# Questions tagged [functional-equations]

The functional-equations tag has no usage guidance.

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### Integration over two functions at different time points

I need to integrate a function of the form:
$H(t) = \int_0^t f(u)g(t-u) du$
The functions aren't simple and so I'm using numerical methods to calculate the value of $H$. However, I need to evaluate ...

**-1**

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39 views

### How to solve this kind of second order equations with variable coefficients?

Let $f(x)$ be a function satisfying the functional equation
$$
c_1(x)f(x)^2 + c_2(x) f(x) + c_3(x) f(x-1) + c_4(x) f(x+1) = 0,
$$
where $c_1, \ldots, c_4$ are known functions. What can be said about $...

**2**

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**1**answer

161 views

### Functional equation $\int_z^{2z} [f(x)-f(z)] dx = 0$

Suppose a continuous function $f:[0,1] \to \mathbb{R}$ satisfies the following equation for all $z \in \left(0,\frac{1}{2}\right)$,
$$\int_z^{2z} [f(x)-f(z)] dx = 0.$$
It is clear that a constant ...

**2**

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35 views

### Integral equation with kernel defined in a rectangle

Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$
Observe that the kernel is not defined on a square.
My question: Can I apply the classical ...

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24 views

### Fredholm integral equation of third kind

Let us consider the following integral equation
$$a(x)u(x) + \int\limits_0^2 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p \in [1,\infty]$ and let $K \in L^q((0,2) \times (0,1))$. Assume ...

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37 views

### A special integral equation of Volterra type

Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation:
$$
f(t)\int_0^t a(s)\,ds + \int_0^t a(t - s) f(s) \, ds = 0
$$
My question is : under what condition ...

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96 views

### Calculating the Taylor series, given a functional equation

I have two functions, whose Taylor series about infinity are given by
$$ f(z) = \frac{1}{z} + \sum_{n=2}^{\infty} \frac{A_k}{z^k}, \quad g(z)=\frac{1}{z} + \sum_{n=2}^{\infty} \frac{B_k}{z^k} $$
and ...

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**1**answer

284 views

### A variant of Cauchy-type functional equation conjecture

Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that
$$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$
Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$
The answer is ...

**0**

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**1**answer

201 views

### Background on the functional equation $F(x+1)+F(x)=f(x)$ [closed]

In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-...

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373 views

### Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$
I think the following system of equations ...

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95 views

### A generalized Cauchy type functional equation

Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$.
Then is it true that $f(x+y)=f(x)...

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138 views

### How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?

Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...

**3**

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**1**answer

120 views

### Solvability of a certain functional equation in simple $C^*$ algebras

For which simple unital $C^*$ algebras does the following functional equation have a solution:
$$ d^2=0,\;{(d+d^*)}^2=1$$
The Calkin algebra and $M_{2n}(\mathbb{C})$ are some examples. It is not ...

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**1**answer

87 views

### 1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation
$$
\begin{split}
\mathbb ...

**1**

vote

**1**answer

71 views

### Equation for a geometrical half surface from folding a flat curved in one direction surface in half [closed]

I cannot not formulate the problem as i do not know how to model this idea. This is an open question not an mathematical exercise so if anybody has a good proposition i'm happy to use it.
Sorry in ...

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**2**answers

72 views

### Find the general solution to the Forsyth/Abel functional equation

Let $R(m,n)$ be defined on all the integers such that $R(m,0)=m, R(0,n)=n, R(m,n)=R(n,m)$ and $R(R(m,n),p)=R(m,R(n,p))$ for all integers $p$. Thus $R$ satisfies the Abel associativity equation. Let $...

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51 views

### Characterisation of functions for which the Fourier transform commutes with a particular operator

Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...

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53 views

### Functional equation with Cosine transform

Defining $f(x)$ as a Cosine transform on $\mathbb{R}^+$: $$f(x)=\int_0^{\infty} S(t) e^{-i \lambda \int_1^{t} S(u)^2 \frac{du}{u^2} } \cos(2 \pi xt) dt $$
Can we find a real function $S(t)$ on $\...

