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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2
votes
2answers
48 views

Cauchy functional equation and the implicit function theorem

The problem statement: Suppose $f:\mathbb{R} \to \mathbb{R}$ is continuously differentiable, $f'(x)$ is strictly increasing, with $\lim_{x \to -\infty}f'(x) = -\infty$, $\lim_{x \to \infty}f'(x) = ...
4
votes
3answers
103 views

Solve $\sin \frac{\alpha - \beta}2 + \sin \frac{\alpha - \gamma}2 + \sin \frac{3\alpha}2 =\frac 3 2$

Solve the following trigonometric eqation where $\alpha, \beta, \gamma$ are angles in a triangle ($\alpha + \beta + \gamma = 180$): $$\sin \frac{\alpha - \beta}2 + \sin \frac{\alpha - \gamma}2 + \sin ...
4
votes
0answers
57 views

Infinite Double Exponential Sum, with Functional Equation $g(x) = g(\sqrt{x})$

What is a closed form for $$ \sum_{-\infty}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)} $$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + ...
1
vote
1answer
71 views

Functions such that $f(f(n))=n+2015$ [duplicate]

Is there a function $f:\mathbb N \to \mathbb N$ such that $\forall n \in \mathbb N, f(f(n))=n+2015$ ? Here's what I've done: Assuming such a function exists, ...
3
votes
1answer
41 views

$(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)$ and $f(1)=2$

Let $f$ be a differentiable function such that $(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)$ and $f(1)=2$. Then area enclosed by $$\frac{|f(x)-x|^{1/3}}{17}+\frac{|f(y)-y|^{1/3}}{2}\le\frac14$$ I rearranged ...
0
votes
1answer
20 views

$(r^2-s^2)^2-(5\cdot\min\{r,s\})=2015$. Find all positive integer solution of this equation.

I know the $\min\{x,y\}$ means the minimum value of $x$ and $y$. and it can be expressed as, $\min\{x,y\}= \frac12\left( x+y-\sqrt{(x-y)^2}\right)$
-3
votes
2answers
77 views

Functional equation $f(ax)=bf(x)$

What are all the solutions to the functional equation $f(ax)=bf(x)$, where $a,b>0$, and $f$ is continuous, strictly monotone and increasing, and $x$ ranges over the reals? references? proof? ...
2
votes
0answers
53 views

Is there a way to graphically show that a solution is the minimum or stationary solution to a functional?

I'm looking for the functional analogue to the visual representations of function optimization you most commonly see. To illustrate, if we have some function: $$ f(x) = (x-1)^2+1 $$ We can look at ...
2
votes
1answer
35 views

General Solution to Almost Riccati Like Equation

Consider the differential equation $$ y' = a_0(x) + a_1(x)y + a_2(x)\frac{1}{y}$$ I am attempting to find the general solution to this. One thing I can note is that the entire equation can be ...
5
votes
1answer
42 views

Can $f(x+1) = f(x)^{\ln(x)}$ be expressed as integral transform $\int g(x,t) dt $?

Let $x$ be a real number. Can some real-analytic function $f$ that satisfies for $x>3$ :$f(x+1) = f(x)^{\ln(x)}$ be expressed by standard functions as an integral transform : $$f(x) = ...
5
votes
0answers
110 views

Show that f is periodic if $f(x+a)+f(x+b)=\frac{f(2x)}{2}$?

Suppose $a$ and $b$ are distinct real numbers and $f$ is a continuous real function such that $\frac{f(x)}{x^2}$ goes to 0 when $x$ goes to infinity or minus infinity. Suppose that$ ...
0
votes
1answer
25 views

Difficult Functional Equation Problem, Non-Standard Type

Find all functions, $f:\mathbb{N} \to \mathbb{N}$, for which $f(1) = 1, f(2n) < 6f(n)$, and $$3f(n)f(2n+1) = f(2n)(3f(n)+1).$$ My first approach is to try to play around and set values equal to ...
2
votes
2answers
50 views

A monotonic multiplicative integer functional equation.

Let $ f:\mathbb N \to \mathbb N $ be such that $ f (x)> f (y)$ if $x> y$. $ f (xy)=f (x) f (y) $. $ f (3) \geq 7$. Find the smallest value of $ f (3) $ My attempt:if we can define the ...
3
votes
5answers
268 views

How are Zeta function values calculated from within the Critical Strip?

