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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2answers
154 views

What do logarithms distribute over?

I notice that division distributes over addition Root extraction distributes over multiplication What operator do logarithms distribute over: ie: what non-constant function $H \in C^2 \rightarrow C$ ...
6
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2answers
244 views

Solving the functional equation $f(x) = f(x / \phi) f(x / \phi^2 - 1)$

I'm trying to find a function $f(x)$ such that the spacing between consecutive roots looks like the infinite Fibonacci word: $$1, \phi^{-1}, 1, 1, \phi^{-1}, 1, \phi^{-1}, 1, 1, \phi^{-1}, 1, 1, ...
5
votes
2answers
142 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ ...
5
votes
2answers
382 views

Real Analysis Proofs: Additive Functions

I'm new here and could really use some help please: Let $f$ be an additive function. So for all $x,y \in \mathbb{R}$, $f(x+y) = f(x)+f(y)$. Prove that if there are $M>0$ and $a>0$ such that ...
5
votes
2answers
85 views

$f,g:[a,b]\to [a,b]$ are continuous, $f\circ g=g\circ f$ and $f$ is 1-1. Show that $\exists t\in [a,b]$ so that $f(t)=g(t)=t$

Using the Intermediate Value Theorem on $h(x)=f(x)-x$ and $r(x)=g(x)-x$ I can easily show that $\exists t_f,t_g\in[a,b]$ so that $f(t_f)=t_f$ and $g(t_g)=t_g$. It remains to show that $t_f=t_g$. Since ...
2
votes
2answers
39 views

function equation with translation of independent variable

The following has come up in some work I'm doing: If $\frac{f(x+a)}{f(a)}=g(x)$ , where $g(x)$ is given and $a \geq 0$ is a constant, what is $f(x)$ ? We can assume that $g(x)>0 ~ \forall x$ . Of ...
1
vote
2answers
50 views

A scaling functional equation

I want to find a closed form of a function satisfying $$G(4z)=\frac{G(z)}{2z},$$ unfortunately I am not experienced with scaling problems, so I have no idea and would be thankful for any hints. ...
1
vote
2answers
89 views

Formula for solving for Cx and Cy…

I'm trying to create a formula to find the third point in a triangle based on two known points and three known sides. Known Sides: $AB, BC, AC$ Known Points: $A(x, y), B(x, y)$ Unknown Points: ...
0
votes
2answers
46 views

Some functional equations in two variables

I have two questions. i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation $$ |f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|} $$ has a solution ...
7
votes
1answer
610 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
102 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
63 views

submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
4
votes
1answer
56 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
63 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
56 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
132 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
votes
1answer
52 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
78 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
79 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
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
41 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
32 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
45 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)} ...
1
vote
1answer
74 views

A fairly difficult differential equation

I was thinking what if I had a differential equation of the form: $$\frac{d^2u}{dx^2} + vu(x) = 0 $$ where $v(y(x))$, that is $y$ is a function of $x$. What are the possible restrictions that I can ...
1
vote
1answer
49 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
1
vote
1answer
69 views

Functional equation, find functions

Find all of the functions defined on the set of integers and receiving the integers value, satisfying the condition: $$f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$$ for each pair of integers $(a,b)$.
1
vote
1answer
165 views

Analytic solution of a certain functional equation

Here's my question. Let $b_2$, ..., $b_d \in \mathbb{C}$ ($d$ is an integer greater than 2), and consider the functional equation $$V(z^d)=dz^{d-1} V(z)+(b_2 z^{d-2} + b_3 z^{d-3}+\ldots + b_d)$$ ...
0
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1answer
37 views

Understanding an Equation and how to implement it

A common method for linking language with psychological variables involves counting words belonging to manually-created categories of language. One counts how often words in a given category are used ...
0
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1answer
46 views

Functional equations and cubes

Problem $10728$ from Amer. Math. Monthly "Preserving the sum of three cubes" says: Determine all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying $f(x^3+y^3+z^3)=(f(x))^3+(f(y))^3+(f(z))^3$ ...
0
votes
1answer
24 views

Functional Equation and totally multiplicative functions

Find all $f:\mathbb{C}\rightarrow\mathbb{C}$ s.t. $\forall a, b\in \mathbb{C}, f(ab) = f(a)f(b)$. I could deduce a lot of things about what happens at the roots of unity, and 0, but I can't find out ...
0
votes
1answer
48 views

$f(x)=xf(1)$ Doubt

I just started learning Functional Equations and I was working on a problem that asked me to find all functions satisfying a certain condition, and at some point I got $f(x)=xf(1)$, is there a way to ...
0
votes
1answer
31 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
0
votes
1answer
70 views

A functional equation for homogeneous functions of degree zero

Let a function $F: \textbf{R}^2_{++} \rightarrow \textbf{R}$ satisfy the relation: if $F(x,y)=F(x',y')$ then $F(x,y)=F(x+x',y+y')$. It is easy to prove that under the additional assumption of ...
0
votes
1answer
47 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 ...
0
votes
1answer
110 views

More general reflection formula for Gamma function

It is know that $\Gamma(z)\, \Gamma{(1-z)}=\pi \csc( \pi z)$. Is there any formula for $\Gamma{(a+z)}\Gamma{(a-z)}$ where $a$ is a rational number, i. e., $a=p/k$ with $p, k$ integers and $z$ is ...
0
votes
1answer
51 views

inverse function , asymptotics ..

let be a function given by $ f^{-1} (x) = \sqrt x + G(x) $ with $ G(x)= \sum_{n=0}^{N}a_{n} \sin(nx+ \pi/4) $ finite fourier series with N big my questio is , if for $ x \rightarrow \infty$ the ...
0
votes
0answers
47 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 $a_1, a_2, a_3, \cdots, a_n$ ...
0
votes
0answers
19 views

Search Space Function:

given a set of integers: ${x_1, x_2, ... x_n}$ Is is possible to construct a generic function $f$ such that there exists $u_1 .... u_n \in R$ where $f(u_k) = x_k$ and: $$f(x+y) = f(x) + f(y)$$ ...
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0answers
84 views

Equation of a conic-IIT JEE Problem

A chord is drawn to a conic section and its mid point lie on the director circle of the conic which of the following can not be its eccentricity (A) sin2θ (B) tan2θ (C) cosec2θ (D) sec2θ – ...
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0answers
133 views

Solving an equation with 2 unknowns

I've been trying to solve this problem and was wondering if there is a more accurate / efficient way to do it. For the following equation $$y = a \times \left(1 - ...
0
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0answers
44 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
votes
0answers
295 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
79 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
votes
0answers
70 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
32 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
votes
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
votes
0answers
58 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
votes
0answers
119 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 ...
0
votes
0answers
76 views

Additivity of averages

Let us suppose that we have a system $(A,B)$ consisting of two independent subsystems $A$ and $B$. Suppose that $A$ has $m$ states having energies $E_1^{(A)}, \dots, E_m^{(A)}$ with probabilities ...
0
votes
0answers
217 views

$ f(nx)=f(x), \qquad n \in \mathbb Z^+$

Let $f$ satisfy the following equation, $$ f(nx)=f(x), \qquad n\text{ a positive integer}.$$ Then I know that the most general solution is $$ f(x)= C_{+}x^{2\pi i m/\log n}+C_{-}x^{-2\pi i m/\log ...