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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1answer
20 views

How can one solve this equation in $Z^2$?!

Ho can one solve the egality $2x+3y=xy$ ? I have to find a value of $x$ in fonction of $y$ so ? I have to add somthing and substrate it I added -2xy then $2x(1+y)-3y(1+x)=0$ Here im suck Can ...
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2answers
24 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
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2answers
43 views

Deterministic condition for the nature of one real root of a cubic equation

A cubic equation $ax^3+bx^2+cx+d=0 \space$ where, $a\neq 0$ always has one real root. Is there any direct condition for determining the nature i.e. sign of one real root for sure? Is it possible ...
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1answer
48 views

How to solve a equation with special conditions?

I have this equation: $z = 11n + 13m$ Conditions: $z < 2015$ $z$, $n$ and $m$ must be natural numbers ($>0$). How many options are possible for $z$?
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1answer
72 views

Find the function satisfying the given condition

If $f(xy)=e^{xy-x-y}[e^y f(x) +e^x f(y)]$ and $f'(1)=e$. $f'$ denotes the derivative of function $f(x)$. Find $f(x)$. I could find that $f(0)=0$ and $f(1)=0$ and then found the derivative the got ...
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0answers
48 views

Find all differentiable functions $f$ such that $f(f(x))=f'(x)$ [duplicate]

Here is a problem I made up: Find all differentiable functions $f$ from the reals to the reals such that $f(f(x))=f'(x)$ for all real $x$.
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0answers
37 views

Approach on solving limit equation systems and finding some f given assymptotes?

This is a "reverse" question of finding the asymptote of a function Recently, I am interested in doing some sort of modelling which involve equations of the form $$@(t)=1-f(t)$$ where $f(t)$ is ...
4
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2answers
155 views

Solve differential equation $f'(z) = e^{-2} (f(z/e))^2$

I'm curious if there's a simple closed solution to the following DE and, if so, what it is. $$\begin{align} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align}$$
2
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0answers
77 views

We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
1
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1answer
45 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
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0answers
17 views

Iterative Functional Equation? (discrete, increment, logarithm)

Say $f(n)$ is defined as discrete iterative process, $n \in [0,1,2,\ldots,N]$, and $g(x)$, $x \in [-\infty,+\infty]$ is "small": $|g(x)|<1, \forall x$: $$f(n+1)=f(n) + \log(1 + g(f(n)))$$ ...
5
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1answer
90 views

Find $f(x) $ given that: $f'(x)=\frac{f(x)-x}{f(x)+x}$ [closed]

I would appreciate if somebody could help me with the following problem: Find $f(x)$ given that: $f \colon \mathbb{R^+} \rightarrow \mathbb{R^+}$, $f$ is differentiable function, and ...
7
votes
4answers
170 views

$f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous , and $f(x+1)+f(x)=x^2$

I would appreciate if somebody could help me with the following problem: Find $f(x)$ ($f(x)$ is not Polynomial function), given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is ...
4
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0answers
100 views

Integer functional equation $f(f(f(n)))=f(n+1)+1$

Can you find all functions $f:\mathbb N\rightarrow\mathbb N$ satisfying the functional equation $$ f(f(f(n)))=f(n+1)+1 $$
3
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2answers
140 views

Solving functional equation for all real numbers.

The functional equation to be solved is $ f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$. Domain:Reals,Codomain:Reals.I found about 4 possible solutions to the equation but ran into a fundamental problem with all ...
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0answers
58 views

Iterative (functional) roots of integer functions (functions on $\mathbb{Z}$)

A function $g:A\to A$ is called a $k$-th iterative root of another function $f:A\to A$ ($A$ an arbitrary set and $k\in\mathbb{N}$) iff $f=g^k$, where $g^k(x)=g\circ g\circ\ldots\circ g(x)=g(g(\ldots ...
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0answers
36 views

Functional equation similar to Babbage's equation

I'm interested in the potential solutions $f: R_{+} \rightarrow R_{+}$ to the functional equation: $ \forall x \in R_{+}, \quad f(s(x) - f(x)) = ax $ where $a>0$ is a constant. $s: R_{+} ...
1
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1answer
23 views

Extension of the additive Cauchy functional equation

Let $f\colon (0,\alpha)\to \def\R{\mathbf R}\R$ satisfy $f(x + y)=f(x)+f(y)$ for all $x,y,x + y \in (0,\alpha)$, where $\alpha$ is a positive real number. Show that there exists an additive function ...
1
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1answer
28 views

hyperplane in $L^2$

Consider $R^2$ valued functions $f,g \in L^2([0,1],\mu, R^2)$ where $f=(f_1,f_2)$,$g=(g_1,g_2)$ and $\sqrt{\langle f, g\rangle}=\int_{[0,1]} (f_1(x)g_1(x)+f_2(x)g_2(x)) d\mu(x)$ Suppose for a given ...
2
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1answer
72 views

What are all pairs of functions f and g so that $f(x)f(y) = g(x+y)$?

