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 ...

learn more… | top users | synonyms

1
vote
0answers
110 views

where can I find a good Proof for Cauchy's functional equation

Do you know where can I find good proofs for Cauchy's functional equation?
1
vote
2answers
121 views

Get equation for a curve which intersects x at seemingly randomly distributed points?

Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last? I mean, not like a trig ...
8
votes
1answer
642 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 ...
14
votes
3answers
1k views

Find all functions with $f(x + y) + f(x - y) = 2 f(x) f(y)$ and $\lim\limits_{x\to\infty}f(x)=0$.

Determine all functions $f \colon\mathbb{R}\to\mathbb{R}$ satisfying the following two conditions: (a) $f(x +y) + f(x - y) = 2 f(x) f(y)$ for all $x, y\in\mathbb{R}$; (b) ...
4
votes
2answers
276 views

Strategies to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy a given functional equation

My question is as follows: What methods can be used to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ satisfying a certain functional equation. An example of a case where this applies is the ...
6
votes
1answer
894 views

$f(m + f(n)) = f(f(m)) + f(n)$

I found this one in the list of IMO'96 (3) problems and decided to have a go at it, but could not complete the solution. So $m$ and $n$ are non-negative integers and $f$ takes values in the same set: ...
14
votes
2answers
881 views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere. If there exists $c \in \mathbb{R}$ ...
1
vote
0answers
660 views

Solving a functional discrete equation.

I was to solve the following functional discrete equation (I arguing that $a_k$ is a discrete function): \begin{equation}f\left[a_{k+1}\right]-f\left[a_{k}\right]=0\end{equation} where ...
4
votes
3answers
998 views

If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$

For $f:\mathbb{R}^n \to \mathbb{R}^m$, if $f(x + y) = f(x) + f(y)$ for then for rational $c$, how would you show that $f(cx) = cf(x)$ holds? I tried that for $c = \frac{a}{b}$, $a,b \in \mathbb{Z}$ ...
10
votes
1answer
632 views

Riemann's thinking on symmetrizing the zeta functional equation

In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as ...
0
votes
1answer
377 views

Functional equation $g(x+y) = g(x)g(y)$ [duplicate]

Possible Duplicate: continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$ Let $g: \mathbf{R} \to \mathbf{R}$ be a function which is not identically zero and which satisfies the ...
1
vote
1answer
755 views

continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$

Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$. $g(0)=1$. If $a=g(1)$,then $a>0$ ...
1
vote
1answer
371 views

A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere

Let $g: \mathbf R \to \mathbf R$ be a function which is not identically zero and which satisfies the equation $$ g(x+y)=g(x)g(y) \quad\text{for all } x,y \in \mathbf{R}. $$ Show that $g(x)\gt0$ for ...
3
votes
1answer
74 views

How to solve a functional equation of the form $1-g(f(s))=m(1-g(s))$?

I have arrived to this equation in several contexts within branching processes. It arises from textbook exercises, so it must be solvable somehow. Here $f$ is a probability generating function which ...
1
vote
3answers
145 views

Functional equation and Riemann function $ \xi(s) $

Is there any theorem or proof that if a function satisfy the functional equation $ f(1-s)=f(s)$ and $ f(s) >0$ for each real $s$ then $ f(s)= \xi(s)$ or $ f(s)= \operatorname{const}$?
1
vote
1answer
788 views

I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive. [duplicate]

Possible Duplicate: Proving that an additive function $f$ is continuous if it is continuous at a single point Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…) ...
2
votes
1answer
298 views

solve functional equation: $[f(x)]^2-[f(y)]^2$=$f(x+y)f(x-y)$

i am trying to solve following problems and please guys help me suppose that,there is given following equation $[f(x)]^2-[f(y)]^2$=$f(x+y) \cdot f(x-y)$ there was said that,it requires some knowledge ...
1
vote
0answers
115 views

solution of d’Alembert’s equation.

i know that equation for d’Alembert’s equation. is looking so $g(x+y)+g(x-y)=2*g(x)*g(y)$ so am trying to find actual solution for this equation,first i took $x=y=0$ and i got $2*g(0)$=$2*g(0)^2$ ...
2
votes
2answers
1k views