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267 views

### Seeking proof to an asymptotics of a recursion or functional equation

My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion
$$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\...

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72 views

### Functional equation involving integrals and exponential

Can we find on $\mathbb{R}^+$ a real positive function $f(x)$ (in $C^{\infty}$) such that:
$$\int_0^{\infty} f(x) e^{\lambda \int_1^{x} f(t)^2 dt} dx=0$$
where $\lambda$ is a complex number (with $0&...

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79 views

### A functional equation with a quadratic solution

I have the following problem. I have a function $v(x, \theta)$ that can be expressed in two ways, for all $x, \theta \in \Re$:
$v(x, \theta) = u(x - \theta)$, where $u$ is strictly concave and ...

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52 views

### Existence of a couple of functions solution of a differential equation (with additional constraint)

I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$):
$$...

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365 views

### What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?

What is the solution, $f(n)$, of the following functional equation:
$$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$
where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...

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367 views

### Solution of the functional equation $f(x+1)=g(x)f(x)$

In the paper (http://www.sciencedirect.com/science/article/pii/S0022247X97953439), Webster obtained a unique solution of the functional equation $f(x+1)=g(x)f(x)$
(where $f,g:\mathbb{R}^+\rightarrow \...

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102 views

### Analytic continuation of $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s}$. Possibly related to Riemann Zeta function $\zeta(s)$?

I'm trying to find the analytic continuation for
$\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s} ,$
with $a\in \mathbb{N^+}$ and $s<1$. I need most likely only the values for $s=\frac{1}{2}-m$...

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137 views

### Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? [closed]

Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that
$$f(x^3+y^3)=f(x+y)((f(x-y))^2+f(xy)),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? I did ...

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174 views

### Solve functional equation: $\frac{F(x)}{F(1)} = 2F\left(\frac{(1+x)^2}{4a}\right)-F(x/a)$

Solve this functional equation:
$$\frac{F(x)}{F(1)} = 2F\left(\frac{(1+x)^2}{4a}\right)-F(x/a)$$
for $F(x)$ where $a > 0$ is a parameter. I know there is a trivial constant solution, $F(x) = 1$. ...

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143 views

### modularity of Theta functions attached to Hecke characters

Let $K/\mathbb{Q}$ be a quadratic imaginary field, and let $\chi$ be a Hecke character on $K$. Using Poisson summation, one can show that the theta function
$$
\theta(z):=\sum_{I\subseteq \mathcal{O}...

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**1**answer

98 views

### Are there solutions for this functional equation?

The distribution $g(x)$ has the following properties:
$$\int_{-\infty}^{\infty}g'(x)f(x)dx=f(\pi g(0)-1/2)-f(\pi g(0)+1/2)$$
for any analytic $f(x)$.
How can I find $g(x)$?

**3**

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**1**answer

218 views

### Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property

First, define a sequence $F_0,F_1,\dots$ of functions by
$$F_0(x,z) = z,$$
$$F_k(x,z)=x\exp\left(F_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$
So, for example,
$$F_1(x,z) = x e^z, \quad F_2(x,z)=xe^{...

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188 views

### Linearizing a power series by conjugation

Let $\mathfrak{I}:=\big\{ \, f:=\sum_{k=0}^\infty f_k z^k \in\mathbb{C}[[z]]\; : \text{s.t. }\; f_0=0 \;\text{ and }\; f_1=1\big\}$. A most basic result about linearization states that, for any $f\...

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47 views

### Variational Problems with Subsidiary Conditions

I am studying from
Gelfand, I.M.; Fomin, S.V., Calculus of variations. Transl. from the Russian and edited by Richard A. Silverman., Mineola, NY: Dover Publications. vii, 232 p. (2000). ZBL0964....

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107 views

### Which functions $f: \mathbb{R} \to \mathbb{R}$ is injective over some subinterval of $(x,y)$ whenever $x<y$ and $f(x) \ne f(y)$?