It can be shown, and is a problem in Rudin's Principles of Mathematical Analysis (Chapter 8), that when $f$ is continuous, and $f(x)f(y) = f(x+y)$, $f$ is a function of the form $e^{cx}$. Must this ...
2
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4answers
83 views

Solving for $f(2004)$ in a given functional equation

Given that $$f(1)=2005$$ and $$f(1)+f(2)+...f(n) = n^{2}f(n)$$ for all $n>1$. Determine the value of $f(2004)$. My progress: I first substituted $n-1$ into the equation to get ...
2
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1answer
44 views

A function $\psi(z)$ satisfies the following functional equation…

I was given the following question to solve: "Given a function $\psi_0(z)$ satisfies the following functional equation: $\psi_0(z+1)=\frac{1}{z}+\psi_0(z)$ Prove that ...
0
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5answers
63 views

Boolean Simplification of AB + A'+B'

Is there any way to simplify this function? Or is this the simplest equation? : AB + A'+B'
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0answers
63 views

Extension of the Cauchy functional equations

Let $f:(0,a)\to\mathbb{R}$ satisfying $$f(x+y)=f(x)+f(y)$$ for all $x,y,x+y\in(0,a)$, where $a$ is a positive real number. Show that there exists an additive function $A:\mathbb{R}\to\mathbb{R}$ such ...
2
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1answer
39 views

A problem on solving functional equations

If $f(x),\forall x\in\mathbb{R}$ is continous and differentiable, and satisfies: $f(x_1+x_2)+f(x_1-x_2)=2f(x_1)f(x_2),\forall x_1,x_2\in\mathbb{R}$ $f\left(1\right)=\dfrac{3}{2}$ How to prove: ...
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0answers
66 views

Proving existence and uniqueness of solutions to the functional equation $f(n) = r \cdot f(n-1)$

Suppose I have a functional equation $f(n) = r \cdot f(n-1)$ where $r$ is a constant. This represents a geometric progression and a known solution is $g(n) = ar^n$ where $a = g(0)$. By intuition, ...
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0answers
19 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 ...
5
votes
2answers
178 views

$f(\alpha x) = f(x)^{\beta}$ under different constraints

With $\alpha > 0,\, \beta \in \Bbb R^*,\, \alpha, \beta \neq 1$ and $f : \Bbb R \to \Bbb R_+^*$, let's consider the functional equation $$ f(\alpha x) = f(x)^{\beta} \tag{$\Xi$}$$ or equivalently ...
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1answer
58 views

Follow-up to $f(x)^2 = f(\sqrt2 x)$

This is a follow-up to: Solving $(f(x))^2 = f(\sqrt{2}x)$ . So $f : \Bbb R \to \Bbb R$ is $\mathcal C^2$ and verifies $\forall x,\, f(x)^2 = f(\sqrt2 x)$. We already know that $f(0) \in \{0,1\}$ and ...
10
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3answers
267 views

Solving $(f(x))^2 = f(\sqrt{2}x)$

I would like to know how to solve this equation : $$f(x)^2 = f(\sqrt{2}x)$$ We assume that $f : \mathbb R \to \mathbb R$ is $\mathcal C^{2}$. The answer should be $f(x)=e^{-x^{2}/2}$, but I don't ...
1
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1answer
54 views

Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}$

Find all continuous $f:\mathbb{R}\to\mathbb{R}$ satisfying $$\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}.$$ I believe the original question was $$\frac{f(x)}{3+f(x)}=\frac{4+x^2}{x^2},$$ which has a ...
1
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1answer
27 views

How to solve the delay algebraic equation $xf(x) + \alpha f(x - {x_0}) - \alpha f(x + {x_0}) = 0$?

In the process of solving a problem, I am faced with the problem of finding a non-zero function $f:\mathbb{R} \to \mathbb{R}$ which satisfies the equation $$xf(x) + \alpha f(x - {x_0}) - \alpha f(x + ...
14
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1answer
1k views

Prove that function is continuous without knowing the function explicitly

Let $f\colon \mathbb R^+\to\mathbb R$ be a function that satisfies the following conditions: $$\tag1 \lim_{x\to 1}f(x)=0 $$ $$\tag2f(x_1)+f(x_2)=f(x_1x_2)$$ Show that $f$ is continuous in its domain. ...
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0answers
26 views

Calculus of Variations for discrete functionals

Question: How does one determine the optimal function f that either maximize or minimizes: $$\int_{x_1}^{x_2} L\left(x, f, D_{h,x}[f ]\right) dx$$ Whereas: $$ D_{h,x}[f] = \frac{f(x + h) - ...
5
votes
0answers
101 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
0
votes
0answers
32 views

Does this problem have analytic/approximate analytic solution?

