R}$, with $f$ being a continuous function?
3
votes
1answer
41 views

Solve functional equation $f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a))$

Solve functional equation $$f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a)),(*)$$ where the following conditions are satisfied: 1)$f:\mathbb{R}_{+}\cup \{0\}\rightarrow [0,1] ,$ 2)$f(0)=1,$ ...
0
votes
1answer
11 views

Criteria for elevating a rational FE solution to reals

I have a functional equation which I have solved for rational numbers. Now to 'elevate' my solution to reals, I use the given fact that f is continuous. I could have done the same process, if I knew ...
2
votes
3answers
47 views

Solve the funtional equation $f(xf(y)+x)=xy+f(x)$

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ so that $f(xf(y)+x)=xy+f(x)$. If you put $x=1$ it's easy to prove that f is injective. Now putting $y=0$ you can get that $f(0)=0$. $y=\frac{-f(x)}{x}$ ...
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votes
1answer
91 views

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

Problem: Let $f(x): \mathbb{R} \to \mathbb{R}$ such that: $$f(x+y)+f(x)=2f(x-y)+2f(y) \ \ \ \forall x,y \in \mathbb{R}$$ This is a problem in my analytics exam, I can't find it if $f$ is not a ...
2
votes
2answers
31 views

To find an extremal of a given functional

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
1
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0answers
21 views

the family of functions with alternating derivatives in different orders

I'm wondering whether there's a name for the family of functions with alternating derivatives in different orders. Formally, $f(x)\in C^\infty:\mathbb{R^+}\rightarrow\mathbb{R}$. For all x, ...
5
votes
1answer
42 views

Can $f(x+1) = f(x)^{\ln(x)}$ be expressed as integral transform $\int g(x,t) dt $?

Let $x$ be a real number. Can some real-analytic function $f$ that satisfies for $x>3$ :$f(x+1) = f(x)^{\ln(x)}$ be expressed by standard functions as an integral transform : $$f(x) = ...
0
votes
0answers
15 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which could be either: a convolution kernel or a difference of two convolution kernels: $\varphi=\varphi_1-\varphi_2$. I am ...
1
vote
1answer
23 views

Is $Tf=x_0+\int_0^tf(t)(1-f)\,dt$ a Lipschitz function?

Consider the Banach space $C[0,1]$ with the uniform norm and the operator given by $(T(f))x=x_0+\int_0^xf(t)(1-f(t))\,dt$ for $x\in[0,1]$ and $x_0\geq0$. I want to show that there is a number ...
2
votes
1answer
28 views

Existence solution pendulum equation using the contraction principle

How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach). The numbers $g$ and $\ell$ are fixed constants ...
0
votes
1answer
10 views

Proving that $(Sf)(x)=1+\int_0^x t^2f(t)\,dt$ is $K$-Lipschitz

I want to show that $S:C[0,1]\to C[0,1]$ where $$(Sf)(x) = 1+\int_{0}^{x} t^2\cdot f(t) \,dt$$ is a Lipschitz map for $x\in[0,1]$. There I use the Euclidean norm on $\mathbb{R}$ and the uniform norm ...
9
votes
3answers
303 views

Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?

A UC Berkeley prelim exam problem asked whether an additive function $f\colon {\mathbb R} \to {\mathbb R}$, i.e. satisfying $f(x + y) = f(x) + f(y)$ must be continuous. The counterexample involved ...
1
vote
1answer
44 views

Is there a $f\in C[0,1]$ such that $f(x)=\frac12 \sin (f(1-x))$

Is there a $f\in (C[0,1],\lVert\cdot\rVert_{\infty})$ such that $f(x)=\frac12 \sin (f(1-x))$? I feel like you need to apply Banach´s fixed point theorem, which means it suffices to prove that ...
2
votes
2answers
43 views

Is $T:C[0,1]\to C[0,1]: Tf(x)=\frac12 x f(x^2)+1$ a contraction?

I am a non-mathematician (a physicist) who wants to find a continuous function on $[0,1]$ that satisfies the relation $f(x)=\frac12 x f(x^2)+1$ with $0\le x\le 1$. I need it for one of my models. ...
19
votes
2answers
767 views

Which $f$ satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...
2
votes
1answer
35 views

General Solution to Almost Riccati Like Equation

Consider the differential equation $$ y' = a_0(x) + a_1(x)y + a_2(x)\frac{1}{y}$$ I am attempting to find the general solution to this. One thing I can note is that the entire equation can be ...
3
votes
0answers
47 views

Problem in Putnum competition? [closed]

Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$ is a continuous function and $f(2x^2 -1)=2xf(x)$ for all $x\in \mathbb{R}$. Prove $f(x)=0\,\,\text{for all} \, x\in [-1,1].$
4
votes
1answer
61 views

Is distributivity sufficient to define composition?

Function Composition has the property of distributivity: $$(f\star g)\circ h = (f\circ h)\star(g\circ h)\;\forall f,g,\star \in\{+,-,\times,\div\}$$ I was wondering if these properties uniquely ...
2
votes
1answer
34 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
1
vote
1answer
38 views

Entire $f,g$ such that $f(f(z)) = p(g(z))$

Let $p(z)$ be a given polynomial of degree $3$. How to find nonconstant entire functions $f,g$ such that $f(f(z)) = p(g(z))$ or prove they do not exist ? I considered the related equation $f(f(z)) ...
2
votes
2answers
75 views

Continuous function with differentiation

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function with $f(x+y) = f(x) +f(y),\forall x,y\in \mathbb{R}.$ Find $\frac {df} {dx} $, if it is exist?
2
votes
0answers
43 views

Functions that hasn't any root

we say that a function like $f:X \to X$ has root if exists a function like $g:X\to X$ that for every $x \in X$: $$f(x) = g(g(x))$$ what is a necessary and sufficient condition for $f$ that it has a ...