# Tagged Questions

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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### 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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### Can this Delay Differential Equation (DDE) be solved in closed form?

$$\frac{\mathrm{d}\psi}{\mathrm{d}x}=\frac{(x+\mu)^2-\lambda^2}{\mu}\psi(x+\mu)$$ where $\mu\geq0$, $\lambda\in\mathbb{C}$ are constants and $\psi(x)\to0$ as $|x|\to\infty$.
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### Does this functional equation have a (unique) solution?

Does the following equation have a (unique) solution for $b_1$? b_1 = \dfrac{1}{2b_1} \left\lbrace -Var(p) + Cov \left(\left[ (p + b_1 + b_0 + \alpha)^2 -4b_1(p + b_0) \right]^{0.5},...
I want to solve a functional differential equation of this kind $$\int d t'' dt''' a(t,t'') b(t'',t''') \frac{\delta u(t'',t';[x])}{\delta x(t''')}+\int d t'' c(t,t'') x(t'') u(t'',t';[x])+u(t,t';[\... 0answers 22 views ### Solving equation over expectation I have an expression (1 + n EX^k)p^{-k} which I would like to minimize over k. Here n and k are positive numbers, while X and p are in [0,1]. Since the expectation converges absolutely, ... 0answers 20 views ### Interval between two solutions of an equation We are in \mathbb{R} and x\geq 0. I have an equation: (1-\alpha)x^{\frac{1}{1-\alpha}}-W(y)x=g(y)W(y) Where \alpha \in (0,1), y is a parameter 0<y<1, W and g are constants in x and ... 0answers 30 views ### Solving the functional equation F(z+1)=Q(z)F(z) I want to solve the functional equation F(z+1)=Q(z)F(z). The F(z) is a matrix function. Q(z) is also a matrix function. But its compoents are all rational functions. i.e.  F(z)=\begin{matrix} ... 0answers 11 views ### Pacejka formula - extracting the curve peak I don't know if many of you knows the Pacejka Magical Formula, but it looks like this: D\sin(C\arctan(Bx - E(Bx-\arctan(Bx)))) As you can see, the formula reaches a peak point(in this case at ~0.6)... 0answers 22 views ### How to complete this? Let f a function defined on ]0,+\infty[ and checking :$$\forall x,y>0,\ f(xy)=f(x)+f(y)$$Assume that f is bounded on ]1-\eta,1+\eta[(\subset]0,+\infty[). Let \alpha =\underset{x\in ... 0answers 37 views ### Functional analysis with KKT conditions I want to solve an optimization problem min F(x_{ik})  subject to x \in X. F here is a function or functional that I wish to determine. I want my optimal ... 0answers 26 views ### Solving a functional convolution equation I have a very weird functional equation that I am trying to solve. Let f:\mathbb{R}^n \to \mathbb{R} be a non-negative Lebesgue integrable function such that \int f=1. Let \tilde{f}(x)=f(-x) ... 0answers 49 views ### Find all functions f: \Bbb Q \rightarrow \Bbb Q Find all functions f:\Bbb Q \rightarrow\Bbb Q that satisfy the conditions: a) f(f(2016))=0; b) f(x+y)= f(x)+f(y)+f(2016) for all x,y \in \Bbb Q. I tried to solve the problem using the ... 0answers 26 views ### A simple PDE related to symmetric functions This question was posed on mathoverflow but was put on hold. So far nobody has been able to give any hint either to the solution or to why it is trivial. Disclaimer: as far as I can tell this is not ... 0answers 29 views ### Functional Equation of Probability Distributions Suppose you have a real random variable X that has probability distribution f_X meaning$$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$Now assume \Phi(f_X) is also a ... 0answers 26 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 mean,... 0answers 25 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 ... 0answers 50 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 ... 0answers 56 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 ... 0answers 30 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: x_s=\displaystyle\frac{1+{... 0answers 20 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 ... 0answers 22 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 ... 0answers 43 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 '' (1)... 0answers 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 ... 0answers 41 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 a}... 