Define $F[i]=F[i-1]-F[i-1]^c$ with $F[0]=r>0,c\in(0,1)$. Define $R(r,c)=\min\{i:F[i]<1\}$. What is solution to $c\in(0,1)$ in $R(r,c)=r^{1/k}$ with some given $k>1$?
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0answers
31 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
0
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1answer
27 views

Defining a rectangular prism using a formula and complex numbers.

I recently read that a line can be defined using the formula $$ A = O + dL $$ where $A$ represents any point on the line, $O$ represents the vector origin of the line, $L$ represents the direction ...
3
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0answers
392 views

What special role plays the function $\pi^{\frac x\pi}$ in analysis?

I have tried to redefine some special functions in the most "natural" way, that is the way which allows to simplify the relations the most. I would call these functions "parelementary". The ...
3
votes
4answers
135 views

A functional equation: $4f(x)^3 +f(3x)=3f(x)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$4f(x)^{3}+f(3x)=3f(x)$$ I know of 2 functions that satisfy the equation but I do not know how to prove that they are the only ones. ...
7
votes
3answers
267 views

Does there exist a function such that $f(a)f(b)=f(a^2b^2)?$

Given $S=\{2,3,4,5,6,7,\cdots,n,\cdots,\} = \Bbb N_{>1}$, prove whether there exists a function $f:S\to S$, such that for any positive $a,b$: $$f(a)f(b)=f(a^2b^2),a\neq b?$$ This is 2015 ...
0
votes
2answers
40 views

Continuous, 1-periodic $f$ with $f(x+y) = f(x) f(y)$ for $x, y \in \mathbb{R}$

The problem is to find all $f : \mathbb{R} \to \mathbb{C}$ that is continuous, has $f(x) = f(x+1) \forall x$, and $$f(x+y) = f(x) f(y) \quad x, y \in \mathbb{R}$$ Plug in $y=0$, we find $f(x) = ...
0
votes
1answer
53 views

What function satisfies the following equation?

$$f(x)e^{-x}\Gamma(x/\pi)=f(\pi/2-x)e^{x-\pi/2}\Gamma(1/2-x/\pi)$$ I think it should be similar to Zeta function, but what is it exactly?
1
vote
1answer
58 views

Solving the functional equation $f(x)=f(f(x-p))+q$

I can see that $f(x) = x + (p-q)$ is a solution. Is this the only possible solution?
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votes
3answers
277 views

Find all real functions that satisfy the functional equations $f(x+y) = f(x) + f(y)$ and $f(xy)=f(x)\,f(y)$ [closed]

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the two functional equations $f(x + y) =f(x)+f(y)$ and $f(xy)=f(x)\,f(y)$.
0
votes
0answers
13 views

Closed form of chained Stationary Functions

Consider the following operator $$D_{(a-1)x,x}\left[ f \right] = \frac{f(ax) - f(x)}{(a-1)x} $$ It's pretty easy to see that a nontrival function $g(x)$ such that $$D_{(a-1)x,x}[g] = g$$ Is given ...
0
votes
0answers
14 views

Recurrence relation involving ordinary generating function

Let $f_1,f_2,\ldots$ be a given infinite sequence of functions. Define the sequence of functions $F_1,F_2,\ldots$ by the recurrence relation $$F_n(x)=f_n(x)\sum_{k=0}^\infty F_{n+1}(k)x^k$$ or ...
2
votes
2answers
78 views

What is the formula and the name of the mathematical-phenomenon seen at the ending of “Around the World in Eighty Days”?

Spoiler in brief for those who don't know the ending yet: At the end, Phileas Fogg alongside with his companions realize that they have arrived back to London a day earlier than expected due fact the ...
9
votes
3answers
1k views

If $ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x $, then $f(x)=x$

Let $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ be a function which has the following property: $$ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x \;,\; \forall \; x, y \in \mathbb{Q} $$ Prove that $ f(x) = ...
3
votes
6answers
145 views

Functional equation $ f(x)+f(x+1)=x$

What functions satisfy $f(x)+f(x+1)=x$? I tried but I do not know if my answer is correct. $f(x)=y$ $y+f(x+1)=x$ $f(x+1)=x-y$ $f(x)=x-1-y$ $2y=x-1$ $f(x)=(x-1)/2$