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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3
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3answers
344 views

A simple but weird functional equation

Let $f$ be a function $f:\mathbb R\to\mathbb R$. Find all functions $f$ that satisfy: $$f(x^2+x+3)+2f(x^2-3x+5)=x^2-x+ \frac{18}{4} + \frac{111}{444} + \frac{222}{333}$$ Maybe the question is ...
2
votes
1answer
73 views

Find all continous functions satistying $ f(x) = f(x^{2} + \frac{x}{3} + \frac{1}{9})$ for all $x \in \mathbb{R}$

The problem I am trying to solve now is to find all continous functions satistying $f(x) = f(x^{2} + \frac{x}{3} + \frac{1}{9})$ for all $x \in \mathbb{R}$ It is the first time for me to face this ...
1
vote
2answers
89 views

when does $f(a)^{f(b)}=f(a^b)$?

First $\text{f}\left( 1 \right)=1$ beacause $\text{f}\left( a \right)^{\text{f}\left( 1 \right)}=\text{f}\left( a \cdot 1 \right)$, and $\log_{\text{f}\left( a \right)} \text{f}\left( a \right)^{\...
3
votes
1answer
305 views

Equation for finding maze solvability

I am programming a game where users can edit the state of a maze. The state of each vertical and horizontal wall (present/not present, on/off, 1/0, etc...) is stored in a database and then referenced ...
4
votes
2answers
366 views

Polynomials $P(x)$ satisfying $P(2x-x^2) = (P(x))^2$

I am looking (to answer a question here) for all polynomials $P(x)$ satisfying the functional equation given in the title. It is not hard to notice (given that one instinctively wants to complete the ...
3
votes
1answer
227 views

A Functional Differential Equation

I was having a play with some trig. identities and noticed the following: $\cos{x}=\frac{\sin{2x}}{2\sin{x}}$ Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential ...
4
votes
2answers
231 views

Many other solutions of the Cauchy's Functional Equation

By reading the Cauchy's Functional Equations on the Wiki, it is said that On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely ...
1
vote
1answer
119 views

$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$

Find all continous functions $f\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(x)=f(x^2+ 1/4)$ What I've tried so far: suppose that $f$ is one-one thus $x=x^2+1/4$ ... $x=1/2$ then $f(x)=f(1/...
4
votes
1answer
68 views

A functional equation over a circle

I am interested in the functional equation $$f(r \cos \phi)+f(r\sin \phi)=f(r),\qquad r\geq 0,\ \ \phi\in[0,\pi/2].$$ Let's assume that $f:[0,\infty)\to\mathbb R$ is monotone. Clearly, $f(x)=ax^2$ is ...
2
votes
2answers
49 views

why is it constant?

$f$ is a function from $\Bbb R$ to $\Bbb R$: $\frac{f(x+y)}{(x+y)} - (x+y)^2 = \frac{f(x-y)}{(x-y)} - (x-y)^2$ for all $x$ and $y$ the solution book just says "Thus $\frac{f(x)}{x} - x^2$ is a ...
1
vote
0answers
73 views

How to calculate straight line into graph having variety of different results

How to calculate straight line into graph having variety of different results. What I mean for example let say we have this kind of results (measuring persons weight ...
15
votes
4answers
433 views

Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds: $$f(x+y)=f(x^{2}+y^{2}).$$
2
votes
2answers
128 views

Functional equation (show that)

Show that there does not exist a function $f:\mathbb N\to \mathbb N$ which satisfy a) $f(2) = 3$ b) $f(mn) = f(m)\cdot f(n)$ for all $m,n \in \mathbb N$ c) $f(m) < f(n)$ whenever $m < n$
10
votes
3answers
302 views

Functional Equation (no. of solutions): $f(x+y) + f(x-y) = 2f(x) + 2f(y)$

Find all the functions $f\colon\mathbb{Q} \to \mathbb{Q}$ such that $f(x+y) + f(x-y) = 2f(x) + 2f(y)$, for all rationals $x,y$.
2
votes
0answers
71 views

Finding every $n$ such that there exists a $n$-th degree polynomial which satisfies $f(x^2+1)={f(x)}^2+1$ [duplicate]

I'm interested in functional equation. I've been thinking about the following functional equation: $$f(x^2+1)={f(x)}^2+1\ \ \ \cdots(\star).$$ I found several functions such as $f(x)=x, x^2+1, (x^2+1)...
1
vote
1answer
71 views

