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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47 views

regarding a set of integro-PDE

Let say I have a minimal example that contains most of the mathematics I am looking for. The example is as follows: \begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 ...
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72 views

How to calculate straight line into graph having variety of different results

How to calculate straight line into graph having variety of different results. What I mean for example let say we have this kind of results (measuring persons weight ...
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47 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
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70 views

A functional power series equation

I am interested in solving the following functional equation: $$(1-w-zw)F(z,w)+zw^nF(z,w^2)=z-zw\,.$$ Here $n\geq2$ is a fixed integer and $F(z,w)$ is a power series in two variables with complex ...
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302 views

iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations. For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear ...
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44 views

Program to determine the relationship of one variable to several possible variables

Suppose I have a system with several variables a, b, c, d, and x. I am trying to solve for the unknown x. I don't know exactly which of those variables x is dependent on, or exactly how the function ...
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139 views

Is there a function $f$ such that $\Gamma (c+x)=\Gamma (c-f\left( x \right) )$?

I was just looking at Euler's reflection formula for Gamma function which states $$\Gamma (1-z)\Gamma (z)=\frac { \pi }{ \sin { (z\pi ) } } $$ but it seems to me that one more reflection formula ...
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112 views

Periodicity of a function.

If we have two functions $f:\mathbb{R} \to \mathbb{R}$ and $g :\mathbb{R} \to \mathbb{R}$ such that the period of $f$ is 7 and that of $g$ is 11, then the period of $F\left ( x \right ) = f( ...
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69 views

Functional equation inspired by moment generating function

A user gave the following nice answer http://math.stackexchange.com/a/161584/5031 My question is that although it is clear $\log M_X(t)=Ct^2$ is a form that satisfies the condition ...
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137 views

Question about ratios and combinatorics

In this question that I posted yesterday (11/15): I am solving a programming puzzle that consists of finding all the possible ways to build a brick wall of $48$" $\times$ $10$" (width $\times$ height ...
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70 views

What is an affine function?

Consider a functional, what is meant by a minimal sequence consistent of 'piecewise affine functions'?
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80 views

$f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$

Let $f(x)$ be a non-constant real-analytic function and for real $x$ it satisfies : $f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$ Before you ask if this simplifies by writing $2^x = y$ note that ...
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56 views

solving this recurrence equation

is it possibel to solve the equation g$(x)= \sum_{n=1}^{\infty}f(x/n) $ for $ f(x)$ with other methods different from taking the Mellin transform on both sides ?? thanks.
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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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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 $ ...
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907 views

Solving a functional discrete equation.

I was to solve the following functional discrete equation (I arguing that $a_k$ is a discrete function): \begin{equation}f\left[a_{k+1}\right]-f\left[a_{k}\right]=0\end{equation} where ...
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126 views

solution of d’Alembert’s equation.

i know that equation for d’Alembert’s equation. is looking so $g(x+y)+g(x-y)=2*g(x)*g(y)$ so am trying to find actual solution for this equation,first i took $x=y=0$ and i got $2*g(0)$=$2*g(0)^2$ ...
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494 views

functional derivative of an integral of the function itself

I have the following $$ \frac{d}{dn(x)} \int_{x \in \cal{R}^3} {n(x) dx} $$ I know that this additional relationship holds $$ \int_{x \in \cal{R}^3}{n(x) dx} = N $$ where N is a constant. My ...
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24 views

Functional Equation of Probability Distributions

Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning $$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$ Now assume $\Phi(f_X)$ is also a ...
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23 views

Class Scatter Fitness Function Calculation

I get a fitness function for class scatter, the equation: Click here to see the Fitness Function Where : T = Transpose of Matrics Mi = class mean Mo = grand mean The equation is based from ...
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52 views

unique function on a hilbert space

unique function on a hilbert space How do you show that with Ω=(-1,1) there exists a unique function u such that the equations in the picture is correct?
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21 views

Extension to a continuous linear functional

Extension to a continuous linear functional May this functional be extended to a continuous linear functional on these Hilbert spaces?
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18 views

Easy computations using the functional equations for Riemann and Gamma functions

Let $\zeta(z)$ the Riemann Zeta function and $\Gamma(z)$, the Gamma function. I've deduced easily an equation involving these functions. I don't known if it is useful, if there are mistakes in my ...
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111 views

Functional equation $f(f(x)+3y)=12x + f(f(y)-x)$

I found this problem on a French exchange forum : Find all the $f : \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)+3y)=12x + f(f(y)-x)$ In fact I solved the problem when $f$ is supposed to be ...
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49 views

two integral equations

I'm trying to solve the two following integral equations : 1) $y(x)=2+\int_1^x\frac{1}{ty(t)}\,dt$, $x>0$ 2) $y(x)=4+\int_0^x2t\sqrt{y(t)}\,dt$ It really looks like an ODE but I'm a bit clueless ...
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53 views

The functional equation $ f(x-f(y))=f(f(y))+xf(y)+f(x)-1$

I came across the functional equation: $f(x-f(y))=f(f(y))+xf(y)+f(x)-1$ So far I tried plugging $x=f(y)$ and got $f(x)=\frac{f(0)-x^2+1}{2}$ which holds for every $x = f(y)$. I suppose that $f(0)=1$ ...
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25 views

Time delay equation

If $x(t)=\displaystyle\frac{1+(x(t-T))^m}{(x(t-T)))^m}$ for all $t$ where $T$ is constant and $x(t)=x_s$ is the solution to the above equation, why can I write that: ...
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19 views

