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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8
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1answer
625 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 ...
5
votes
1answer
109 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 ...
4
votes
1answer
61 views

Solve the functional equation, $f(x+y+1)= (\sqrt{f(x)} + \sqrt{f(y)})^2$

Solve the functional equation, find all real valued functions $f(x)$ s.t. $$f(x+y+1)= (\sqrt{f(x)} + \sqrt{f(y)})^2$$ given $f(0)=1$. I reached to a point that $4f(x)=f(2x+1)$
3
votes
1answer
40 views

Are there algorithms for solving simple functional equations?

So somebody posted yesterday asking a question for continuous solutions $f$ satisfying $f(x+y) = f(x)f(y)f(xy)$. Continuity could be used for a simpler proof but then somebody posted a solution ...
3
votes
1answer
68 views

Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R $ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
3
votes
1answer
57 views

Weird ordinary differential equation

I am searching for the functions $f,g:[0,\infty) \to [0,\infty)$ which are both increasing, $f'$ is strictly increasing, $g$ is the inverse of $f$ and $$ g(x)^2=f(x)^2\cdot g'(x)$$ My approach was ...
3
votes
1answer
154 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 ...
2
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1answer
34 views

Soultion of a particular functional differential equation

I need to solve the following functional differential equation: $(4\mu-\lambda r - \lambda x+\lambda)f'(x) = \lambda( f(x) - f(1-x-r))$, where $x\in (0, 1-r)$, $r\in (\frac{1}{2}, 1)$ Or, a more ...
2
votes
1answer
20 views

Characterizing functions which satisfy de Moivre's theorem

Let $f$ and $g$ be two non-zero functions $ \mathbb R \to \mathbb R $ which are continuous and differentiable everywhere. Furthermore, say that for all integer $n$: $$ (f(\theta) + i g(\theta))^n = ...
2
votes
1answer
60 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 ...
2
votes
1answer
55 views

Prove That $f(n+f(n))=n$

if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function such that $f(1)=1$ and $$f(n)=n-f(f(n-1))\,\,\: \forall n \ge 2$$ Prove That $$f(n+f(n))=n$$
2
votes
1answer
81 views

Functional Equation $f(n) = 2 f(n / f(n) )$

I have the following functional equation: $f(n) = 2 \cdot f\left(\frac{n}{f(n)}\right)$ Under the precondition that $ f(n) = \omega(1) $, monotinic and a initial value $ f(1) = \Theta(1) $ one can ...
2
votes
1answer
81 views

Functional/Differential Equation

In the midst of a calculation too long (and too irrelevant) to describe here, I've been forced to confront the following equation: $$1-p-f(f(p))-f(p)f'(f(p))=0$$ Here $f$ is a differentiable ...
1
vote
1answer
22 views

Additive & Multiplicative Link

Is there an isolative property about the following multivariable equation: $$f_1(x)g_1(y)=f_2(x)+g_2(y)$$ That is, is this equation rearrangable such that it may be put in the form ...
1
vote
1answer
38 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
1
vote
1answer
56 views

Solution of functional equation

i know the solutions of the well known Cauchy-functional-equation $f(x+y)=f(x)+f(y)$ But what does it change if i have the following form $f(x+g(y))=f(x)+f(g(y))$ ? what can i say about g? ...
1
vote
1answer
27 views

Linear after change of variables

Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, ...
1
vote
1answer
46 views

Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $ f(1)$ is an integer and $xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for ...
1
vote
1answer
36 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants ...
1
vote
1answer
47 views

What is the family of functions that satisfies

Let $f$ be a continuous function in $\Re$, differentiable (at least one time). Then, which functions can satisfy: $$ \frac {f(x)} {f'(x)}= - \frac {f(x-\frac {f(x)} {f'(x)})}{f'(x-\frac {f(x)} ...
0
votes
0answers
20 views

Methods of showing that a non-trivial solution exists for a functional equation?

I am just looking at a one variable functional equation, I won't put it down here because it is university related, and I keep thinking that 0 is the only possible solution so it got me wondering how ...
0
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0answers
11 views

Measuring availability of a service

I have a task to measure availability of some IT-components. The problem is that I need to create a equation for measuring this. For now the availability is going to be measured through life-cycle ...
0
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0answers
37 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
0
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0answers
29 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 !
0
votes
0answers
25 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
0
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0answers
27 views

Solving a functional equation with a boundary condition (involving probabilites)

Ok, I am getting a functional equation in $z$ domain given by $F(z)= F(G(z))$ where $G(z)= e^{-a(1-z)}$. I want to get $f(n)$ ($F(z)$ is the $z$ transform of $f(n)$) where $f$ is some pmf, hence we ...
0
votes
0answers
14 views

Validation of proof,for a functional equation

Sorry for bad English in advance,I had to translate most of things from another language that's why it's pretty messy,I feel like I made a mistake about this Given a natural number $k$.Let ...
0
votes
0answers
26 views