Proof read of functional equations

My teacher gave me this functional equation as an excercise $$f(x+f(y))=x+f(f(y))\,\, \forall\,\, x,y \in \mathbb{R}$$ If $f(2)=8$, calculate $f(2005)$ So my solution was For every $y$, let $f(y)=c$...
0
votes
1answer
151 views

Function $f(x)$ such that $f(x-i)+f(x)=\frac{1}{x^2}$

Help me find a function $f(x)$ such that $$f(x-i)+f(x)=\frac{1}{x^2}$$ where $i$ is the imaginary unit.
1
vote
2answers
113 views

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$

Determine the polynomials $p(x)$ satisfying $x\cdot p(x-1) = (x-26)\cdot p(x)$. My Solution: Put $x=0$, we get $p(0) = 0$, Similarly put $x=26,$ we get $p(26) = 0$. That means $x=0,26$ are two roots ...
16
votes
6answers
887 views

$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.

Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$. I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying ...
32
votes
7answers
6k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the ...
1
vote
0answers
48 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
4
votes
1answer
277 views

Mean density of the nontrivial zeros of the Riemann zeta function

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
1
vote
0answers
73 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
0
votes
1answer
55 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in C[0,1]}=\int_0^{s^{-1}(...
5
votes
2answers
279 views

Iterative roots of sine

Is there an analytical function $f(z)$ such that $f(f(z)) = \sin(z)$? More generally, an analytical function such that f applied $n$ times to $z$ gives $\sin(z)$? Is there a general theory for ...
1
vote
2answers
87 views

Solve this equation: $f(s)=P(s)\exp(Q(s))$

Let $f,P,Q$ three analytic functions. Here $P$ is a polynomial. I want to solve this equation: $$f(s)=P(s)\exp(Q(s)).$$ The unknown here are $P, Q$ and $f$ is known.
0
votes
3answers
80 views

Solve this functional equation: $h(-s)=a-h(s)$

Let $h$ be an analytic function. My question is : Solve this functional equation: $$h(-s)=a-h(s)$$ holds true for all $s∈ℂ$. Here, $a∈ℂ$, $a≠0$.
3
votes
1answer
40 views

A question on functional equations.

Question: If it is given that $$ e^xf(x) = 2 + \int_0^x\sqrt{1+x^4}\,dx $$ then what is the value of $ \dfrac {d} {dx} \Big(f^{-1}(x)\Big)\Bigg|_{x=2} $ Where I am stuck: Now, since we are to ...
1
vote
0answers
312 views

iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations. For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear ...
2
votes
1answer
85 views

How prove this sequences have limts and find this value?

let $f:R\longrightarrow R$,and $f(x)=x-\dfrac{x^2}{2}$,and $$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$ and let $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ ...
-3
votes
2answers
527 views

$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$ [duplicate]

$g(x+y)=g(x)g(y)$ for all $x,y \in\mathbb{R}$. If $g$ is continuous at $0$, prove that $g$ is continuous on $\mathbb{R}$.
2
votes
0answers
123 views

Unicity of solution to an integral equation

Suppose we have the following integral equation $$f(x)=\int_0^\infty f(y)(x+y)e^{-x^2/2-xy}\text{d}y$$ where $f: [0,\infty)\rightarrow[0,\infty)$. The function $f(x)=Ce^{-x^2/2}$ solves the equation. ...
2
votes
1answer
108 views

Find $f(x)$ from $2f'(x)-3f'(1/x)=x$

Find $f(x)$ given that $2f'(x)-3f'(1/x)=x$ Also, is it possible to do this without integration?
7
votes
0answers
117 views

Any chance for a solution of $f[f(x)]=x^2+1$? [duplicate]

Not sure if this is a closed or open question. But the question is suppose $f[f(x)]=x^2+1$ then what is $f(x)$? Though the question does not refer to the domain let us suppose it is defined on $\Bbb R$...
2
votes
2answers
113 views

Solve the function equation $g^2(x)-g(x+1)-\dfrac{x^2+2x-6}{4}=0$

let $g(x)\in \Bbb R$ and for any $x\in \Bbb R$ such that $$g^2(x)-g(x+1)-\dfrac{x^2+2x-6}{4}=0, g(0)=0$$ find $g(x)$ my idea let $x\longrightarrow x+1$, then we have $$g^2(x+1)-g(x+2)-\dfrac{(x+1)...
1
vote
2answers
83 views

Follow up to previous functional equation question.

