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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132 views

Applications of mathematics to some kinds of sporting strategies

I am a rather newbie maths person. Haven't studied maths in a while and so not sure what things are called was hoping to get some information to push me in the right direction so I know what it is I ...
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1answer
155 views

Find the general solution to $f(z)=f(z/2)f(z-1)$

Find the general solution to $f(z)=f(z/2)f(z-1)$ where $z$ is a complex number.
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71 views

Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
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1answer
62 views

Solution of $z(t+a) = h(a)z(t)$

I'm reading Howard Georgi online book on The Physics of Waves and found the following argument. Given the functional equation $$ z(t+a) = h(a)z(t) $$ he makes the following derivation (I'm citing ...
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1answer
150 views

Symmetric homogeneous functions of degree 1

Suppose: $cf(x,y)=f(cx,cy)$ $f(x,y)=f(y,x)$ If $f$ is a polynomial, then $f(x,y)=c(x+y)$ because by Euler's homogeneous function theorem, $f(x,y)=xf_x(x,y)+yf_y(x,y)$ where $f_x,f_y$ are ...
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2answers
92 views

About $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$

Suppose $M$ is map from vector space $X$ to vector space $Y$, $M(0) =0$, and $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$. Does this mean that $M$ is a linear map? If not, could someone please give ...
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5answers
139 views

Help solving a functional equation

Is there a function $f(x)$ on the real domain and real constants $a$ and $b\neq 0$ for which the following is true: $$f(x)-f(x-\delta)+a+bx^2=0$$ for some real $\delta\neq 0$? EDIT: I missed a very ...
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0answers
115 views

Defining Oblique Lines

Is it correct to classify a line which is neither vertical nor horizontal as oblique. I am trying to classify lines in a plane based on the quadrants through which they pass.
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2answers
512 views

Are all multiplicative functions additive?

Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$. Proof: There is some $c$ such that $y=cx$. Then ...
6
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2answers
286 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, ...
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1answer
162 views

two functions $ f(x) $ and $ g(x) $

let be two functions $ f(x) $ and $ g(x) $ with an infinite set of roots $ a_{n} $ and $b_{n} $ so $ f(a_{n}) =0= g(b_{n}) $ also they satisfy the same functional equation $ f(1-s)=f(x) $ and $ ...
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2answers
211 views

About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$

On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
19
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4answers
757 views

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

I was trying to find functions $f:(0,+\infty)\to(0,+\infty)$ satisfying the following functional equation $$ f(x)^2+f(y)^2=f(x+y)(f(f(x))+f(y)) $$ The problem is that I can't find here any reasonable ...
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0answers
101 views

Convexity conditions for $f$ and $\dfrac {1} {f}$

Let $f:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}$ be a real function. Find all conditions on $f$ under which $f$ is convex on $\left( a;b\right) \subset \mathbb{R} _{+}$$\Leftrightarrow$ $\dfrac ...
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1answer
268 views

find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$

Find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$ Find all continuous function $f:\mathbb R\to\mathbb R$ satisfy $\forall a<b, \exists c \in (a,b): ...
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1answer
112 views

Delta function question

Given the functions $$f(x)= \delta (x-a)$$ $$g(x)= \frac{1}{a} \delta \left(x- \frac{1}{a}\right)$$ for a real constant $a\gt0$, is there a relationship between $f$ and $g$? I believe that $ ...
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2answers
271 views

Cauchy's functional equation for $\mathbb R^n$

Suppose $f(x+y)=f(x)+f(y)$. If $f:\mathbb R\to \mathbb R$ and is measurable, then $f(x)=cx$. This is referred to as Cauchy's functional equation. Suppose $f:\mathbb R^n\to \mathbb R^n$ instead. Does ...
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4answers
679 views

The functional equation $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$

How to find the all continuous functions $f\colon \mathbb{R}\to\mathbb{R}$ such that $f(x)+f(y)+f(z)+f(x+y+z)=f(x+y)+f(y+z)+f(z+x)$
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1answer
129 views

Reverse engineer a Bayesian estimate?

