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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4
votes
3answers
89 views

Retrieving the Taylor series for $\log$ from its functional equation

Consider the unique continuous function $\mathbb{R}^+\to\mathbb{R}$ such that: $$f(xy)=f(x)+f(y),\qquad f(e)=1$$ where $\displaystyle e=1+\sum_{n=1}^{\infty} \frac{1}{n!}$. Assuming $f$ has a ...
5
votes
5answers
129 views

Is there analytic solution to $x^y=y^x\land x\neq y$ as $y(x)$?

Equation $x^y=y^x\land x\neq y$ has trivial solution $ y(x) = x$. Is there non trivial solution given say in terms of elementary or special functions as $y(x)$? A solution that would yield $y(2) = 4$ ...
0
votes
1answer
27 views

Finding polynomials sattisfying $P\bigr(-c + K/(u+c)\bigl) (u+c)^2/K =P(u)$

Is there any simple way to find the polynomials satisfying the functional relation \begin{align*} P\left(-c + \frac{K}{u+c}\right) \frac{(u+c)^2}{K} = P(u) \tag{*} \end{align*} Where $K = ...
4
votes
2answers
126 views

How to solve $(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$

$$(f'(x+1)+f'(x-1))f(x)-(f(x+1)+f(x-1))f'(x)=0$$ I don't have any ideas about the solution of this problem. How can I solve this differential equation?
17
votes
1answer
338 views

Find $f(x)$ where $ f(x)+f\left(\frac{1-x}x\right)=x$

What function satisfies $ f(x)+f\left(\frac{1-x}x\right)=x$ ?
1
vote
2answers
103 views

show that $f(x)=c\log x $ for some $c$

Let, $f$: $\mathbb{R^+}$$\rightarrow$$\mathbb{R}$ be a continuous function satisfying $f(xy)=f(x)+f(y)$. Prove that, $f(x)=c\log x$ for some $c>0$.
1
vote
0answers
104 views

Galois Theory for Differential Equations?

Consider the set of algebraically primitive functions: that is the set of functions who can be created through some recursive expression involving arithmetic operators. Example $$y = x +1$$ $$y = ...
3
votes
5answers
86 views

Simple functional equation

I have a simple functional equation: $$ \alpha(x) = {1 \over 2}\left[\alpha\left(x - 1\right) + \alpha\left(x + 1\right)\right]\,, \qquad \alpha\left(0\right) = 1\,,\quad\alpha\left(m\right) = 0 $$ I ...
2
votes
1answer
90 views

How find this function $f(2^m)=?$

show that:there exists unique function $f:N^{+}\to N^{+}$,such $$f(1)=f(2)=1$$ and $$f(n)=f(f(n-1))+f(n-f(n-1))),n=3,4,\cdots$$ and Find $f(2^m),m>2,m\in N$ My try: let $n=3$ ,then we have ...
7
votes
1answer
175 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
18
votes
2answers
564 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to 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 ...
2
votes
1answer
128 views

Characterization of arithmetic mean

Let $f_m$: $\mathbb{R}_{\geq 0}^m \to \mathbb{R}_{\geq 0}$ be a series of functions that satisfy symmetry (when permuting indices), strong monotonicity (in every entry), homogeneity of degree 1, ...
1
vote
2answers
68 views

functional equations with restricted domain

How can I find all discontious solutions of functional equation $f(xy)=f(x)f(y)$ on $[0,1]$. Similar question is to find all solutions of the equation $f(x+y)=f(x)+f(y)$ on $[0,\infty)$. Can we still ...
1
vote
6answers
204 views

Solutions of $f(x)\cdot f(y)=f(x\cdot y)$ [duplicate]

Can anyone give me a classification of the real functions of one variable such that $f(x)f(y)=f(xy)$? I have searched the web, but I haven't found any text that discusses my question. Answers and/or ...
6
votes
1answer
237 views

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $f(xy)+f(x+y)=f(xy+x)+f(y)\quad\forall x,y \in \mathbb{R}$

Find all continuous functions $f$ : $\mathbb{R} \rightarrow \mathbb{R}$ satisfied that: $$f(xy)+f(x+y)=f(xy+x)+f(y)$$ $\forall x,y \in \mathbb{R}$ I have tried that : $P(y;x)-P(x;y)$: ...
9
votes
3answers
140 views

