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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11
votes
1answer
707 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 formal-power-...
6
votes
1answer
145 views

Uniqueness of solution of functional equation

I have a function $f(x_1,x_2) \colon \mathbb{R}^2_{+} \to \mathbb{R}_{+}$ positive homogenous: $$ f(\lambda x_1, \lambda x_2) = \lambda f(x_1,x_2), \; \forall \lambda > 0 $$ and such that $f(x_1,...
5
votes
1answer
82 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 $f$ ...
4
votes
1answer
69 views

Find all solutions to $f\left(x^2+xf(y)\right)=xf(x+y)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^2+xf(y)\right)=xf(x+y)$$ for all $x,y\in\mathbb{R}$. This is somewhat related to this question, but with an $xf(y)$ term instead ...
4
votes
1answer
80 views

If $f$ is a continuous function from $R \rightarrow R$ and $f(x)=f(x+f(x))$ then prove that $f$ is constant.

If $f$ is a continuous function from $R \rightarrow R$ and $f(x)=f(x+f(x))$ then prove that $f$ is constant. I could prove that $f(x)=f(x+f(x))=..=f(x+nf(x))$ after $n$ iterations.Then , how will I ...
4
votes
1answer
131 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
3
votes
1answer
35 views

Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$.

Find all functions $f$ and $g$ for which $f(x+y) = g(xy)$. Is there anything wrong with this? We see that $f(1) =g(0)$ and $f(0) = g(0)$ so $f(1) = f(0)$. Also, $f(x) = g(0)$ and therefore $f(x) = ...
3
votes
1answer
50 views

Can this Delay Differential Equation (DDE) be solved in closed form?

$$\frac{\mathrm{d}\psi}{\mathrm{d}x}=\frac{(x+\mu)^2-\lambda^2}{\mu}\psi(x+\mu)$$ where $\mu\geq0$, $\lambda\in\mathbb{C}$ are constants and $\psi(x)\to0$ as $|x|\to\infty$.
0
votes
0answers
22 views

Does this functional equation have a (unique) solution?

Does the following equation have a (unique) solution for $b_1$? \begin{equation} b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},...
0
votes
0answers
28 views

Functional differential equation

I want to solve a functional differential equation of this kind $$ \int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') u(t'',t';[x])+u(t,t';[\...
0
votes
0answers
22 views

Solving equation over expectation

I have an expression $(1 + n EX^k)p^{-k}$ which I would like to minimize over $k$. Here $n$ and $k$ are positive numbers, while $X$ and $p$ are in $[0,1]$. Since the expectation converges absolutely, ...
0
votes
0answers
20 views

Interval between two solutions of an equation

We are in $\mathbb{R}$ and $x\geq 0$. I have an equation: $(1-\alpha)x^{\frac{1}{1-\alpha}}-W(y)x=g(y)W(y)$ Where $\alpha \in (0,1)$, y is a parameter $0<y<1$, W and g are constants in $x$ and ...
0
votes
0answers
30 views

Solving the functional equation F(z+1)=Q(z)F(z)

I want to solve the functional equation $F(z+1)=Q(z)F(z)$. The $F(z)$ is a matrix function. $Q(z)$ is also a matrix function. But its compoents are all rational functions. i.e. $ F(z)=\begin{matrix} ...
0
votes
0answers
11 views

Pacejka formula - extracting the curve peak

I don't know if many of you knows the Pacejka Magical Formula, but it looks like this: $D\sin(C\arctan(Bx - E(Bx-\arctan(Bx))))$ As you can see, the formula reaches a peak point(in this case at ~0.6)...
0
votes
0answers
22 views

How to complete this?

Let $f$ a function defined on $]0,+\infty[$ and checking : $$\forall x,y>0,\ f(xy)=f(x)+f(y)$$ Assume that $f$ is bounded on $]1-\eta,1+\eta[$($\subset]0,+\infty[$). Let $\alpha =\underset{x\in ...
0
votes
0answers
37 views

Functional analysis with KKT conditions

I want to solve an optimization problem min $F(x_{ik}) $ subject to $x \in X$. $F$ here is a function or functional that I wish to determine. I want my optimal ...
0
votes
0answers
26 views

Solving a functional convolution equation

I have a very weird functional equation that I am trying to solve. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a non-negative Lebesgue integrable function such that $\int f=1$. Let $\tilde{f}(x)=f(-x)$ ...
0
votes
0answers
49 views

Find all functions $f: \Bbb Q \rightarrow \Bbb Q$

Find all functions $f:\Bbb Q \rightarrow\Bbb Q$ that satisfy the conditions: a) $f(f(2016))=0;$ b) $f(x+y)= f(x)+f(y)+f(2016)$ for all $x,y \in \Bbb Q.$ I tried to solve the problem using the ...
0
votes
0answers
26 views

A simple PDE related to symmetric functions

This question was posed on mathoverflow but was put on hold. So far nobody has been able to give any hint either to the solution or to why it is trivial. Disclaimer: as far as I can tell this is not ...
0
votes
0answers
29 views

Functional Equation of Probability Distributions

Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning $$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$ Now assume $\Phi(f_X)$ is also a ...
0
votes
0answers
26 views

Class Scatter Fitness Function Calculation

I get a fitness function for class scatter, the equation: Click here to see the Fitness Function Where : T = Transpose of Matrics Mi = class mean Mo = grand mean The equation is based from mean,...
0
votes
0answers
25 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
0
votes
0answers
50 views

two integral equations

I'm trying to solve the two following integral equations : 1) $y(x)=2+\int_1^x\frac{1}{ty(t)}\,dt$, $x>0$ 2) $y(x)=4+\int_0^x2t\sqrt{y(t)}\,dt$ It really looks like an ODE but I'm a bit clueless ...
0
votes
0answers
56 views