Graph of discontinuous linear function is dense

$f:\mathbb{R}\rightarrow\mathbb{R}$ is a function such that for all $x,y$ in $\mathbb{R}$, $f(x+y)=f(x)+f(y)$. If $f$ is cont, then of course it has to be linear. But here $f$ is NOT cont. Then show ...
12
votes
3answers
218 views

What was this theorem called

Back at the university we have proven (lot of work) that if $$S(X)C(Y)+C(X)S(Y) = S(X+Y)$$ and $$C(X)C(Y)-S(X)S(Y) = C(X+Y)$$ then $S(X)$ is $\sin(x)$ and $C(X)$ is $\cos(x)$ (or constant $0$, meh). ...
1
vote
1answer
172 views

Analytic solution of a certain functional equation

Here's my question. Let $b_2$, ..., $b_d \in \mathbb{C}$ ($d$ is an integer greater than 2), and consider the functional equation $$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$ ...
1
vote
2answers
142 views

Find $f$ such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$

Find all continuous function such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$ for all $x, y\in\mathbb{R}$. Denote $\{x\}=x-[x]$ in which $[x]$ is the largest integer number does not exceed x.
3
votes
2answers
251 views

Recurrence relations on a continuous domain

While attempting to read Shannon's paper I came across the following (p. 3): suppose $N\colon \mathbb{R} \to \mathbb{R}$ is a function, which for some fixed (given) set of values $t_1, t_2, \dots, ...
5
votes
1answer
2k views

Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…)

My lecturer was talking today (in the context of probability, more specifically Kolmogorov's axioms) about the additive property of functions, namely that: $$f(x+y) = f(x) + f(y)$$ I've been trying ...
6
votes
4answers
321 views

3rd iterate of a continuous function equals identity function

If $ f: \mathbb{R} \to \mathbb{R} $ is continuous, and $\forall x \in \mathbb{R} :\;(f \circ f \circ f)(x) = x $, show that $ f(x) = x $. The condition that $f$ is continuous on $\mathbb{R}$ is ...
2
votes
0answers
71 views

Given that $f(x^2)=f(x)^2$ and $f(x+1)=f(x)+1$, try to find $f$ [duplicate]

Possible Duplicate: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that for all reals $x$ $f(x^2)=f(x)^2$ and $f(x+1)=f(x)+1$ Can we show that ...
0
votes
1answer
46 views

In which cases iterative equations can be reduced to finite-difference equations?

In which cases iterative equations can be reduced to finite-difference equations and when they can't?
11
votes
3answers
680 views

$f(x^2) = xf(x)$ implies that $ f(x) = mx$?

Suppose a function $f : \mathbb{R} \to \mathbb{R} $ satisfies the relation $$f(x^2) = xf(x) \ \ \forall x$$ Does this imply $f$ must be a straight line, $f(x) = mx$? If so, why? If not, are there ...
2
votes
0answers
416 views

Finding inverse of function without knowing function?

This question has a programming application, but I thought it would be more appropriate (and educational) to ask here first and get some basic understanding. So my use-case will refer to some ...
13
votes
3answers
769 views

very elementary proof of Maxwell's theorem

Maxwell's theorem (after James Clerk Maxwell) says that if a function $f(x_1,\ldots,x_n)$ of $n$ real variables is a product $f_1(x_1)\cdots f_n(x_n)$ and is rotation-invariant in the sense that the ...
11
votes
1answer
286 views

Is there a real-valued function $f$ such that $f(f(x)) = -x$?

Is there a function $f\colon \mathbb{R} \to\mathbb{R} $ such that $ f(f(x)) = -x$ ?
1
vote
0answers
123 views

Could anyone derive a formula for this?

Edited: I want to get a sentiment score of various sentences and I've tried coming up with an equation that could satisfy the conditions that are inherent to each sentence (It's estimated mood as ...
0
votes
2answers
82 views

General aggregation functions

Is there a way to find all or some functions which "aggregate" numbers and are non-isomorphic to addition. I mean functions which are commutative and associative: $f(x,y)=f(y,x)$ ...
4
votes
1answer
593 views