Multiplicative functional equation on Gaussian integers

Using polar form of complex numbers, it can be checked that all solutions $f:\Bbb C \to \Bbb R$ of the functional equation $$ f(zw)=f(z)f(w) \tag 1 $$ are of the form $$ ...
0
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0answers
10 views

Quicker way of solving Fout = Fref * A / B / C for a given Fout and Fref

I've got a chip device that calculates frequency based on the following equation: $Fout = Fref * A / B / C$ Where: $Fout$ is the output frequency $Fref$ is a reference frequency 0 < $A$ <= ...
0
votes
0answers
19 views

Logarithmic functional equation on positive rational numbers

Let $\Bbb Q^+$ be the set of positive rationals and $f:\Bbb Q^+ \to \Bbb R$ be a continuous function satisfying $$ f(rs)=f(r)+f(s) $$ for all $r, s\in \Bbb Q^+$. Then $f$ has only the form $$ f(r)=c ...
0
votes
0answers
27 views

Solving differential functional equations with a restricted solution

I have a variable vector $X=\{x_1,x_2,...,x_n\}$, and a constant vector $V=\{v_1,v_2,...,v_n\}$. $f(x_i,X)$ is a function that takes X and xi as the parameter, for example: $f(x_i,X) = ...
0
votes
0answers
52 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
23 views

Linear-like function

Suppose we need to find all the functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $\forall x,y,z\in \mathbb{R} ~~ f(x+z) - f(x) = f(y+z) - f(y)$ It can be shown that $~~\forall r \in ...
0
votes
0answers
25 views

Mellin transform and a proof of the functional equaton for the $ \zeta (s) $

i would like to obtain a proof of the functional equation for the RIemann zeta function to do so i would like to know if there is a function or a distribution so $$ \int_{0}^{\infty}f(t)t^{s-1}dt = ...
0
votes
0answers
49 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
82 views

Probability distribution satisfying constraints?

Continuing from this question. Given two random variables $X$ and $Y$ where $X \sim \operatorname{Beta}(a, b)$ and $Y \sim \operatorname{Beta}(c, d)$, I'm looking for a random variable $Z$ with a ...
0
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0answers
40 views

Symmetric expressions for homogeneous functions

Suppose $f(x_1, \ldots, x_n)$ is a homogeneous function, i.e. a function such that $$f(\lambda x_1, \ldots, \lambda x_n) = \lambda^d f(x_1, \ldots, x_n)$$ for all $\lambda$ and for some positive ...
0
votes
0answers
24 views

Question on 2 functional equations.

Let $z,x$ be complex numbers. Im looking for analytic functions $f(z)$ such that : $$1) \exp(\ln^{5} (f(x))=\sum_i a_i f(b_ix)$$ $$2)f(x)^5=\sum_j c_j f(d_jx)$$ holds for all $x$ and where both ...
0
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0answers
55 views

What is the analytical solution to a Volterra integral equation?

I need to solve a following equation: \begin{equation} r_{k+1} = -\sum\limits_{l=0}^{k-1} r_l \cdot (k-l) \cdot \left(\frac{\omega}{t_c - l}\right)^{2 \beta} + \delta_{k,0} \end{equation} subject to ...
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0answers
66 views

Numerically solving delay differential equations

I understand that the Dickman Function can be solved numerically by converting it to a delay differential equation. The Mathworld link above describes how this may be achieved for $$\rm ...
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0answers
48 views

Integral Functional Equation

If $f(x) = 1 + \int_{x/c}^1 f(t) dt$, find $f(x)$ in terms of $c$. This problem was inspired by trying to find the expected value of the longest increasing subsequence $a_1, a_2, a_3, \cdots, a_n$ ...
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48 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) $(2x)^2f(0)=0$ ...
0
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0answers
515 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
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0answers
80 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 ...
0
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0answers
74 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 ...
0
votes
0answers
37 views

Proving Join Dependencies in MVD

I have a question regarding natural join operations in multivalued dependencies. I know that a join operator joins two tables on similair attributes, however I have a hard time to figure out how to ...
0
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0answers
72 views

What kind of (differential) equation is this?

This may be a silly question, but I am confused with the following. To my knowledge, in general any initial value problem we have a differential equation of the form $\dot{y}(x)=f(x,y(x))$ plus an ...
0
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0answers
61 views

properties about number of groups of linearly independent solutions in the general solutions of linear functional-differential equations

Are there any effective methods to determinate the number of groups of linearly independent solutions in the general solutions of linear functional-differential equations of the form ...
0
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0answers
120 views

Could anyone derive a formula for this?

Edited: I want to get a sentiment score of various sentences and I've tried coming up with an equation that could satisfy the conditions that are inherent to each sentence (It's estimated mood as ...
-1
votes
0answers
39 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) ...