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ s.t $$f(x^2+f(y))=(x-y)^2f(x+y)$$ Putting $x=y$ yields $f(x^2+f(x))=0$. (1) Putting $y=-x$ yields $f(x^2+f(-x))=(2x)^2f(0)$. (2) By (1) $(2x)^2f(0)=0$ ...
5
votes
2answers
95 views

Finding all $f:\mathbb{R} \rightarrow \mathbb{R}$ s.t $f(x^2+f(y))=(x-y)^2f(x+y)$

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ s.t $$f(x^2+f(y))=(x-y)^2f(x+y)$$ I don't want people to solve this one for me I'd just like to know whether one of my steps is legitimate. So I put $...
4
votes
2answers
113 views

Finding a function from the given functional equation .

The question asks us to find the function $f(x)$ with the given information Let $f:\mathbb R \rightarrow \mathbb R$ such that $f'(0)=1$ and $f(x+y)=f(x) + f(y) + (e^{x+y})(x+y)-xe^x-ye^y+2xy$ ...
2
votes
1answer
76 views

functional equation (conti-function $f(x)$)

I would appreciate if somebody could help me with the following problem Q: Find conti-function $f(x)=?$ $$4(1-x)^{2} f \left({1-x\over 2} \right)+16f \left({1+x\over 2} \right)=16(1-x)-(1-x)^{4}$$
6
votes
1answer
215 views

Functional Equation - Am I right?

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$x^2+y^2+2f(xy)=f(x+y)(f(x)+f(y))$$ So here's my solution, If $x=y=0$, $2f(0)=2f(0)^2$ $\implies f(0)=0$ or $f(0)=1$. Case $1$: ...
4
votes
2answers
151 views

Cauchy functional equation over the complex field

It is known that the only measurable solutions to the Cauchy functional equation $f(x+y) =f(x)+f(y)$ are the linear ones ($x,y\in \mathbb{R}$). Does the same hold if we take $x,y \in \mathbb{C}$? Edit:...
0
votes
0answers
1k views

An example of the Sequential Quadratic Programming (SQP)

Given an objective function $f(x,y)=(x+6)^2+(y-10)^2$, we want to minimize the function $f$ under a constrained condition $xy-5\leq0$. Obviously it is a constrained optimization, a good algorithm to ...
1
vote
1answer
72 views

Functional equations very like the Taylor Series

Let $g(x,y)=0$ be a closed curve, that means, any point inside that curve satisfies $g(x,y)<0$ and any point outside that curve satisfies $g(x,y)>0$. Given a point $(a,b)$ outside the curve ($g(...
8
votes
5answers
310 views

Functional equation: $R(1/x)/x^2 = R(x) $

The following can be shown without much hassle. Suppose $R$ is a rational function satisfying the following functional equation. \begin{align} \frac{1}{x^2} R\left( \frac{1}{x} \right) = R(...
4
votes
2answers
135 views

Deriving the series representation of the digamma function from the functional equation

By repeatedly using the functional equation $ \displaystyle\psi(z+1) = \frac{1}{z} + \psi(z)$, I get that $$ \psi(z) = \psi(z+n) - \frac{1}{z+n-1} - \ldots - \frac{1}{z+1} - \frac{1}{z}$$ or $$\...
17
votes
4answers
2k views

About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$

Consider the functional equation $$f(x+y) = f(x)g(y)+f(y)g(x)$$ valid for all complex $x,y$. The only solutions I know for this equation are $f(x)=0$, $f(x)=Cx$, $f(x)=C\sin(x)$ and $f(x)=C\sinh(x)$. ...
7
votes
2answers
238 views

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$

Find all real polynomials $p(x)$ that satisfy $\sin( p(x) ) = p( \sin(x) )$ and prove they are indeed all. Is there an easy way to prove this?
0
votes
3answers
59 views

Solve the equation about matrix

The equation is $x^2 = x$, which $x$ is a $2\times2$ matrix. Anyone can give me some hint? Thanks!
1
vote
1answer
66 views

Functional equation problem

I've been trying to find a function that satisfies this to solve a separate problem, but I'm finding it difficult and no polynomial seems to work. $f(x) + \frac{1}{x+1} = f(x+1)$
2
votes
3answers
208 views

How find the $f(x+f(y))+f(y+f(z))+f(z+f(x))=0,$

:$f:R\longrightarrow R$ ,and is continuous such that $$f(x+f(y))+f(y+f(z))+f(z+f(x))=0,$$ find all $f$