Solutions for the Functional Equation $f(x^2)=f(x)^2$

Suppose a continuous function $f:[0,1]\rightarrow [{0,1}]$ satisfies the functional equation $f(x^2)=f(x)^2$. Then I conjecture that we must have $f(x)=0$ or $f(x)=x^r$ for some real number $r\geq 0$. ...
2
votes
2answers
39 views

function equation with translation of independent variable

The following has come up in some work I'm doing: If $\frac{f(x+a)}{f(a)}=g(x)$ , where $g(x)$ is given and $a \geq 0$ is a constant, what is $f(x)$ ? We can assume that $g(x)>0 ~ \forall x$ . Of ...
5
votes
3answers
72 views

Simple Functional Equation $\frac{f(a)-f(b)}{a-b}\cdot(-a)+f(a)=-ab$

Compute all real-valued functions $f$ so that the line between any two points on the graph $f$ intersects the $x$-axis at the product of those two points' $x$-coordinates times $-1$. (if we do some ...
1
vote
1answer
49 views

Solving limit equations

Notice the following: all first order differential equations take on the form: $$G(x,f(x),\frac{d}{dx}[f(x)]) = 0 $$ notice that we can replace $\frac{d}{dx}[f(x)]$ with the expression $$ ...
8
votes
2answers
155 views

What do logarithms distribute over?

I notice that division distributes over addition Root extraction distributes over multiplication What operator do logarithms distribute over: ie: what non-constant function $H \in C^2 \rightarrow C$ ...
4
votes
2answers
71 views

finding the value of $f(\frac{1}{7})$

$f$ is a function mapping positive reals between $0$ and $1$ to reals. Let $f$ be given by, $f( \frac{x+y}{2} ) = (1-a)f(x)+af(y)$ where $y > x$ and $a$ being a constant. Also,$f(0) = 0$ and $f(1) ...
11
votes
2answers
156 views

Solve $f^2(x)=x+f(x+1)$

If the function $f(x)$ is such that $$f^2(x)=x+f(x+1),$$ find a closed-form expression for $f$. I found $$f(x)=\sqrt{x+\sqrt{x+1+\sqrt{x+2+\sqrt{x+3+\cdots}}}}$$ is such an $f$. Does anyone have ...
6
votes
2answers
98 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
0
votes
2answers
45 views

Summarize my formula?

I would like to summarize my formula. $p$ and $y$ are constant value, $10000$ and $0.65$. When $n = 3$, my formula recalculate the result of $n = 2$. I don't want to recalculate. Is there way to ...
0
votes
1answer
35 views

functional equation, find $h$ continous on $\mathbf{R}$ such that $h(x) + h(2x) + h(4x) = {x^n}$

I am having trouble finding easily $h$ defined and continuous on $\mathbf{R}$ verifying for all $x$ in $\mathbf{R}$, $$ h(x) + h(2x) + h(4x) = {x^n} $$ where $n$ is a fixed natural number. I have a ...
2
votes
3answers
223 views

Really nice functional equation with second partial deratives.

Show that $f:\mathbb{R}^2\to\mathbb{R}$, $f \in C^{2}$ satisfies the equation $$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{\partial y^2} = 0$$ for all points $(x,y) \in \mathbb{R}^2$ if ...
1
vote
1answer
53 views

What is a function satisfying these constraints?

I'm looking for a monotonically increasing function $f\colon (0,1)\times(0,1)\mapsto(0,1)$ satisfying: $f(x,y)=1-f(1-y,1-x)$ If $x >\frac{1}{2}$, $f(x, y)>y$ $f(\frac{1}{2}, y)=y$ If $x ...
3
votes
3answers
107 views

Solving equation system of complex funtions

Does there exist two complex functions $f$ and $g$ satisfy below equation system? $$ \begin{cases} f=e^g\\ g=e^f \end{cases} $$ What about analytic funtions?
0
votes
2answers
71 views

Is this $f$ a linear function?

My question is related to this as I posted earlier. But this time, we drop certain conditions: Suppose $f:[a,b]\to\mathbb{R}$ be continuous and there exists a sequence $(\alpha_n)_{n=1}^{\infty}$ ...
0
votes
0answers
12 views

Functional equations on $\mathbb{C}^*$

Let $\tau\in\mathbb{C}$ with $\Im(\tau)>0$, $k=\exp(-\pi i\tau)$ and $q=\exp (2\pi i \tau)$. Let $c:\mathbb{C}^*\rightarrow\mathbb{C}^*$ be a holomorphic function. Determine all $c$ such that ...
1
vote
1answer
65 views

Help with a functional equation

I am asked to find all $f : \mathbb{R} \to \mathbb{R}$ such that $f(x - f(y)) = f(f(y)) + 2 x f(y) + f(x) - 1$, which I solved and got that $f(x) = 1 - x^2$ -- a correct solution and everything. What ...
2
votes
0answers
61 views