My apologies if this is a basic question because I am no mathematician. Struck on my work on this, so came here to get some help.I am working on this bayesian estimate explained here.This is a ...
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5answers
319 views

$g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$. For what value of $p$, $g(x+p)=g(x)$.

$g(x)$ is a function such that $g(x+1)+g(x-1)=g(x)$, $x \in \mathbb{R}$.For what value of $p$, $g(x+p)=g(x)$. $g(x+2)+g(x)=g(x+1)$
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1answer
82 views

Show that for this function the stated is true.

For the function $$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$ show that $$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$ Hey everyone, I'm very new to this kind of maths and would really ...
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1answer
467 views

Find all continuous functions $ f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$

Find all continuous functions $ f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$
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2answers
190 views

Seeking a contest question on functional equation on $[-1,1]$

I vaguely remember a question going something like Let $f$ be a function on $[-1,1]$ with $f$ satisfying (something like) $$f(x^2-1)=(2x)f(x).$$ Show that $f$ is identically zero on $[-1,1]$. ...
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1answer
101 views

Determining a function through three equations

I have the following assignment question, and I'm having trouble even getting started: Consider the set of functions $\mathcal{F}=\{f,g\}$, with $f:\mathbb{R}^2\to\mathbb{R}$, and ...
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0answers
103 views

Indeterminacy in second degree equations of Spencer-Brown's “Laws of Form”

I am trying to follow G. Spencer-Brown's argumentation regarding 'Equations of the Second Degree' in the Laws of Form. He shows how infinitely growing self-referential forms can be created by using a ...
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2answers
467 views

Correct order of books for a beginner

what should be the order of the books in which a beginner should do the following books in algebra: -1.E.J. Barbeau POLYNOMIALS -2. Polynomials and Polynomial Inequalities (Graduate Texts in ...
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0answers
55 views

secants, exponentials, quotient structures

Trigonometric functions are . . . . . . somewhat like exponential functions. If $f$ is an exponential function, then $\displaystyle \frac{\prod_i ...
3
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1answer
130 views

Solution space to a functional equation

This question comes from my attempts at understanding an example presented by Bill Gasarch on his blog. The example is of a continuous strictly increasing function whose derivative is zero almost ...
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6answers
2k views

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$?

What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
3
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1answer
158 views

What is this topic (competition questions)?

I discovered the following question by accident, and found it interesting, but I only resolved it by brute force: Problem: Let f: $\mathbb{R} \rightarrow \mathbb{R}$ have the property $(\pi)$ iff ...
3
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2answers
380 views

How to find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$

Find all polynomials $P(x)$ with real coefficents satisfying $P^2(x)-1=4P(x^2-4x+1)$. My solution: Let the first term of $P(x)$ be $ax^n$. We see first term of left side is easily $a^2x^{2n}$ ...
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2answers
168 views

Does anyone recognize this function?

I am looking for a function $f(n)$ that satisfies the following two conditions at the same time $$ \frac{f(n-1)}{f(n)}=(-1)^n\quad ,\quad \frac{f(n+1)}{f(n)}=(+1)^n\equiv 1,\quad \forall ...
3
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1answer
116 views

If $g(x) := \int_1^2 f(xt)dt \equiv 0$ then $f \equiv 0$

Let $f \colon \mathbb R \to \mathbb R$ be a continuous function. Let's define $$ g(x) := \int_1^2 f(xt)dt. $$ Prove that $g \equiv 0 \Rightarrow f \equiv 0$. Well, I show you what I have ...
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1answer
130 views

Functional Equation with Value

If $f$ is a strictly increasing function from the naturals to the naturals, and $f(f(x))=3x$, what are all values of $f(2012)$? I have only proven that $f(3x)=3f(x)$ but that get's nowhere :(
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2answers
152 views

