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
96 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
2
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1answer
62 views

Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...
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5answers
192 views

Solutions to functional equation $f(f(x))=x$

Is there any more solutions to this functional equation $f(f(x))=x$? I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.
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1answer
45 views

Able to adjust a Function/Formula with Weighted Variables to correct for changes in a variable?

I apologize for the somewhat cryptic title as I don't quite know how to word it. I have a somewhat abstract question that may have a simple answer. But I am wracking my mind all over this! So ...
2
votes
1answer
72 views

Solutions to functional equation $f(at+x)+g(x)+h(t,bt+x) =0 $

Let $a \neq 0$ and $b \neq 0$ be fixed constants with $a \neq b$. Find all twice continuously differentiable functions $f:\mathbb{R}\rightarrow \mathbb{R}$, $g:\mathbb{R}\rightarrow \mathbb{R}$ and ...
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2answers
107 views

Functional equation with strange property about irrational numbers

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number r, and any real number x we have $f(x)=f(x+r)$. Show that f is a constant function. It's easy to see any constant ...
2
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1answer
110 views

Functional equation with cyclic function.

Find all functions $f:\mathbb R \to \mathbb R$ that satisfy: $$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x.$$ Some progress: I plugged-in $\dfrac{x-1}{x}$ and $\dfrac{1}{1-x}$, got a system of ...
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1answer
58 views

Uniqueness of Pexider's functional equation

Let $f:\mathbb{R}\rightarrow\mathbb{R}$, $g:\mathbb{R}\rightarrow\mathbb{R}$, and $h:\mathbb{R}\rightarrow\mathbb{R}$ and consider Pexider's equation, $$ f(x) + g(y) = h(x + y) \qquad \qquad (1) $$ ...
3
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0answers
36 views

Convergence of sum of antiderivative and derivative

This question is inspired by this question: Solutions for $ \frac{dy}{dx}=y $?. It makes me wonder if there are any function where the sum of all antiderivative and derivative converges. The ...
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3answers
1k views

I am searching for an unusual real-valued function.

It needs to fulfill following condition for all positive $x,y ∈ ℝ$: $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ any ideas?
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0answers
32 views

Need a equation that defines a certain number

Im programming a function but I just cant structure the equation. I think this is the right place to ask since the problem is completely mathematics. Let me explain three scenarios. There are 4 ...
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vote
0answers
90 views

Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
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1answer
44 views

Extending $f(p^k)$ where $p$ is prime

If we have a function $f(x)$, for which we know that $f(p^k)=(p^s+1)^k p^{sk}$ where $p$ is prime, $k$ is a real number, and $s$ is a constant, how do we find $f(x)$? My try: let $k=\log_p(x)$, so ...
5
votes
2answers
205 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
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2answers
72 views

Equation Theory: A polynomial with specific remainders when divided by specific divisors. What is the remainder when divided by BOTH divisors

Again, for my Equation Theory class, I have the subject question.$p(x)$ has a remainder of 3 when divided by $x-1$ and a remainder of 5 when divided by $x-3$. What is the remainder when $p(x)$ is ...
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4answers
124 views

Polynomials that satisfy $(x-1)(p(x+1))=(x+2)(p(x))$ where $p(2)=12$?

I am taking a graduate class on Equation Theory and one of my homework questions asks me to "Determine all polynomials $p(x)$ such that $(x-1)(p(x+1))=(x+2)(p(x))$ and $p(2)=12$. A provided hint is to ...
9
votes
2answers
131 views

Functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$

I was looking for examples of real valued functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$. Preferably, I'd like them to be continuous, differentiable, etc. Of course, there are the constant functions ...
7
votes
1answer
162 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
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votes
0answers
30 views

An equation with a nested function

I'm trying to find the function $\eta(x)$ such that $\eta(x) F(\eta(x))-G(\eta(x)) = \eta(x)H(x) - G(x)$ but I have no idea how to go about it, or where to look. Thanks for the inputs. All ...
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votes
0answers
48 views

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$?

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$ for all $y\in \mathbb{R}$? Under what conditions would this imply that $f(x,y)=g(x,y)$?
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2answers
57 views

What functions satisfy the condition $f(x,y)=g(x)$?

Are there any functions $f(x,y)$ and $g(x)$ that satisfy (1) $f(x,y)=g(x)$ for all $x \in \mathbb{R}$ and $y\in \mathbb{R}$ (2) $f(x,y)$ is not constant in $y$ for each $x$ (i.e. for each $x$ there ...
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0answers
58 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
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0answers
56 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ...
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0answers
38 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
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1answer
89 views

Solving a second level functional equation over all functions $g$

I am trying to find a closed form expression $f$ such that $$f(g(x+1) - g(x)) + f(g(x) - g(x-1)) = f(g(x))$$ For all functions $g$ I have concluded that for polynomials $$2^{n+1}f(0) = f(a_0 + ...
4
votes
3answers
149 views

Solve functional equation $f(f(f(x)))+f(x)=2x$

Please help me solve this functional equation: find $f(x)$ given that $$f(f(f(x)))+f(x)=2x$$ Thanks very much.
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votes
1answer
80 views