On a specific non-linear partial differential equation

Given an $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$ and the functions $h_i: \mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\dots,l$, we would like to find a solution of the following equation: ...
8
votes
1answer
429 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
9
votes
2answers
276 views

How to prove that $f(f(x))=-x$ implies that $f$ is not continuous? [duplicate]

I am trying to prove that: Given an $f:\mathbb{R} \rightarrow \mathbb{R}$, if $f(f(x))=-x$ then $f$ is not continuous? any help? Thank you!
0
votes
1answer
70 views

A functional equation for homogeneous functions of degree zero

Let a function $F: \textbf{R}^2_{++} \rightarrow \textbf{R}$ satisfy the relation: if $F(x,y)=F(x',y')$ then $F(x,y)=F(x+x',y+y')$. It is easy to prove that under the additional assumption of ...
3
votes
1answer
67 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$. I know how to show it is continuous but no clue how to show ...
0
votes
0answers
47 views

Integral Functional Equation

If $f(x) = 1 + \int_{x/c}^1 f(t) dt$, find $f(x)$ in terms of $c$. This problem was inspired by trying to find the expected value of the longest increasing subsequence $a_1, a_2, a_3, \cdots, a_n$ ...
1
vote
0answers
33 views

Solution of equation with special function using maths

I have the following equation with β∈[0,1] and δ∈[0,1] are 2 parameters and $\mu$, $\sigma$ are the mean and standard deviation of a random variable. Is it possible to use software to get the explicit ...
7
votes
3answers
173 views

Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
1
vote
0answers
49 views

Differential Equation for CDF

Consider the following differential equation $$F(cx) = F(x) + x F'(x)$$ for $c>1$. Does this differential equation belong to a some well known class? Is there a way to find all the solutions ...
2
votes
4answers
106 views

Solve $\frac{d}{dx}f(x)=f(x-1)$

I am trying to find a function such that $\dfrac{d}{dx}f(x)=f(x-1)$ Is there such function other than $0$ ?
0
votes
1answer
44 views

Convert this scenario into algebra equation

Sales for the month minus the VAT @ 20% = (x). 20% of (x) is profit margin (y). 5% of (y) is commission earned (c). How can I write an equation that demonstrates the above please? I.e x - 20% of y ...
14
votes
1answer
330 views

Condition for an additive function to be continuous

The problem below is Problem 7 from this year's Miklos Schweitzer competition (contest ended Nov 4th). Suppose that $f: \Bbb{R} \to \Bbb{R}$ is an additive function (that is $f(x+y) = f(x)+f(y)$ ...
2
votes
2answers
167 views

Functional equation: $f\left(\frac{x-1}{x}\right)+ f\left(\frac{1}{1-x}\right)= 2- 2x$

There is a function given $f\left(\dfrac{x-1}{x}\right)+ f\left(\dfrac{1}{1-x}\right)= 2- 2x ,f\colon \Bbb R\setminus\{0,1\}\to \Bbb R$ How many fuction exist? I have no idea how to start
3
votes
1answer
106 views

is this expansion possible (f+g)(t)= f(t)+g(t)?

Hi if the functions were polynomials is (f+g)(t)= f(t)+g(t) possible? I am trying to integrate a function of that form
32
votes
7answers
791 views

How to find $f$ if $f(f(x))=\frac{x+1}{x+2}$

let $f:\mathbb R\to \mathbb R$,and such $$f(f(x))=\dfrac{x+1}{x+2}$$ Find the $f(x)$ My try I found $f(x)=\dfrac{1}{x+1}$ because when $f(x)=\dfrac{1}{x+1}$,then ...
1
vote
2answers
91 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
3
votes
2answers
95 views

Finding every possible $f(x), a\in\mathbb R$ such that ${\{f(x)\}}^2=a+f(x^2)$

Let $f(x)$ be a rational expression of $x$, and let $a$ be a real number. Then, I'm facing difficulty for finding every possible $a, f(x)$ such that $${\{f(x)\}}^2=a+f(x^2).$$ Here, suppose that ...
3
votes
2answers
158 views

Do there exist functions such that $f(f(x)) = -x$? [duplicate]

I am wondering about this. A well-known class of functions are the "involutive functions" or "involutions", which have that $f(f(x)) = x$, or, equivalently, $f(x) = f^{-1}(x)$ (with $f$ bijective). ...
3
votes
2answers
185 views

Functions minimized at the median of their arguments

I am doing research on problems of location of a public facility on a network which lead me to the following question. Is there an interesting way to characterize the class of functions $f : ...