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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16 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": $|g(x)|<1, \forall x$: $$f(n+1)=f(n) + \log(1 + g(f(n)))$$ ...
2
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1answer
45 views

Polar to cartesian equation conversion

I have a polar equation defined as: $r = ae^{θ tan (m)}$ where, $a$ and $m$ are constants, θ is the angle between the horizontal axis from the origin (xc,yc) to the coordinate. $e$ refers to ...
5
votes
1answer
87 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
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4answers
167 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
90 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
votes
0answers
91 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
53 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
23 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
21 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
votes
1answer
68 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
votes
4answers
81 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
votes
1answer
42 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 ...
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5answers
55 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
57 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
votes
1answer
36 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: ...
1
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0answers
49 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
15 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
173 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
57 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 ...
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3answers
254 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 ...
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1answer
53 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
vote
1answer
24 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
votes
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
21 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
87 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
31 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
25 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
votes
1answer
17 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 ...
6
votes
0answers
261 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
133 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
255 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
39 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
52 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?
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1answer
56 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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3answers
247 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 ...
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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 ...
1
vote
2answers
34 views

Simplifying $\frac{1/(\frac{1}{z_1}(1-t)+\frac{1}{z_2}t) - z_1}{(z_2 - z_1)}$

This drives me mad! I am not very good in math but thought I could at least do basic things like this one, but can't figure it out and I spent a day on it. I am trying to simplify: ...
2
votes
2answers
77 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
142 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. $f(x)=y$ $y+f(x+1)=x$ $f(x+1)=x-y$ $f(x)=x-1-y$ $2y=x-1$ $f(x)=(x-1)/2$
6
votes
2answers
203 views

Find $f(x)$ satisfy $f(2x)=2f(x)+x$

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 continuous at $x=0$, and ...
0
votes
1answer
31 views

monotonic function. I need to show ots linear

If a real additive function f is monotonic, then it is linear. I need to show that the monotonic function f that satisfies caushy additives functional equation is linear
1
vote
1answer
76 views

Show that f is linear

Let $f : \mathbb R \to \mathbb R$ be a solution of the additive Cauchy functional equation satisfying the condition $$f(x) = x^2 f(1/x)\quad \forall x \in \mathbb R\setminus \{0\}.$$ Then show that ...
1
vote
1answer
69 views

analitycs solutions to the equation $f'(x)=f(x)f(x-1)$

As the title says I'm serching for functions ($C^n$ or analitycs $f$) that satisfies $f'(x)=f(x)f(x-1)$ some details: I've come at this equation after looking for a function $g$ satisfying for ...
1
vote
0answers
33 views

Generalized Riesz theorem of operator value function

I am reading a book Gustafson-Rao's Numerical Range and come across a problem that I really don't understand. In theorem 2.1-2 of the book, it asserts that for an operator valued function ...
2
votes
0answers
42 views

What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= ...
7
votes
0answers
96 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
1
vote
2answers
81 views

Show that $f$ is a Cauchy function

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a solution of the functional equation $$|f(x + y)| = |f(x)| + |f(y)| \quad \forall x,y \in\mathbb{R}.$$ Show that $f$ is an additive function.