We note that for $Re(s) > 1$ $$ \zeta(s) = \sum_{i=1}^{\infty}\frac{1}{i^s} $$ Furthermore $$\zeta(s) = 2^s \pi^{s-1} \sin \left(\frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)$$ Allows us to ...
2
votes
1answer
44 views

General Solution to functional equation

I was wondering how to derive the solution to $$ \frac{f(x + (1-2x)) - f(x)}{1-2x} = f(x)$$ Which can be simplified to $$\frac{f(1-x) - f(x)}{1-2x} = f(x)$$ One idea is as follows. Consider the ...
5
votes
1answer
152 views

Prove that a $1$-periodic function $\phi$ is constant if $\phi(\frac{x}{2}) \phi(\frac{x+1}{2})$ is a constant multiple of $\phi$

For $0 < x< \infty$, let $\phi (x)$ be positive and continuously twice differentiable satisfying: (a) $\phi (x+1) = \phi (x)$ (b) $\phi(\frac{x}{2}) \phi(\frac{x+1}{2}) = d\phi(x),$ where ...
5
votes
0answers
67 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, ...
4
votes
2answers
82 views

Solve functional equation $(x+y)(f(x)-f(y))=f(x^2)-f(y^2)$

Find all real functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ so that $(x+y)(f(x)-f(y))=f(x^2)-f(y^2)$. Can someone at least find the value of $f(1)$ if it is possible, it would help me.
5
votes
2answers
228 views

Defining sine and cosine

We know the following are true about sine and cosine (and that they can be proven geometrically): $\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)$ $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ ...
5
votes
1answer
67 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 $f$ ...
0
votes
2answers
38 views

Is there any standard terminology for this property?

Let $f$ be a map whose domain is $X$. If $f$ satisfies the property that for all $x\in X$, $$f(f(x))=f(x)\text{,}$$ is there any standard name for such a function? Not sure if "projection" is the ...
1
vote
1answer
25 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 ...
0
votes
1answer
31 views

Difficult Functional Derivative

I've been working on this problem set for a little bit now and I've finally made it to the last question. I'm now left with this monster and I don't quite understand where to proceed. All the ...
1
vote
0answers
21 views

Commutivity of second functional derivatives

In multivariable calculus we learn that partial derivatives are commutative: i.e. $\partial _{xy}F=\partial_{yx}F$ When dealing with multiple functional derivatives this doesn't seem to be the case. ...
4
votes
0answers
95 views

Symmetry and the zeta function

It is an incontestable fact that symmetry is one of, if not the, most powerful tools a mathematician can have at his or her disposal. What I want to know, is if there are any good reasons as to why it ...
0
votes
0answers
12 views

Meta Euler Lagrange

Consider the following generalized standard Euler Lagrange Problem which is to maximize the quantity $$ \int_{x_1}^{x_2} L(x,y,y', y'' ... y^{(n)}) dx $$ It can be solved by first determining a ...
0
votes
1answer
32 views

Manipulating an equation

I am working on one of my assignment questions and am having difficulty manipulating the equation. The equation is as follows, where I have to solve for T, temperature, and the rest of the variables ...
0
votes
1answer
33 views

Solution to functional equation and Cauchy

I want to show that the only continuos solutions to the functional equation $g(x)+g(y)=g(\sqrt{x^2+y^2})$ is $g(x)=cx^2$ where c is a constant i.e. c=g(1). I think that I maybe could rewrite the ...
0
votes
0answers
42 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
1answer
34 views

Problem involving distances on a parametrically defined curve

SO the problem is stated below: A curve is defined paramterically by: $$x(t)=a\cosh t$$ and $$y(t)=b\sinh t$$ where a and b are positive constants and $-\infty < t < \infty$ The expression ...
2
votes
1answer
26 views

Newton method why the error is proportional to the square for the error of the last one?

We have learned that the Newton method is used to solve different equations. As I know, this method is iterative, which means that using an estimate point and using a loop, we can get closer and ...
8
votes
4answers
156 views

Solve $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{x+3}{x+1}\right)=x$ $\forall x\neq -1$

Given function $y=f(x)$ such that $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{x+3}{x+1}\right)=x \quad\forall x\neq -1$$ find $f(x)$ and $f(2007)$.
4
votes
2answers
72 views

Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation

For learning purposes, I'm trying to prove that the shortest function passing through the two points $(x_1, y_1)$, $(x_2, y_2)$ is a straight line, without using the Euler-Lagrange equation. My ...
2
votes
1answer
49 views