Limit of a function satisfying an inequality

If $f(x)+f(y)\leq f(x+y)$ and $f:\mathbb{R}\to\mathbb{R}$, then can we find $\lim_{x\to 0} \frac {f(x)}{x}$? I am not sure whether the question is correct.Thank you.(I tried this idea: ...
1
vote
0answers
394 views

functional derivative of an integral of the function itself

I have the following $$ \frac{d}{dn(x)} \int_{x \in \cal{R}^3} {n(x) dx} $$ I know that this additional relationship holds $$ \int_{x \in \cal{R}^3}{n(x) dx} = N $$ where N is a constant. My ...
5
votes
2answers
1k views

How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable

Suppose $f(x)$ is integrable in any bounded interval on $\mathbb R$, and it satisfies the equation $f(x+y)=f(x)+f(y)$ on $\mathbb R$. How to prove $f(x)=ax$?
11
votes
2answers
598 views

Which trigonometric identities involve trigonometric functions?

Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled functional equation gave the identity $$ \sin^2\theta+\cos^2\theta = 1 ...
5
votes
1answer
961 views

Examples of functions where $f(ab)=f(a)+f(b)$

What are some examples of continuous (on a certain interval) real or complex functions where $f(ab)=f(a)+f(b)$ (like $\ln x$?)
3
votes
0answers
249 views

A function $f(x)$ such that $f(f(x))=\ln(x)$

How to find a continuous function $f(x)$ for $x > 12$, such that $f(f(x))=\ln(x)$? Preferably analytic too.
0
votes
1answer
112 views

Bijective functions $f(n)=f(f(n-1))=f^n(1)$

How can I find a function (or more) which satisfy $f(n)=f(f(n-1))=f^n(1)$, defined for positive integers n, such that it is a bijection between the positive integers? And which satisfy i) For every ...
2
votes
1answer
464 views

About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How to show that ...
4
votes
3answers
252 views

Solving the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$

I am trying to solve the equation $f(x+t)=f(x)+f(t)+2\sqrt{f(x)}\sqrt{f(t)}$ - as in find a function that satisfies this equation. I notice that the RHS is $({\sqrt{f(x)}+\sqrt{f(t)}})^2$ but I am ...
2
votes
1answer
3k views

What does it mean for a functional equation to have a unique solution?

It is my thinking that unique conventionally means special or one of its kind. But in the context of solving functional equations*, I am confused what it means to have a unique solution... *e.g. Find ...
1
vote
2answers
240 views

Is there a continuous {permutation,duplicate,translation,stretch}-invariant function on ordered sets of vectors that returns a vector?

Is there any example of such a function $f$, preferably one defined on all $V^n$ and all positive integers $n$ where $V$ is some vector space? It must satisfy the following: $f(T) = f(\sigma(T))$ ...
0
votes
3answers
286 views

What is the solution of $f(x)\cdot f(-x) = 1$

What is the general solution of the equation? $$f(x) \cdot f(-x) = 1$$ I know that $f(x) = A^{k \cdot x}$ is a solution, and I am feeling this is the general solution, but I don't have any proof. ...
7
votes
3answers
2k views

Proving that an additive function $f$ is continuous if it is continuous at a single point

Suppose that $f$ is continuous at $x_0$ and $f$ satisfies $f(x)+f(y)=f(x+y)$. Then how can we prove that $f$ is continuous at $x$ for all $x$? I seems to have problem doing anything with it. Thanks in ...
1
vote
5answers
718 views

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

Is there any function $f$ which would satisfy $f(x)=f(x+1)$ and $f(-1/x)=f(x)$ for every $x$ or at least positive $x$? For the widest possible domains of $x$? If I could turn this functional equation ...
7
votes
1answer
424 views

Entire function with $f(z)=\sin(f(z))$ must be constant?

I'm trying to show why an entire function with the property $f(z)= \sin(f(z))$ everywhere must be constant. Is it sufficient to say that when taking the derivatives, we will get $f'(z)=f'(z) ...
2
votes
1answer
250 views

How to solve polynomial functional equation $P(x,y)=P(\frac{x-y}{2},\frac{y-x}{2})$?

Given $P(x,y)$ which is a polynomial function, satisfying $P(x,y)=\displaystyle P(\frac{x-y}{2},\frac{y-x}{2})$. Then why should $P(x,y)$ be $\displaystyle\sum^n_{i=0}a_i(x-y)^i$? Is it unique?
3
votes
0answers
149 views

Analytic function that provide $f^2(z)=z$

I am trying to solve this problem: Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$. I have seen a solution that ...