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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7
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73 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really ...
7
votes
0answers
229 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 ...
7
votes
0answers
62 views

What are the densities of branches of the euclidean tree?

The Euclidean algorithm shows how all coprime pairs of positive integers can be uniquely obtained from the pair $(1,1)$ by applying the two operations $(a,b) \to (a+b,b)$ and $(a,b) \to (a,a+b)$. (or ...
6
votes
0answers
98 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 ...
6
votes
0answers
101 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
6
votes
0answers
144 views

Solution(s) to $f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y))$

I would like to know the solutions of the functional equation: $$f(x+f(y))+f(y+f(x))=2f(f(x))f(f(y)), \forall x,y\in\mathbb{R}$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$. I have already determined ...
5
votes
0answers
81 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
5
votes
0answers
109 views

Width of the Eiffel Tower as a function of height?

In the preface of Advanced Engineering Mathematics, 2nd Ed. by Zill and Cullen, it is claimed that the function relating the width of the Eiffel Tower as to the distance from its top, $x \mapsto ...
4
votes
0answers
77 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
4
votes
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238 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
4
votes
0answers
34 views

Wronskian different from zero and solutions of ODE.

Let $a_0, \ldots , a_{n-1}$ continuous functions in an interval $I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let $\phi_1, \phi_2, \ldots,\phi_n$ $n$ are ...
4
votes
0answers
82 views

$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$

Consider the equation: $f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for $a \geq 0$ and $b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ...
4
votes
0answers
184 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...
3
votes
0answers
46 views

Evaluation of $\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d\!}x$ for $\mu>0$

I'm trying to evaluate the integral $$ \Psi(\mu,\nu)=\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d}\!x\qquad(\text{for}\; \mu>0)\tag 1 $$ where $\nu\in\Bbb R$ and $f$ ...
3
votes
0answers
30 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 ...
3
votes
0answers
76 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
3
votes
0answers
52 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 ...
3
votes
0answers
73 views

Order of Recursion?

Define an extended algebraic function $f(a)$ as a function on $a$ that utilizes any combination of recursive extensions and inverses of sequentiation. Example: $a + 1$ , sequentiation. $a + a$, ...
3
votes
0answers
49 views

secants, exponentials, quotient structures

Trigonometric functions are . . . . . . somewhat like exponential functions. If $f$ is an exponential function, then $\displaystyle \frac{\prod_i ...
3
votes
0answers
246 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.
3
votes
0answers
146 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 ...
3
votes
0answers
211 views

What is known about functions of type $f(n+1) = f(n)^ {f(n)}$?

Update : having looked at Knut's Double Arrow Notation ( Thank you DJC), it seems that this question is nothing more than a frivourless wondering that should be undertaken by whom ever wonders it, it ...
3
votes
0answers
77 views

Function Shape Reference

I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague. I would like to have a ...
2
votes
0answers
44 views

Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
2
votes
0answers
28 views

The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
2
votes
0answers
59 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
2
votes
0answers
51 views

If $f(2x-f(x))=x$ . Find all bijective functions.

It is given that $f :[0,1] \rightarrow [0,1] $ and it is bijective. If $f(2x-f(x))=x$ , find all such f. Is my solution correct? My attempt $f(x)$ is bijective. thus there exists g(x) which is the ...
2
votes
0answers
54 views

Correct math notation to show min and max of a list of numbers and sign?

I am working on a paper that uses a calculation from DIN 15018. The calculation is only described in text and not as a math equation. I would like to display as an actual equation if possible. The ...
2
votes
0answers
23 views

Is determining a non-constant solution to a functional inequality with polynomial arguements decidable?

So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say integer constants and $p_i$ are polynomials, say with ...
2
votes
0answers
70 views

Creating a monotonic function

I have $n$ functions $f_i(x) \{i = 1 ,...,n\} $that does not preserve the monotonic mapping order. i.e. if $x_1 < x_2$, then in general, $f_i(x_1)$ is not less than $f_i(x_2)$ (for all $i = 1 ... ...
2
votes
0answers
56 views

Solving functional equation

Problem:find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$$
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: ...
2
votes
0answers
63 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
0answers
71 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 $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ ...
2
votes
0answers
89 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
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, ...
2
votes
0answers
66 views

How to solve the functional equation : $T(n)=(\log n)T(\log n)+n$

I want to solve the following functional equation using any ways: $$T(n)=(\log n)T(\log n)+n$$
2
votes
0answers
25 views

Multiple integrals satisfying functional equations

If $f:\mathbb{R}_+\to\mathbb{R}_+$ is a positive function which is locally $L^1$ with $\int_0^t{f(s)ds}=at^n$ for all $t\geq0$ then, by differentiating, it follows that $f(t)=a$ if $n=1$ and $f=0=a$ ...
2
votes
0answers
100 views

Convexity conditions for $f$ and $\dfrac {1} {f}$

Let $f:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}$ be a real function. Find all conditions on $f$ under which $f$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$$\Leftrightarrow$ $\dfrac ...
2
votes
0answers
80 views

Indeterminacy in second degree equations of Spencer-Brown's “Laws of Form”

I am trying to follow G. Spencer-Brown's argumentation regarding 'Equations of the Second Degree' in the Laws of Form. He shows how infinitely growing self-referential forms can be created by using a ...
2
votes
0answers
57 views

Matrix functional equation

Could someone give me some hint about a possible method to find the function $f$ which solve this equation: $$f(H^2)=\alpha f(H)$$ where $\alpha$ a constant with $\alpha \in C$ and: ...
2
votes
0answers
197 views

How to find a function which satisfies such functional equation?

How to find a function which satisfies: $$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$ where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
1
vote
0answers
19 views

Commutivity of second functional derivatives

In multivariable calculus we learn that partial derivatives are commutative: i.e. $\partial _{xy}F=\partial_{yx}F$ When dealing with multiple functional derivatives this doesn't seem to be the case. ...
1
vote
0answers
20 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...
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vote
0answers
137 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
1
vote
0answers
47 views

How to demonstrate a particular functional equation solution

In order to find a prior probability distribution I have to solve the following functional equation: $$af\left(\frac{a\theta}{1-\theta-a\theta}\right)=(1-\theta+a\theta)^2f(\theta)$$ the solution of ...
1
vote
0answers
17 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 ...
1
vote
0answers
84 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 ...
1
vote
0answers
45 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) + ...
1
vote
0answers
25 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 ...