Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation(s) and possibly other conditions; e.g., the goal can be to find all continuous solutions. Solving the equation means finding all functions satisfying the given equation(s) and any additional conditions.This is different from the more common use of the word "equation", where the solutions are numbers. It is also different from the more common use of the word "functional", referring to a mapping from a space into the reals or complexes. For basic questions about functions use more suitable tags like or .

A common technique used in solving functional equations is finding some properties of satisfying functions by substituting variables for certain values in the equation. Proving properties of satisfying functions is also helpful - finding that a function is injective, surjective, involutive, and so on, is often a key step in finding all possible solutions. Other techniques such as exploiting symmetry, considering fixed points, and even using certain properties of domains (e.g. well-ordering) sometimes help.

Some well-known functional equations are:

More information can be found at Wikipedia.

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Solve the functional equation $ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}. $

Let $f(x),h(x)$ be two differentiate on $\mathbb{R}$ functions, $f(0)=h(0)=1$. Solve the functional equation $$ q \, \frac{f(x+1)}{f(x)}=\frac{h(x+1)}{h(x)}, $$ here $q$ is a constant. For $q>0$ it is easy to find a solution: $f(x)=e^{ax},…
Leox
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Find a function $\Phi$ such that $ \Phi(x)^{T}\Phi(y)=\exp(-\|x-y\|^2/(2\sigma^2))$

It's a question from HW: Suppose we have $ \Phi:\mathbb{R}^p \to \mathbb{R}^\infty $ that satisfies: $$ \Phi\left(x\right)^{T}\Phi\left(y\right)=\exp\left(-\frac{\left\Vert x-y\right\Vert ^{2}}{2\sigma^{2}}\right) $$ Find $ \Phi $. First I noticed…
HUO
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Functional equation and Riemann function $ \xi(s) $

Is there any theorem or proof that if a function satisfy the functional equation $ f(1-s)=f(s)$ and $ f(s) >0$ for each real $s$ then $ f(s)= \xi(s)$ or $ f(s)= \operatorname{const}$?
Jose Garcia
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Transcendental Functional Equations

Given $f^k(x) = f \circ f \circ f\circ ...(x)$ composed $k$ times, Do the functional equations $f^k(x) = g(x)$, where $g(x)$ is a basic transcendental elementary function, for example, the inverse hyperbolic trigonometric functions, have elementary…
Jack Tiger Lam
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Generalized Functional Equation

$ f\circ f\circ f(x) = f(3x) $ Ignoring trivial (constant) solutions, I am not sure what I can try as an initial guess. Also, how does this generalise? i.e. If $f^k(x) = f \circ f \circ f\circ ...(x)$ composed $k$ times, then what are the…
Jack Tiger Lam
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solution of d’Alembert’s equation: $g(x+y)+g(x-y)=2g(x)g(y)$

I know that equation for d’Alembert’s equation is looking so: $g(x+y)+g(x-y)=2g(x)g(y)$. So I am trying to find actual solutions for this equation. First I took $x=y=0$ and I got $2g(0)=2g(0)^2$. From here $g(0)=0$ or $g(0)=1$. If I take $x=y$, then…
dato datuashvili
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Solving the functional equation $f(x + y) + g(x-y) = \lambda g(x) f(y)$

Let $\lambda$ be a nonzero real constant. Find all functions $f,g : \mathbb R \rightarrow \mathbb R$ that satisfy the functional equation for all $x,y \in\Bbb R$: $$f(x + y) + g(x-y) = \lambda g(x) f(y)$$ I tried this: $y=0$ implies…
M'smary
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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 by simply observing only the signs of coefficients…
Harish Chandra Rajpoot
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Find $f$ such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$

Find all continuous function such that $\{f(x+y)\}=\{f(x)\}+\{f(y)\}$ for all $x, y\in\mathbb{R}$. Denote $\{x\}=x-[x]$ in which $[x]$ is the largest integer number does not exceed x.
Takasima Senko
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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 $f(x) = f(0)f(x)$. We write down $f = 0$ as a solution, and…
MT_
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Monotonic additive function. I need to show it's 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
M'smary
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Technique to iterative functional equations $f_{(r)}(x)=2^{(\log x)^c}$ and $g_{(r)}(x)=2^{x^\frac{1}{c}}$

What is function $f,g:\mathbb R^+\to\mathbb R$ sought that satisfies $$\forall x\in\mathbb N,\,f_{(r)}(x)=\underbrace{f(f(\dots(f(f(x)))\dots))}_{r\text{ times}}=2^{(\log x)^c}$$ $$\forall x\in\mathbb…
Turbo
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Calculation of Sigma (Multiple Sigmas)

When I was studying Game Theory, I came across this equation: $$F_i(q_1, \ldots, q_n) = \sum_{s_1 \in S_1} \: \ldots \: \sum_{s_n \in S_N} \big\{ \prod_{j=1}^n q_j(s_j) \big\} \: \: f_i(s_1, \ldots, s_n)$$ How can one interpret this…
user51966
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Solutions to functional equation $ \gamma(s,t)=f(t \cdot g(s))+h(t) $

Let $$ \gamma(s,t)=f(t \cdot g(s))+h(t) $$ where $\gamma$ is a known function of $s \in \mathbb{R}$ and $t \in \mathbb{R}$ while $f$, $g$, and $h$ are unknown functions. Assume $f(0)=f'(0)=h(0)=h'(0)=0$ and $f''(0)=1$. Find all functions $f$, $g$,…
user103828
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Criteria for elevating a rational FE solution to reals

I have a functional equation which I have solved for rational numbers. Now to 'elevate' my solution to reals, I use the given fact that f is continuous. I could have done the same process, if I knew that f is monotone. What I would like to know is…
Anach
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