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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0
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1answer
149 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
100 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 ...
12
votes
5answers
600 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 ...
27
votes
7answers
4k 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 ...
1
vote
0answers
45 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
197 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
62 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
51 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 ...
5
votes
2answers
268 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
86 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
79 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
39 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
274 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
0answers
73 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
200 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
96 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
104 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
110 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 ...
2
votes
2answers
109 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 ...
0
votes
0answers
52 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
86 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 ...
3
votes
2answers
82 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
74 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
179 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 $1$: ...
0
votes
0answers
679 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
70 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 ...
8
votes
5answers
264 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) = ...
4
votes
2answers
116 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 ...
16
votes
4answers
1k 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
222 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
56 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
62 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
199 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 $f$
1
vote
1answer
94 views

Solving Functional Equations

How would I go about solving $f(r+1) - f(r) = r^3$? I know the answer is $f(r) = c + \frac{1}{4}r^2(r-1)^2$, but I have no idea what method can be used to solve it. I have another functional ...
1
vote
0answers
40 views

Program to determine the relationship of one variable to several possible variables

Suppose I have a system with several variables a, b, c, d, and x. I am trying to solve for the unknown x. I don't know exactly which of those variables x is dependent on, or exactly how the function ...
3
votes
0answers
53 views

Could we compute $P(t^2)$?

Let $P$ be an operator such that $P(kx)=kP(x)$, $k \in \mathbb{C}$, $x$ is a variable, $P(xy)=P(xP(y))+P(P(x)y)-P(x)P(y)$, $x, y$ are variables. All variables commute. Let $P(t)=t$. Then ...
20
votes
1answer
2k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
2
votes
0answers
28 views

A question about scaling

One wants the function $\Delta ^2$ to be such that, $\Delta^2(k,\tau) = \Delta^2(\frac{k}{\lambda ^{\frac{4}{n+3}}}, \lambda \tau )$. Now from this how does this follow that, the following holds, ...
4
votes
2answers
225 views

Is a function $f$ satisfying $f(x+1) = f(x) + 1$ and $f(x^2) = (f(x))^2$ odd or even?

The problem statement, all variables and given/known data 1) $f(x+1)=f(x)+1$ 2) $f(x^2) =(f(x))^2$ Let a function $f \colon \mathbb{R} \to \mathbb{R}$ satisfy the above statements. Then prove ...
1
vote
2answers
70 views

Playing with a functional equation

I was playing with a functional equation and proved the result below: Let $f$ be such that $$f(f(z))=z$$ If $f^{-1}$ exists then $$f(z)=f^{-1}(z)$$ If $f'$ exists then as ...
0
votes
2answers
2k views
8
votes
2answers
93 views

A functional equation with inequality

Find all (at least one) functions $f\colon \mathbb{R}\to \mathbb{R}$ (or show there is none), such that $$ f(x^3+x)≤x≤f(x^3)+f(x), \quad \text{for all $x\in \mathbb{R}$}. $$ This is a problem asked ...
9
votes
1answer
216 views

The value of the trilogarithm at $\frac{1}{2}$

From the functional equation $$\text{Li}_{3}(z) + \text{Li}_{3}(1-z)+ \text{Li}_{3} \Big( 1 - \frac{1}{z} \Big) = \zeta(3) + \frac{\ln^{3} (z)}{6}+ \frac{\pi^{2} \ln (z) }{6}- \frac{\ln^{2} (z) ...
6
votes
2answers
433 views

Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?

To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that ...
7
votes
0answers
262 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...
10
votes
2answers
474 views

find functions f such that $f(f(x))=xf(x)+1$,

let $f:R\longrightarrow R$, and $f$ is continous,and such that $f(f(x))=xf(x)+1$, find all this $f$? follow is my some idea:(but I don't have solution) We have $f(f(0)) = 1$, so there is your $c = ...
63
votes
7answers
2k views

Does a non-trivial solution exist for $f'(x)=f(f(x))$?

Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind ...
-2
votes
2answers
133 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
3
votes
1answer
179 views

Find all polynomials $P(x)$ satisfying this functional equation

Find all polynomials $P(x)$ which have the property $$P[F(x)]=F[P(x)], \quad P(0) = 0$$ where $F(x)$ is a given function with the property $F(x)>x$ for all $x\geq 0$. This is an exercise ...
2
votes
2answers
159 views

Equation $f(x,y) f(y,z) = f(x,z)$

How to solve the functional equation $f(x,y) f(y,z) = f(x,z)$?