Under what conditions on a function $f: \mathbb{R} \to \mathbb{R}$ can we say that given any real numbers $x,y$ with $x<y$ if $f(x) \ne f(y)$ then there is a sub-interval $S_{(x,y)}$ of $(x,y)$ ...

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### Is research of the Hyers-Ulam stability of this functional equation already conducted?

The functional equation in question is of the type $f(g(x))=g(f(x))$, where $f$ is the unknown function. Are there existing research already conducted on the Hyers-Ulam stability of this generalized ...

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1k views

### For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant?

Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true :
$$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - ...

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471 views

### Intuition behind the Riemann $\zeta$ functional equation

Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann ...

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37 views

### Necessary additive and multiplicative properties to characterize a mildly growing function

Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we ...

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**1**answer

91 views

### Does this equation have an explicit solution? [closed]

For a positive constant $C$:
\begin{align}
y(x)+C\ln y(x)=f(x).
\end{align}
At least from specific $f(x)$, such as piece-wise linear function, is there an explicit solution for $y(x)$?

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124 views

### Reflection Formulas for the $\Gamma$ Function

We have
$$\begin{align}
&\Gamma\Big(1~+~x\Big)~\cdot~\Gamma\Big(1-x\Big)~=~\frac{\pi x}{\sin\pi x}
\\\\
&\Gamma\Big(\tfrac12+x\Big)~\cdot~\Gamma\Big(\tfrac12-x\Big)~=~\frac\pi{\cos\pi x}
\\\...

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52 views

### Solving a nonlinear integral equation for a distribution function

I have an equation of a probability distribution function $F$ on $[0,\infty)$, $$F(x)=e^{-\eta [1-\int_0^x F(x-y)g(y)\mathrm{d}y]},\quad x\in[0,\infty), $$
where $g$ is a probability density function, ...

**0**

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**2**answers

97 views

### A simple equation with a normal distribution [closed]

Let $f$ be the distribution of a normal variable $\mathcal{N}(0,1)$, ie
$$ f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$
I have to solve the equation:
$$x+y = f(x) - f(y)$$
I was working with mathematica ...

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**1**answer

239 views

### On a generalization of the classical Cauchy's functional equation

I start with some known preliminaries on the problem:
Classical result. The one-dimensional Cauchy functional equation
$$
\forall x,y \in \mathbb{R}, \,\,\,f(x+y)=f(x)+f(y)
$$
with $f:\mathbb{R}\to \...

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**0**answers

134 views

### Zappa-Szép products of the group of integers with itself

Since my previous question didn't get much attention and I couldn't make any relevant progress on it, I thought it would be a good idea to "simplify" it by replacing monoids by groups. That is:
...

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87 views

### Zappa-Szép products of the monoid of integers with itself

Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...

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**1**answer

435 views

### A conjecture about certain values of the Fabius function

The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations
$$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$
and
$$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\...

**9**

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**1**answer

241 views

### Some nice functional equations for $q$-continued fractions

Given $\large q=e^{2\pi i \tau}$. Define,
$$\alpha(\tau) = \sqrt2\,q^{1/8}\prod_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$
$$\beta(\tau) = q^{1/5}\prod_{n=1}^\infty\frac{ ...

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**2**answers

458 views

### Total progeny of a Galton-Watson branching process - standard textbook question

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly.
...

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389 views

### On the consistency of the definition of the conductor for automorphic forms

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conudctor associated to $\pi$ can be defined in two usual manners:
By its ...

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**0**answers

198 views

### General solution of the Multiplicative symmetry equation $f(xf(y))=f(f(x)y)$ in nonabelian groups

As we know, the functional equation $f(xf(y))=f(f(x)y)$ was completely solved in abelian groups (by J. G. Dhombres, Solution... $f(x\ast
f(y))=f(y\ast f(x))$, Aequationes Math. 15 (1977), 173--193, ...

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**1**answer

2k views

### $f'=e^{f^{-1}}$, again

This question is a spin-off of this one, in which the OP asks whether there is a solution $f:\mathbb R\to\mathbb R$ of the functional equation (not exactly an ODE) $f'=e^{f^{-1}}$, where $f^{-1}$ is ...