Find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(mx+c)=mf(x)+c$

Find all the functions $f:\mathbb{R}→\mathbb{R}$ such that $f(mx+c)=mf(x)+c$, $m≠1$. I know that $f(x)=x$ and $f(x)=c/(1-m)$ are two solutions. But to completely solve it I have no idea. Can we ...
5
votes
1answer
183 views

find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $ [duplicate]

This is a very hard functional equation. the problem is this : find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $ to solve it i have no idea! can we solve it ...
3
votes
1answer
152 views

General Solution of Functional Equation

What is the general solution to: $$ \frac{f(x+h)-f(x)}{h} = \frac{1}{x} \tag{1} $$ Obviously the solution to this for the limiting case of $h\to 0$ is $f(x) = \ln(x) + c$ Attempting to solve the ...
3
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2answers
156 views

calculation of all function which satisfy $f(x – y) = f(x) f(y) – f(a – x) f(a + y)\;,$ where $f(0)=1$

A real valued function $f(x)$ satisfies the functional equation $f(x – y) = f(x) f(y) – f(a – x) f(a + y)$ Where $a$ is a given constant and $f(0) = 1\;,$ Then prove that $f(2a – x) = -f(x)$. ...
3
votes
0answers
74 views

Continuous functions such that $f(x)+f(x^2)=x$ [duplicate]

Find all continuous $f :[0,1]\to \mathbb R$ such that $\forall x\in [0,1], f(x)+f(x^2)=x$ I suspect there are none. I made little progress so far, but it's worth noticing that $f(0)=0$ and ...
4
votes
2answers
179 views

Find $f$ such that $\frac{d^2}{dx^2}f(x)=f(\sqrt{x})$

Which non-constant functions $f$ (if any) satisfy $\dfrac{d^2}{dx^2}f(x)=f(\sqrt{x})$ for $x>0$? I suspect there is no $f$ which satisfies the differential equation, but I cannot prove this.
8
votes
1answer
194 views

Functional equation and fixed points

Let $f$ be strictly increasing and such that $f(x)+f^{-1}(x)+1=e^x$. Is it true that $f$ has at most one fixed point? I am told the answer is yes, but I am having trouble proving it. It's obvious ...
11
votes
1answer
340 views

Additive functional inequality

The function $f:R_+\to R_+$ is continuously differentiable and increasing. Also, $f(0)=0$ and $f(\infty)=\infty$. Continuity and differentiability of higher orders can be assumed if necessary. ...
8
votes
4answers
222 views

Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
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0answers
42 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
9
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4answers
192 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
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1answer
44 views

A logarithm-like functional equation

Suppose we are given that a monotonically decreasing smooth function $f$ on $(0,\infty)$ obeys the functional equation $f(x) = -f(\frac{1}{x})$, and satisfies $f(\frac{1}{3}) = \frac{1}{2}$ and ...
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1answer
43 views

Understanding an Equation and how to implement it

A common method for linking language with psychological variables involves counting words belonging to manually-created categories of language. One counts how often words in a given category are used ...
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16 views

Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
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2answers
353 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
6
votes
2answers
146 views

Problem about functional equation (Bulgarian selection team test)

This problem was taken from the bulgarian selection team test for the 47th IMO and appeared in a chinese magazine, I came across it in my own training. http://www.math.ust.hk/excalibur/v10_n4.pdf ...
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3answers
175 views

functions satisfying $f(x)=2f(2x)$

How can I prove every smooth real function f satisfying $f(x)=2f(2x)$ is of the from $f(x)=k/x$ where $k$ is a constant? I have tried this for integers that can be divided by $2^n$, but then I cannot ...
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votes
0answers
39 views

A functional equation for a Dirichlet serie

I'm looking for a functional equation for the following Dirichlet serie $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ? Thank you !
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0answers
26 views

What is the function $f$ verifying : $f(\frac{x}{2}+\frac{x}{2}\cos(\frac{v\pi}{x}))=\frac{x}{2}\sin(\frac{v\pi}{x})$

What are the solutions to the functional equality (for a constant $v$): $$ \forall\, x > 0, \ \ \ \ ...
0
votes
1answer
63 views

Functions such that $f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$

What are the continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for every $x,y\in \mathbb{R}$ $$f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$$
5
votes
4answers
158 views

Find all functions $f(x)$ such that $f(x^2-y^2)=(x-y)(f(x)+f(y))$.

find all functions $f(x):R\to{}R$ such that $f(x^2-y^2)=(x-y)(f(x)+f(y))$. I have derived this clues:- $f(0)=0$, $f(x^2)=xf(x)$, $f(x)=-f(-x)$ but now I am confused. I know solution will be ...
1
vote
1answer
57 views

Functional equations and cubes

Problem $10728$ from Amer. Math. Monthly "Preserving the sum of three cubes" says: Determine all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying $f(x^3+y^3+z^3)=(f(x))^3+(f(y))^3+(f(z))^3$ ...
0
votes
2answers
80 views

Find all $f:\mathbb R\to\mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $xf(y)+yf(x)=(x+y)f(x)f(y)$.

Find all $f:\mathbb R\to \mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$ My try: whenever $y=0$, we have $$x\cdot f(0)\cdot(1-f(x))=0$$ ...
8
votes
1answer
131 views

All functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$

I just thought about the following question: Find all functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$ for all $m,n\in\mathbb{N}$. Clearly every polynomial $g(X)\in\mathbb{Z}[X]$ in ...