# 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).

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### 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 formal-power-...
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### Find a function such that $f^{-1}=f'$

Let $f:\Bbb{R}^+\rightarrow\Bbb{R}^+$ be a differentiable bijection and let $f$ satisfy: $f'=f^{-1}$ (where $f^{-1}$ denotes the inverse of $f$). Find $f$. This comes from a facebook page "...
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I want to find equation of conformal map (= Fatou function $\Psi : z \to u$ ) which: maps some region of complex plane ( attracting petal) to right half of complex plane in u coordinate $Re(u) > ... 0answers 84 views ### Functional division$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$As the title suggests, the problem here is: Find all functions$f:\mathbb{Z}\to\mathbb{N}$such that, for every$x,y\in\mathbb{Z}$, we have $$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$$ I ... 0answers 382 views ### Infinite Double Exponential Sum, with Functional Equation$g(x) = g(\sqrt{x})$What is a closed form for $$\lim_{n\to-\infty}\sum_{i=n}^{\infty}\frac{x^{2^i}(x^{2^i}-1)}{(x^{2^{i+2}}+1)}$$ The series has the form: $$... \frac{x^{\frac{1}{4}}(x^{\frac{1}{4}}-1)}{x+1} + \frac{... 1answer 173 views ### Uniqueness of solution of functional equation I have a function f(x_1,x_2) \colon \mathbb{R}^2_{+} \to \mathbb{R}_{+} positive homogenous:$$ f(\lambda x_1, \lambda x_2) = \lambda f(x_1,x_2), \; \forall \lambda > 0 $$and such that f(x_1,... 0answers 50 views ### Does there exist a function f_{\Box,\Box}(\Box) making the formula a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c) true? Let a and b denote the resistances of two resistors. If they're put in series, the total resistance is a+b. If they're put in parallel, the total resistance is$$a \oplus b := \frac{1}{\frac{1}{... 0answers 157 views ### Solution of advanced functional differential equation Statement Consider an advanced functional differential equation $$Lf(x) = f(2x+\pi)+f(2x-\pi),\quad L\equiv\frac{d^2}{dx^2}+1. \tag{1}$$ Let's construct a solution of Eq.$(1)$with finite ... 2answers 75 views ### Determine all functions$f : \mathbb{N} \rightarrow \mathbb{N}$such that, for every positive integer$n$, we have:$2n+2001≤f(f(n))+f(n)≤2n+2002$. Determine all functions$f : \mathbb{N} \rightarrow \mathbb{N}$such that, for every positive integer$n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ... 0answers 81 views ### Does there exist a function$f$such that$f(x)$is an integer for only finitely many values of$x$? Define$f : \mathbb{R} \to \mathbb{R}$such that$f(x)\leq f(y)$whenever$x\leq y$and$f^{2018} (z) \in \mathbb{Z}\forall z \in \mathbb{R}$. Does there exist a function$f$such that$f(x)$is an ... 0answers 149 views ### A nontrivial solution for$ f(f(x)) = \exp(\exp(x)) $Consider the equation $$f(f(x)) = \exp(\exp(x))$$ Valid for all real$x$,$f(x) \neq \exp(x)$(not identically equal everywhere),$f(x)$is analytic and $$f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 +... 1answer 96 views ### Find all functions f: \mathbb{R}\setminus\{0\} \to \mathbb{R} satisfying 3f(x) - f\left(\frac{1}{x}\right) = x^2 Find all functions f: \mathbb{R}\setminus\{0\} \to \mathbb{R} satisfying 3f(x) - f\left(\frac{1}{x}\right) = x^2. What I did was first plug in x = 1 to get 2f(1) = 1 \implies f(1) = \frac{1}{... 0answers 110 views ### Increasing function f(x) such that f(\gcd(x,y))=\gcd(f(x),f(y)) This problem was largely inspired by this problem here. There were many counterexamples given to the problem, such as a multiplicative function that maps primes to a permutation thereof. However, if ... 0answers 192 views ### Convergence of the solution of Volterra integral equation with convergent kernel. Consider the following Volterra integral equation$$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$where g(t) and K_n(t,s) are known(continuous) and K_n(t,s)\geq K_{n+1}(t,s) for all t,s. Moreover, ... 0answers 873 views ### Width of the Eiffel Tower as a function of height? In the preface of Advanced Engineering Mathematics, 2nd Ed. by Zill and Cullen, it is claimed that the function relating the width of the Eiffel Tower as to the distance from its top, x \mapsto f(x),... 0answers 387 views ### Vector valued contraction I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like C(X), the space of bounded continuous function f:... 1answer 170 views ### Approximation of \sum_{n=0}^\infty 2^{-n} \tanh(3^n x) in x = 0 I want to (asymptotic function) approximate the series f(x) = \sum_{n=0}^\infty 2^{-n} \tanh(3^n x) around x = 0. I found the equation for f(x):$$f(x) = \dfrac{1}{2}\ f(3x) + \tanh(x)$$for ... 