The functional equation $ f(x-f(y))=f(f(y))+xf(y)+f(x)-1$

I came across the functional equation: $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ So far I tried plugging $x=f(y)$ and got $f(x)=\frac{f(0)-x^2+1}{2}$ which holds for every $x = f(y)$. I suppose that $f(0)=1$ ...
0
votes
0answers
30 views

Time delay equation

If $x(t)=\displaystyle\frac{1+(x(t-T))^m}{(x(t-T)))^m}$ for all $t$ where $T$ is constant and $x(t)=x_s$ is the solution to the above equation, why can I write that: $x_s=\displaystyle\frac{1+{...
0
votes
0answers
20 views

How to convert $\ max \int_{-\infty}^{\infty} f(x) dx =\ min \int_{-\infty}^{\infty} g(x) dx $

Let $f$ or $g$ have a given condition. ( example $\int_{- \infty}^{\infty} f(x) exp(-x^2) dx = 1$ Or $g(g(x)) = x^3$ Or a differential equation for one of $f,g$. ) I want to find a general way - if ...
0
votes
0answers
22 views

Existence of an inverse function in this functional equation

Let $V,W$ each be connected and separable sets. Suppose we have continuous functions $F:\mathbb{R}\times V\rightarrow \mathbb{R}$, $G:\mathbb{R}\times V\rightarrow \mathbb{R}$, $f:V\times W\rightarrow ...
0
votes
0answers
43 views

Uniqueness question from functional equation

Let $f(x)$ and $f_2(x)$ be real and continuous on the interval $[0,\infty[$. Let $f(1) = f_2(1) = 1 , f(2) = f_2(2) = e$. Let $g(x)-g(x-1) = f(x-1)$. Let $ f ' (x) = exp(g(x) - g(1)).$ and $f '' (1)...
0
votes
0answers
13 views

Formulation of a rule

Im new to this forum(hence, formatting issues). I come from CS background. I am trying to figure out a formula to code set of rules. The requirement is as below. Imagine two parameters X and Y ...
0
votes
0answers
41 views

Finding the “hidden values”.

I have this problem but my knowledge of the methods of linear algebra is totally useless if it exists at all (I know what is a matrix at least... maybe). I have a three collections of objects ${\bf a}...
0
votes
0answers
49 views

Cauchy-like functional equation $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$

I am looking for the solution to the following two variable functional equation: (*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$ where: $h$ is some given continuous function, $f, g,$ unknown functions on ...
0
votes
0answers
38 views

$\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$

Let $t,x$ be nonnegative reals. Let $* ^{[k]}$ denote k th iteration. Find real-analytic $f(x)$ such that $\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$ Holds. We require analytic iterations. ( $ f^{...
0
votes
0answers
67 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle a,b\rangle}{(0.99-|b|^2)^{\frac{1}{2}}+(0.99-|...
0
votes
0answers
40 views

Transcendental Functional Equations

Given $f^k(x) = f \circ f \circ f\circ ...(x)$ composed $k$ times, Do the functional equations $f^k(x) = g(x)$, where $g(x)$ is a basic transcendental elementary function, for example, the inverse ...
0
votes
0answers
17 views

Find a derivative of equation that contains Fourier series

I need to find a derivative of follow equation $$ \left(r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] \right)({\sin\phi-k\cos\phi}) - b = 0 $$ I know the derivative of $\left(r_{0} + \sum [a_{i}\...
0
votes
0answers
42 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in C^{\...
0
votes
0answers
21 views

Solving a linear functional equation

Working with Green functions, I have found to solve the following equation $$ -\omega^2G(\omega)-m^2G(\omega)+\kappa\sum_{n=-\infty}^\infty b_nG(\omega-n\omega_0)=1 $$ where $m$, $\kappa$ and $\...
0
votes
0answers
102 views

Differential equation involving composition

I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ...
0
votes
0answers
41 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which is a convolution kernel. I am interested in the following equation: $0=A+B~(\varphi*v)+C~v+D~v~(\varphi*v)$, where $v:\...
0
votes
0answers
84 views

Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant function),...
0
votes
0answers
44 views

Conjecture about $A f(x) = f(g(x)) + f(h(x))$

Let a given real $A$ satisfy $0 < A < 2$. Conjecture : For any real entire nonconstant $f(x)$ there exist real entire $g(x)$ and $h(x)$ such that $A f(x) = f(g(x)) + f(h(x))$ or $ A f(x) =...
0
votes
0answers
32 views

Existence of a function satisfying all the given infima

Given a function $f : \mathbb{R} \to \mathbb{R}$, we can compute its infimum over $A$ for all the Borel measurable $A \subset \mathbb{R}$. I am wondering when we can deduce in the other direction, i....
0
votes
0answers
33 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 ...
0
votes
0answers
63 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 $f(x,y)=g(x,y)$?
0
votes
0answers
43 views

A functional equation for a Dirichlet series

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

Does there exist a nontrivial cumulative distribution function $F$ on $\mathbb{R}_+$ for which $(F^2)'= (F')^{\ast k}$ for some $k > 0$?

This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ...
0
votes
0answers
64 views

Non-constant solution of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$ in rings

The answers to this question prove that over an integral domain the functional equation $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ has only solutions where one of $f$ or $g$ is constant. The proofs make ...
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 ...
0
votes
0answers
84 views

Seeking a function which satisfies a given functional equation.

I wish to find a function $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies: $f(u) \geq 0$ when $0 \leq u < 1$, $f(u)=0$ when $u<0$ or $u \geq 1$, $\int_0^1 f(u) du=1$, and $$f(u)=\frac{f\...
0
votes
0answers
99 views

Interpretation of functional equation of dedekind eta function

It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a ...