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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22 views

Functional equation with only two solutions?

Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation: $$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$ Unfortunately, I was able to show only ...
2
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2answers
66 views

Functional equation with sqrt solutions [closed]

Let $f:(0,\infty)\to(0,\infty)$ so that for all $x,y\in(0,\infty)$ we have $$f\left(\frac {x} {f(y)}\right)=\frac {x} {f(x\sqrt y)}.$$ Find function $f$.
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1answer
21 views

Functional equation: $f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$

Find an algebraic function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\frac x{\prod_{i|x}^{x-1} f(i)}$$ and $$f(1)=1$$ for all $x\in\Bbb N$ I allready know two things: $f(p^k)=p$ where $p$ is prime ...
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1answer
42 views

Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that $f(x)=f(x^y)$

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(x)=f(x^y)$$ for all $x,y\in\mathbb{N}$. I'm not intrested in the trivial solution $f(x)=k$, where $k\in\mathbb{N}$.
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23 views

System of linear Volterra integral equation

Consider the Volterra integral equation $$ f(t) = g(t) +\int_0^t K(t,s) f(s) ds $$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...
7
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1answer
83 views

Functions satisfying 4 out of 5 inner product properties

Let us consider function $s:K^m \times K^m \mapsto K$ (here $K = \mathbb{R}$ or $K = \mathbb{C}$). If $\forall x, y, z \in K, \forall \lambda \in K$ $s(x + y, z) = s(x, z) + s(y, z)$ $s(\lambda x, ...
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2answers
50 views

Polynomial equation

Is it true that if $P \in \mathbb{Z}[X]$, then there exists $Q \in \mathbb{Z}[X]$ such that $P(x)=Q(x+1)-Q(x)$? Can we generalize to other rings than $\mathbb{Z}$? I came up with this by ...
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0answers
33 views

Find $f$ such that $(f(x)-f(y))\left( f\left(\frac{x+y}{2}\right) -f(\sqrt{xy})\right) =0,\forall x,y \in (0,\infty).$

Find all continuous function $f:(0,\infty)\to\mathbb{R}$ such that $$\displaystyle (f(x)-f(y))\left( f\left(\frac{x+y}{2}\right) -f(\sqrt{xy})\right) =0,\forall x,y \in (0,\infty).$$ My try: Assume $...
9
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1answer
66 views

if $f(x + y) = f(x)f(y)$ is continuous, then it has to be injective.

Let $f$: $\Bbb R$ $\rightarrow$ $\Bbb R$ be a non-constant function such that $f(a + b) = f(a)f(b)$ for all real numbers $a$ and $b$. Prove that if $f(x + y) = f(x)f(y)$ is continuous, then it has ...
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0answers
32 views

Show that $f$ is continuous and hence of the form $f(x)=cx$ [duplicate]

Let $f:\Bbb R\to \Bbb R$ be additive i.e. satisfies $f(x+y)=f(x)+f(y)$ for all $x$, $y$. and Lebesgue measurable. Show that $f$ is continuous and hence of the form $f(x)=cx$. In order to show ...
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0answers
28 views

Functional differential equation

I want to solve a functional differential equation of this kind $$ \int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') u(t'',t';[x])+u(t,t';[\...
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0answers
64 views

Solutions for differential equations of the form: $f = f' \circ f''\circ \ldots \circ f^{(n)}$

Which are the n-times differentiable real functions that fit the condition: $f = f' \circ f'' \circ \ldots \circ f^{(n)}$ ? I think I have came up with a tentative solution for $n=2$, which may ...
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0answers
23 views

Solving equation over expectation

I have an expression $(1 + n EX^k)p^{-k}$ which I would like to minimize over $k$. Here $n$ and $k$ are positive numbers, while $X$ and $p$ are in $[0,1]$. Since the expectation converges absolutely, ...
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0answers
20 views

Interval between two solutions of an equation

We are in $\mathbb{R}$ and $x\geq 0$. I have an equation: $(1-\alpha)x^{\frac{1}{1-\alpha}}-W(y)x=g(y)W(y)$ Where $\alpha \in (0,1)$, y is a parameter $0<y<1$, W and g are constants in $x$ and ...
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0answers
30 views

Solving the functional equation F(z+1)=Q(z)F(z)

I want to solve the functional equation $F(z+1)=Q(z)F(z)$. The $F(z)$ is a matrix function. $Q(z)$ is also a matrix function. But its compoents are all rational functions. i.e. $ F(z)=\begin{matrix} ...
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0answers
12 views

Pacejka formula - extracting the curve peak

I don't know if many of you knows the Pacejka Magical Formula, but it looks like this: $D\sin(C\arctan(Bx - E(Bx-\arctan(Bx))))$ As you can see, the formula reaches a peak point(in this case at ~0.6)...
4
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1answer
30 views

Multiplicative even and odd functions?

An even function satisfies $$ f_e(x) = f_e(-x) $$ and a odd function $$ f_o(x) = -f_o(-x) $$ Every function can be split into an even and an odd part $$ f(x) = f_e(x) + f_o(x) = \frac{1}{2}(f(x)+f(-x))...
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22 views

How to complete this?