1answer 76 views ### For fixed n \in \mathbb{R}, what are all the continuous functions that satisfy nf(x) = f(nx)? For fixed n \in \mathbb{R}, what are all the continuous functions that satisfy nf(x) = f(nx)? I thought it would just be functions of the form f(x) = kx but, for example, in the n=2 case we ... 1answer 45 views ### Find all functions such that if I open bounded interval then f(I) is also open of same length Find all functions f : \mathbb{R} \rightarrow \mathbb{R} such that for all I open bounded interval it follows f(I) is also an open bounded interval of same length as I. It's easy to see ... 2answers 176 views ### Functional equation : f(1)^3 + f(2)^3 + \ldots + f(n)^3 = (f(1) + f(2) + \ldots + f(n))^2 Find all function f:\mathbb{N}\rightarrow\mathbb{N}\cup\{0\} satisfying$$ f(1)^3 + f(2)^3 + \ldots + f(n)^3 = (f(1) + f(2) + \ldots + f(n))^2$$\forall n \in \mathbb{N}. Thank you, ... 0answers 62 views ### Functional equations in the form f(x)-f(x-c)=P_n(x) Consider f:\mathbb R\to\mathbb R, and let P_n be a polynomial with degree n\ge 1. Now, if$$f(x)-f(x-c)=P_n(x)$$for some fixed constant c, then what assumptions on f could let us ... 0answers 196 views ### Solving the functional equation f(xf(x)+yf(y))=xy over positive reals Does there exist any function f: \mathbb{R}^+ \to \mathbb{R} such that for all positive real numbers x,y the following is true :$$f(xf(x)+yf(y))=xy$$1answer 125 views ### Tricky composite function Let f: A\to A where A\in (-1,\infty ) and f(x + f(y) + xf(y)) = y+f(x)+yf(x) for all x,y \in A and \frac{f(x)}{x} is strictly increasing for all x \in (-1,0) \cup (0,\infty). Then find ... 0answers 122 views ### Solve this integral equation that results in a linear function I need to find the family of real-valued single variable functions F:\, (0,1) \to [0,1] that satisfy the following integral equation:$$\int_{\theta = 0}^{\pi} F\big(~ x\, \sin\theta ~\big)\text{d} \... 0answers 132 views ### Function with$f(f(n))=f(n-1)f(n+1)-f(n)^2$Let$\mathbb{N}$denote the set of positive integers. Does there exist a function$f:\mathbb{N}\rightarrow\mathbb{N}$such that $$f(f(n))=f(n-1)f(n+1)-f(n)^2$$ for all$n\geq 2$? If$f$is linear, ... 0answers 106 views ### Functional equation:$f(x,y)=f(x+y,y)=f(x,x+y)$Is there a nonconstant continuous function$f:\mathbb{R}^2\rightarrow\mathbb{R}$satisfying the functional equations$f(x,y)=f(x+y,y)=f(x,x+y)$? If the answer is yes, can we characterize all ... 1answer 89 views ### Functional equation with only two solutions? Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation: $$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$ Unfortunately, I was able to show only ... 0answers 67 views ### Solution techniques for f'(x)=f(g(x)) I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type$f'(x)=g(f(x))$are studied for all kind of classes of ... 0answers 105 views ### Is the product rule for logarithms an if-and-only-if statement? If a function$f(x)$is proportional to$\ln x$, then we know $$f(xy) = f(x) + f(y).$$ My question is, is the converse true? If we know that, for an unknown function f, $$f(xy) = f(x) + f(y),$$ ... 1answer 98 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$f$... 0answers 505 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 ... 0answers 173 views ### How to solve this finite-difference equation? How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say$f(x) = f_0(x)$for$x \in [0, \sqrt{2})is a given function. ... 0answers 296 views ### A late-diverging “approximating solution” for a system of functional equations Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} \psi(\alpha)-\... 0answers 63 views ### Wronskian different from zero and solutions of ODE. Leta_0, \ldots , a_{n-1}$continuous functions in an interval$I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let$\phi_1, \phi_2, \ldots,\phi_nn$are ... 1answer 114 views ### How prove this$f=C$if$4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$Question: if$f:\mathbb{Z}^2\to \mathbb{R}$is bounded ,and for any$x,y\in \mathbb{Z}$,we have $$4f(x,y)=f(x-1,y)+f(x,y-1)+f(x+1,y)+f(x,y+1)$$ show that $$f\equiv C$$ where$C$is constant. My try:... 0answers 97 views ###$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$Consider the equation:$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for$a \geq 0$and$b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ... 0answers 85 views ### Asymptotics for$ f(n) = f(n - 1) + f( n - g(n) ) $? Define$g(x)$as : If$f(m) =< x < f(m+1)$for a positive integer$m$then$g(x) = m$. Now we define$f(n)$for strict positive integer$n$. $$f(1) = 1$$ $$f(2) = 3$$ $$f(3) = 7$$ For$n &...
I need an inversion formula with the form $f(r)=\cdots$, from this integral relation: $$g(r)=\frac{1}{2\pi}\int_0^{2\pi}d\theta\,f\left(\sqrt{r^2+r_0^2-2rr_0\cos\theta}\right)$$ where $r_0\geq0$ is a ...