Continuous function $g$ satisfying $g(x + y) = 5g(x)g(y)$

Let $g$ be a continuous function with $g(1) = 1$ such that $$g(x + y) = 5g(x)g(y)$$ for all $x$, $y$. Find $g(x)$.
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1answer
96 views

fancy about $f(x+a)=f(x)$ , where $a$ is any non-real complex number

It is well-known that when $a$ is any non-zero real number, the most general solution of $f(x+a)=f(x)$ should be $f(x)=\Theta(x)$, where $\Theta(x)$ is an arbitrary periodic function with period $|a|$ ...
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3answers
202 views

Solution of functional equation $f(x)=-f(x-a)$

I have a problem with finding solution. I suppose it will be something like $f(x) =G(x)\Re(e^{\frac{x\pi}{a}})$, where $\Re$ is real part of a complex number, $G(x)$ periodic function whith period ...
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1answer
162 views

Solving $f(2011)=2012$, $f(4xy)=2yf(x+y)+f(x-y)$

How to find the all functions $f$ :$ \mathbb{R}\longrightarrow\mathbb{R}$ such that $f(2011)=2012$,for every $x,y\in\mathbb{R}$ then: $$f(4xy)=2yf(x+y)+f(x-y)$$
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652 views

Problem on Euler's Phi function

Let $S(n)$ be $S(n)=\left\{k\;\left|\;\left\{\frac{n}{k}\right\}\right.\geq \frac{1}{2}\right\}$,where $\{x\}$ is the fractional part of $x$ Prove that : \begin{align} \sum_{k\in S(n)} ...
2
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2answers
281 views

Positive twice differential decreasing function, is it convex?

If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being ...
4
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1answer
305 views

A question concerning on the axiom of choice and Cauchy functional equation

The Cauchy functional equation: $$f(x+y)=f(x)+f(y)$$ has solutions called 'additive functions'. If no conditions are imposed to $f$, there are infinitely many functions that satisfy the equation, ...
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0answers
62 views

Matrix functional equation

Could someone give me some hint about a possible method to find the function $f$ which solve this equation: $$f(H^2)=\alpha f(H)$$ where $\alpha$ a constant with $\alpha \in C$ and: ...
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3answers
81 views

even function representation

given any even function $ g(x)=g(-x) $ is it always possible to write it as the product of two functions ? i mean $ g(x) = f(x)f(-x) $ so $ g(x) $ is always an even function even though $ f(x) $ it ...
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0answers
43 views

if I get the asymptotic solution of a certain equation involving $ f(x)$ does it mean that the solution exists

Let's take a complicated functional equation $ f(g(x))=f(1-x)g(x) $. Let us suppose that by using a) Analytic method b) Numerical method I can prove that for example $ f(x) \sim x $ as $ ...
3
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2answers
258 views

Solve functional equation $f(x_1 x_2) = g_1(x_1) g_2(x_2)$

Let $x_1$ and $x_2$ be real positive numbers. The problem is to find all possible triples of $f$,$g_1$,$g_2$ such that $f(x_1 x_2) = g_1(x_1) g_2(x_2)$. I suspect that the only one solution is $f(x_1 ...
9
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1answer
687 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 ...
14
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3answers
1k views

Find all functions with $f(x + y) + f(x - y) = 2 f(x) f(y)$ and $\lim\limits_{x\to\infty}f(x)=0$.

Determine all functions $f \colon\mathbb{R}\to\mathbb{R}$ satisfying the following two conditions: (a) $f(x +y) + f(x - y) = 2 f(x) f(y)$ for all $x, y\in\mathbb{R}$; (b) ...
4
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3answers
342 views

Strategies to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy a given functional equation

My question is as follows: What methods can be used to find the set of functions $f:\mathbb{R}\to\mathbb{R}$ satisfying a certain functional equation. An example of a case where this applies is the ...
7
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2answers
1k views

Continuous and additive implies linear

The following problem is from Golan's linear algebra book. I have posted a solution in the comments. Problem: Let $f(x):\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function satisfying ...
6
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1answer
970 views

$f(m + f(n)) = f(f(m)) + f(n)$

I found this one in the list of IMO'96 (3) problems and decided to have a go at it, but could not complete the solution. So $m$ and $n$ are non-negative integers and $f$ takes values in the same set: ...