Let $f$ a function defined on $]0,+\infty[$ and checking : $$\forall x,y>0,\ f(xy)=f(x)+f(y)$$ Assume that $f$ is bounded on $]1-\eta,1+\eta[$($\subset]0,+\infty[$). Let $\alpha =\underset{x\in ...
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37 views

Functional analysis with KKT conditions

I want to solve an optimization problem min $F(x_{ik}) $ subject to $x \in X$. $F$ here is a function or functional that I wish to determine. I want my optimal ...
5
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3answers
80 views

$f(x)$ is an analytic function in $\mathbb{R}$ such that $f(-x)f(x)=1$. What else can we find out about $f(x)$?

Well, I know that there are some easy things we can say immediately: $f(0)= \pm 1$, follows immediately $f(x)=\pm 1$ is the obvious solution, so let's look for other solutions. Moreover, let's ...
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0answers
27 views

Functional equation that models trigonometric identities

Find, with proof, all continuous functions $f,g : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x)^2 + g(x)^2 = 1$ and $2f(x)g(x)=f(2x)$. I am aware that the solution pair $(f,g)=(\sin{ax},\cos{ax})$...
5
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2answers
70 views

Can a conformal map be turned into an isometry?

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, \...
0
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0answers
26 views

Solving a functional convolution equation

I have a very weird functional equation that I am trying to solve. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a non-negative Lebesgue integrable function such that $\int f=1$. Let $\tilde{f}(x)=f(-x)$ ...
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0answers
56 views

What does non-degenerate mean?

The context is that f is a real-valued function on a plane such that for every non-degenerate square ABCD in the plane, f(A)+f(B)+f(C)+f(D)=0.... What does this word mean here? I've found that "in ...
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0answers
50 views

Find all functions $f: \Bbb Q \rightarrow \Bbb Q$

Find all functions $f:\Bbb Q \rightarrow\Bbb Q$ that satisfy the conditions: a) $f(f(2016))=0;$ b) $f(x+y)= f(x)+f(y)+f(2016)$ for all $x,y \in \Bbb Q.$ I tried to solve the problem using the ...
1
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1answer
47 views

How to solve $2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$

I saw a question today. $$2f(x)+xf(\frac{1}{x})-2f(|\sqrt2 \sin(\pi(x+\frac{1}{4}))|) = 4 \cos^2 \frac{\pi x}{2}+x\cos\frac{\pi}{x}$$ It had options like this (one or more than one may be correct): ...
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1answer
21 views

Finding a possible region where this PDE has a solution

Consider the problem $$xu_t+u_x = 0, \quad u(0,x) = \sin x.$$ We're asked to prove that the problem doesn't have a solution defined in all of $\Bbb R^2$, and to give a possible open set in $\Bbb R^2$ ...
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1answer
22 views

Find the equation of the common part of two objects

How to find the equation of the intersection curve of the ball $ x^2 + y^2 + z^2 = 4a^2 $ (1)and the cylinder $x^2+y^2=2ax(a>0)$(2)? let (1)-(2), we can get $$z^2+2ax-4a^2=0 $$ but this is not ...
2
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1answer
59 views

To find whether the function is one-one and onto and also to proof a function bijective.

$f$ is a function from $\mathbb{N}$ to $\mathbb{N}$. $$ \begin{align} f(x)&=x+1,\text{ if $x$ is odd} \\ &=x-1,\text{ if $x$ is even} \end{align} $$ I have proved it one-one by taking $...
11
votes
3answers
254 views

The function $f (x) = f \left (\frac x2 \right ) + f \left (\frac x2 + \frac 12\right)$

The function $f: [0,1] → \mathbb R $ satisfies the equation $$f (x) = f \left (\frac x2 \right ) + f \left (\frac x2 + \frac 12\right)$$ for every $x$ in $[0,1]$. Can we assert that $f (x) = c (1-2x)...
6
votes
1answer
129 views

Prove that $f(x)=8$ for all natural numbers $x\ge{8}$

A function $f$ is such that $$f(a+b)=f(ab)$$ for all natural numbers $a,b\ge{4}$ and $f(8)=8$. Prove that $f(x)=8$ for all natural numbers $x\ge{8}$
4
votes
1answer
68 views

Find $f(x)$ if for every $x$: $f(x) + f(\frac {2x-3}{x-1}) = x$

I want to find $f(x)$ if for every $x$ (except one and two): $$f(x) + f\left(\frac {2x-3}{x-1}\right) = x$$ I know that the answer goes something like $g(x)= \frac {2x-3}{x-1} $ and in conclusion $...
2
votes
2answers
112 views

Find all functions $f:\mathbb R\to \mathbb R$ such that $f(a^2+b^2)=f(a^2-b^2)+f(2ab)$ for every real $a$,$b$

I guessed $f(a)=a^2$ and $f(a)=0$, but have no idea how to get to the solutions in a good way. Edit: I did what was suggested: from $a=b=0$ $f(0)=0$ The function is even, because from $b=-a$ $f(...
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0answers
49 views

Is the function from the Cauchy functional equation, $f(x+y)=f(x)+f(y)$ injective?