Solve the functional equation $4f(x)=f(2x)$

Solve the functional equation $4f(x)=f(2x)$. As for now I know that one solution is $f(x)=cx^2$, where c is a constant value.
0
votes
2answers
35 views

What is the minimum degree for a polynomial to pass through points with defined slopes [duplicate]

I'm having some difficulty solving this problem. The information I have is the following: What is the minimum degree for a polynomial for it to pass through points $A(x_1,y_1)$ and $B(x_2,y_2)$ with ...
1
vote
2answers
137 views

Find all real real functions that satisfy the following eqation $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$

Find all real functions $f:\Bbb R\rightarrow\Bbb R$ so that $f(x^2)+f(2y^2)=[f(x+y)+f(y)][f(x-y)+f(y)]$, for all real numbers $x$ and $y$. $f(x)=x^2$ is the only solution I think. So far I have got: ...
0
votes
0answers
27 views

$f(x+y)=f(x)+f(y)$ for all $x,y∈\mathbb{R}$. Prove that if $f$ is continuous at some point $x_{0}$, then it is continuous on $\mathbb{R}$ [duplicate]

$f(x+y)=f(x)+f(y)$ for all $x,y∈\mathbb{R}$. Prove that if $f$ is continuous at some point $x_{0}$, then it is continuous on $\mathbb{R}$. I have no idea about this question, does anyone could ...
0
votes
1answer
28 views

Number of possible solutions to equation

I am trying to solve $$x+y+z = 32$$ Where $x$, $y$, and $z$ are positive integers I believe the answer is: $C_{2}^{31}=465$ but I am not sure why. Can someone please explain?
0
votes
1answer
37 views

How do find the distribution of a R.V. from this functional/differential equation?

Assume that $X$ is some R.V. with domain $[0,1]$. A function $U:[0,1]\to[0,1]$ is defined as follows: $$U(x)=x\cdot F_X(x)+(1-x)(1-F_X(x)) = 2x\cdot F_X(x)- x - F_X(x) + 1$$ Where $F_X(x)=\int_0^x ...
7
votes
2answers
110 views

Function such that $f(x) f(\pi/2 - x) = 1$

I'm looking for functions that are smooth ($C^\infty$) between $0 < x < \pi/2$ that satisfy the equation $$f(x)\, f(\pi/2-x) = 1$$ on the inteverval $0<x<\pi/2$. I know that the constant ...
0
votes
0answers
14 views

In which condition related to coefficients, some equation has a solution?

My purpose is to know in which condition related to the coefficients $c>0, \, n\geq 2, 0<p<1,\, a>c\, \, \, \text{and} \, \, b>0$, this equation $$ F(x)= -c x^{n+p} -bx^n + (a-c)x^p ...
3
votes
1answer
41 views

Solve functional equation $f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a))$

Solve functional equation $$f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a)),(*)$$ where the following conditions are satisfied: 1)$f:\mathbb{R}_{+}\cup \{0\}\rightarrow [0,1] ,$ 2)$f(0)=1,$ ...
2
votes
1answer
38 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 ...
-1
votes
2answers
46 views

Solve the functional equation $f(x+a+f(y))=f(f(x))+a+y$ [closed]

So let $a$ be a real number. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $f(x+a+f(y))=f(f(x))+a+y$, for all real $x,y$.
0
votes
1answer
39 views

Cauchy's Functional Equation

Consider Cauchy's Functional Equation $$\phi(t+s)=\phi(t)+\phi(s).$$ Can we say that any right continuous with left limits (cadlag) solution is Borel measurable? Obviously continuous solutions are ...
4
votes
2answers
64 views

Solve the functional equation $2f(x)=f(ax)$ for some $a$.

I am trying to solve the following functional equation, and could use some help.$$ 2f(x)=f(ax)$$ For some $a\in\mathbb{R}$. By repeated adding $2f(x)$ together we notice that $$2nf(x)=f(a^nx).$$ ...
0
votes
1answer
41 views

Find all such functions $f:R\to R$

It's my last question. Just give me advise how to start. Q: Find all such functions $$f:\mathbb R\to \mathbb R,$$ for all real x, y, the equality $$f(yf(x))=x^2y^4$$
5
votes
0answers
82 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
1
vote
1answer
38 views

Nonlinear Functional Equation

Find all functions $f(x)$ such for a given fixed $a\in \mathbb{R}$ such that the following functional equation holds $$f(x)^{2}=f(x/a)$$ I'm not sure how to solve this equation other then using the ...
1
vote
0answers
21 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...