How to convert $\ max \int_{-\infty}^{\infty} f(x) dx =\ min \int_{-\infty}^{\infty} g(x) dx $

Let $f$ or $g$ have a given condition. ( example $\int_{- \infty}^{\infty} f(x) exp(-x^2) dx = 1$ Or $g(g(x)) = x^3$ Or a differential equation for one of $f,g$. ) I want to find a general way - if ...
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21 views

Existence of an inverse function in this functional equation

Let $V,W$ each be connected and separable sets. Suppose we have continuous functions $F:\mathbb{R}\times V\rightarrow \mathbb{R}$, $G:\mathbb{R}\times V\rightarrow \mathbb{R}$, $f:V\times W\rightarrow ...
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32 views

Uniqueness question from functional equation

Let $f(x)$ and $f_2(x)$ be real and continuous on the interval $[0,\infty[$. Let $f(1) = f_2(1) = 1 , f(2) = f_2(2) = e$. Let $g(x)-g(x-1) = f(x-1)$. Let $ f ' (x) = exp(g(x) - g(1)).$ and $f '' ...
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13 views

Formulation of a rule

Im new to this forum(hence, formatting issues). I come from CS background. I am trying to figure out a formula to code set of rules. The requirement is as below. Imagine two parameters X and Y ...
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39 views

Finding the “hidden values”.

I have this problem but my knowledge of the methods of linear algebra is totally useless if it exists at all (I know what is a matrix at least... maybe). I have a three collections of objects ${\bf ...
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44 views

Cauchy-like functional equation $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$

I am looking for the solution to the following two variable functional equation: (*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$ where: $h$ is some given continuous function, $f, g,$ unknown functions on ...
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36 views

$\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$

Let $t,x$ be nonnegative reals. Let $* ^{[k]}$ denote k th iteration. Find real-analytic $f(x)$ such that $\int_0^t f(x) - x dx = f^{[t]}(0) - t - 1$ Holds. We require analytic iterations. ( $ ...
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67 views

How to do this estimate

If $a,b$ are two vectors in $\mathbb R^n$ satisfy the following relation \begin{equation} \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle ...
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38 views

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 ...
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17 views

Find a derivative of equation that contains Fourier series

I need to find a derivative of follow equation $$ \left(r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] \right)({\sin\phi-k\cos\phi}) - b = 0 $$ I know the derivative of $\left(r_{0} + \sum ...
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23 views

A Somewhat Complicated Functional Equation

Given a function on $\mathbb R^2$ $$C(x,y)=e^x\big(e^y N(d_1)-N(d_2)\big) \tag1$$ where $N$ is the cumulative standard normal distribution function (an increasing function) and ...
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31 views

Stability of Cauchy exponential functional equation

If f : R → R is a function satisfying |f(x + y)- a^xy f(x) f(y)| ≤ δ for all x,y ∈ R and for some positive δ, where a is a positive real constant, then show that either the function $f(x) ...
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41 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...
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38 views

The most general solution to the functional equation

Suppose we have the following functional equation $$X(x_1,x_2)Y(x_1,x_3)=Z(x_2,x_3).$$ Is the most general solution given by $$X(x_1,x_2)=A(x_1)B(x_2), Y(x_1,x_3)=\frac{C(x_3)}{A(x_1)}, ...
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136 views

Approach on solving limit equation systems and finding some f given assymptotes?

This is a "reverse" question of finding the asymptote of a function Recently, I am interested in doing some sort of modelling which involve equations of the form $$@(t)=1-f(t)$$ where $f(t)$ is ...
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20 views

Iterative Functional Equation? (discrete, increment, logarithm)

Say $f(n)$ is defined as discrete iterative process, $n \in [0,1,2,\ldots,N]$, and $g(x)$, $x \in [-\infty,+\infty]$ is "small": $|g(x)|<1, \forall x$: $$f(n+1)=f(n) + \log(1 + g(f(n)))$$ ...
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59 views

Functional equation similar to Babbage's equation

I'm interested in the potential solutions $f: R_{+} \rightarrow R_{+}$ to the functional equation: $ \forall x \in R_{+}, \quad f(s(x) - f(x)) = ax $ where $a>0$ is a constant. $s: R_{+} ...
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67 views

Extension of the Cauchy functional equations

Let $f:(0,a)\to\mathbb{R}$ satisfying $$f(x+y)=f(x)+f(y)$$ for all $x,y,x+y\in(0,a)$, where $a$ is a positive real number. Show that there exists an additive function $A:\mathbb{R}\to\mathbb{R}$ such ...
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21 views

Solving a linear functional equation

Working with Green functions, I have found to solve the following equation $$ -\omega^2G(\omega)-m^2G(\omega)+\kappa\sum_{n=-\infty}^\infty b_nG(\omega-n\omega_0)=1 $$ where $m$, $\kappa$ and ...
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35 views

Does this problem have analytic/approximate analytic solution?

Define $F[i]=F[i-1]-F[i-1]^c$ with $F[0]=r>0,c\in(0,1)$. Define $R(r,c)=\min\{i:F[i]<1\}$. What is solution to $c\in(0,1)$ in $R(r,c)=r^{1/k}$ with some given $k>1$?
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57 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
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14 views

Closed form of chained Stationary Functions

Consider the following operator $$D_{(a-1)x,x}\left[ f \right] = \frac{f(ax) - f(x)}{(a-1)x} $$ It's pretty easy to see that a nontrival function $g(x)$ such that $$D_{(a-1)x,x}[g] = g$$ Is given ...