It's obviously not injective in the case of $f(x)=0$. I'm wondering if it's injective in all other cases. The other linear solutions of the form $f(x)=c\cdot x$ where $c$ is some constant are ...
16
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3answers
330 views

If $f:[0,\infty)\to [0,\infty)$ and $f(x+y)=f(x)+f(y)$ then prove that $f(x)=ax$

Let $\,f:[0,\infty)\to [0,\infty)$ be a function such that $\,f(x+y)=f(x)+f(y),\,$ for all $\,x,y\ge 0$. Prove that $\,f(x)=ax,\,$ for some constant $a$. My proof : We have , $\,f(0)=0$. Then , $$\...
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0answers
26 views

A simple PDE related to symmetric functions

This question was posed on mathoverflow but was put on hold. So far nobody has been able to give any hint either to the solution or to why it is trivial. Disclaimer: as far as I can tell this is not ...
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1answer
114 views

Is this graph plot for real?

I came across this equation while surfing through the internet.Is this for real?If it is how is it done?
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1answer
120 views

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f (x+xy+f(y) )= (f(x)+ \frac 12 )\ (f(y)+ \frac 12 \ ).$

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f\left (x+xy+f(y) \right )=\left (f(x)+ \frac 12 \right )\left (f(y)+ \frac 12 \right ).$$ for every $x,y \in \mathbb R$. My work ...
2
votes
3answers
46 views

how to calculate derivative of $f_n(x)=f \circ f … \circ f(x)$? Derivative on $f \circ f_{n-1}$ or $f_{n-1} \circ f$?

Denote $f_n(x)=f \circ f ... \circ f(x)$, the $n$th power of composition multiplication of $f(x)$. Assume $f(x)$ is differentiable for any order. $f(1)=1$, $f^{'}(1)=p$, $f^{''}(1)=q$ Question: Get ...
3
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3answers
61 views

How to show that $f$ is a straight line?

Let $f:\mathbb R\to\mathbb R$ be continuous such that $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}~\forall~x,y\in\mathbb R.$ How to show that $f$ is a straight line?
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0answers
29 views

Functional Equation of Probability Distributions

Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning $$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$ Now assume $\Phi(f_X)$ is also a ...
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1answer
47 views

Error with the proof that all solutions to the Cauchy Functional Equation are linear

If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1, 2,3, 4) However, what is wrong with this ...
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0answers
26 views

Class Scatter Fitness Function Calculation

I get a fitness function for class scatter, the equation: Click here to see the Fitness Function Where : T = Transpose of Matrics Mi = class mean Mo = grand mean The equation is based from mean,...
8
votes
1answer
172 views

Find all $f:\mathbb {R} \rightarrow \mathbb {R}$ where $f(f(x))=f'(x)f(x)+c$

Recently, while studying calculus, I have come across multiples problems which asked the following: If $f(x)$ is a polynomial, find all $f(x)$ that $f(f(x))=f'(x)f(x)+c$, where $c$ is a constant. ...
6
votes
1answer
92 views

A Increasing Multiplicative Functional Equation where $nm$ is a cube

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing function such that $$f(2)=2$$ and $$f(mn)=f(m)f(n)$$ for all positive integers $m,n$ such that $mn$ is a perfect cube. Prove ...
1
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0answers
34 views

$(2+z^2) f(z) + 3 z + 4 = f(z+1)$?

Consider the equation $(2+z^2) f(z) + 3 z + 4 = f(z+1)$ Such that $f(z)$ is analytic near the positive real axis and the functional equation holds for real $z>0$. Can we express Some solutions ...
1
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0answers
42 views

Functional equation $f(f(f(x)f(y)))=f(x)f(y^2)$ for $f: \mathbb R \rightarrow \mathbb R$.

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that $f(f(f(x)f(y)))=f(x)f(y^2)$ for all $x, y \in \mathbb R$. I made this problem myself. It is not hard to do it for $f: \mathbb R_{>...
4
votes
1answer
62 views

Find all function $f:\mathbb{R^+}→\mathbb{R^+}$ that satisfies $f(xy)^{xy} =f(x)^x f(y)^y$

Find all function $f:\mathbb{R^+}→\mathbb{R^+}$ that satisfies the given condition: $$f(xy)^{xy} =f(x)^x f(y)^y$$ If $f:\mathbb{R}→\mathbb{R^+}$ the question would be rather simple, as putting in $...
5
votes
0answers
64 views

Increasing function $f(x)$ such that $f(\gcd(x,y))=\gcd(f(x),f(y))$

This problem was largely inspired by this problem here. There were many counterexamples given to the problem, such as multiplicative function that maps primes to a permutation thereof. However, if $...
7
votes
2answers
88 views

Find all $f(x)$ such that $f(gcd(x,y))=gcd(f(x),f(y))$

How does one find all $f:\mathbb {Z} \rightarrow \mathbb {Z}$ that satisfies the following: $$f(gcd(x,y))=gcd(f(x),f(y))$$ I had suspected that there would be some results concerning this functional ...