0answers 49 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 ... 0answers 38 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. (  f^{... 0answers 67 views ### How to do this estimate If a,b are two vectors in \mathbb R^n satisfy the following relation \frac{|a|^2}{1+(1-|a|^2)^{\frac{1}{2}}}\geq \frac{|a|^2-2\langle a,b\rangle}{(0.99-|b|^2)^{\frac{1}{2}}+(0.99-|... 0answers 40 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 ... 0answers 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 $$I know the derivative of \left(r_{0} + \sum [a_{i}\... 0answers 42 views ### Unique solution of a simple functional equation Let x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R} be two smooth functions (x,y\in C^{\infty}([a,b])). How can I prove that there is a unique function \theta:[a,b]\to\mathbb{R},\ \theta\in C^{\... 0answers 21 views ### Solving a linear functional equation Working with Green functions, I have found to solve the following equation$$ -\omega^2G(\omega)-m^2G(\omega)+\kappa\sum_{n=-\infty}^\infty b_nG(\omega-n\omega_0)=1 $$where m, \kappa and \... 0answers 102 views ### Differential equation involving composition I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ... 0answers 41 views ### A functional equation involving convolution Let's say I have a smooth function \varphi:\mathbb R \to \mathbb R which is a convolution kernel. I am interested in the following equation: 0=A+B~(\varphi*v)+C~v+D~v~(\varphi*v), where v:\... 0answers 84 views ### Existance of solution to integral Fredholm equation I am a bit confused with the existence/non existence of a solution to the following equation:$$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$where y(t)=C (a constant function),... 0answers 44 views ### Conjecture about A f(x) = f(g(x)) + f(h(x)) Let a given real A satisfy 0 < A < 2. Conjecture : For any real entire nonconstant f(x) there exist real entire g(x) and h(x) such that A f(x) = f(g(x)) + f(h(x)) or  A f(x) =... 0answers 32 views ### Existence of a function satisfying all the given infima Given a function f : \mathbb{R} \to \mathbb{R}, we can compute its infimum over A for all the Borel measurable A \subset \mathbb{R}. I am wondering when we can deduce in the other direction, i.... 0answers 33 views ### An equation with a nested function I'm trying to find the function \eta(x) such that \eta(x) F(\eta(x))-G(\eta(x)) = \eta(x)H(x) - G(x) but I have no idea how to go about it, or where to look. Thanks for the inputs. All ... 0answers 63 views ### What functions satisfy \int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx? What functions satisfy \int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx for all y\in \mathbb{R}? Under what conditions would this imply that f(x,y)=g(x,y)? 0answers 43 views ### A functional equation for a Dirichlet series I'm looking for a functional equation for the following Dirichlet series$$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$where q is a rational number. Any help ? 0answers 60 views ### Does there exist a nontrivial cumulative distribution function F on \mathbb{R}_+ for which (F^2)'= (F')^{\ast k} for some k > 0? This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ... 0answers 64 views ### Non-constant solution of f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x) in rings The answers to this question prove that over an integral domain the functional equation$$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$has only solutions where one of f or g is constant. The proofs make ... 0answers 1k views ### An example of the Sequential Quadratic Programming (SQP) Given an objective function f(x,y)=(x+6)^2+(y-10)^2, we want to minimize the function f under a constrained condition xy-5\leq0. Obviously it is a constrained optimization, a good algorithm to ... 0answers 84 views ### Seeking a function which satisfies a given functional equation. I wish to find a function f: \mathbb{R} \rightarrow \mathbb{R} which satisfies: f(u) \geq 0 when 0 \leq u < 1, f(u)=0 when u<0 or u \geq 1, \int_0^1 f(u) du=1, and$$f(u)=\frac{f\...
It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z})$ converges for $z$ in the upper half plane to a ...