# Questions tagged [functional-equations]

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### Imaginary delay

I'm looking at a particular delay differential equation: Suppose we have $x'(t) = x(t - i \pi),$ with initial condition $\phi(t) = x_0(t),$ where $i \pi$ is imaginary. Well, uh, how do we derive a ...
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### What are the process and fundamentals of solving delay differential equations?

I have a lot of confusion around delay/difference differential equations, I'm hoping someone can clear it up. This is a very niche, specialized subject, so I need to ask here among people who have ...
1 vote
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### Invariant polynomials under a non-standard group action

There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...
306 views

### How can I derive functional properties of (the solutions of) this simple functional differential equation?

I've not yet finished a course in functional analysis so I'm unsure how to go about this, but I've always been fascinated by a simple functional differential equation I concocted for almost no reason. ...
1 vote
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### Is there a systematic procedure to Solve Abel's, Böttcher's, or Schröder's Equation

I've been interested greatly in the study of functional equations for some time now, I've learnt many different techniques for their solution. Currently I have been studying superfunctions and ...
1 vote
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### Proving the simple form of a function from statistical mechanics

I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
89 views

### Finding minimal $\gamma$ that satisfies the integral equation

I have a function $f(z) \in [0,\infty]$ that satisfies $-1 < f(z) < 0$. I would like to find the minimal $\gamma$ that satisfies: $$\int_0^{\gamma} f(z)dz = \log(1+f(0)).$$ Clearly, I cannot ...
1 vote
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### Can I apply $q$-Lagrange Inversion formula?

Now I have equation $F(x) = x \sum_{k\ge 0} g_k F(x) F(qx) \cdots F(q^{k-1} x)$, I need to get the coefficient of $x^n$ in $F(x)$, can I apply $q$-Lagrange Inversion formula to this? Moreover, I have ...
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### Fermat's Last Theorem (FLT) in standard model space corresponding to an infinite Blaschke product

Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the ...
Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that $$\lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^... 0 votes 1 answer 169 views ### Is it possible to numericaly solve functional equation Given a functional equation of form f(f(x))=T(x) is there any good ways to solve it numerically? If not then at least approximate in some small region x\in(-a;a). E.g. with the equation f(f(x))=x+... 2 votes 0 answers 142 views ### Square root of a function on a finite set Let S be a finite set and f \colon S \to S be an arbitrary function. How can we find all functions g \colon S \to S with f = g \circ g? If f and g are both required to be invertible, the ... 2 votes 2 answers 183 views ### Functional equations and normal distribution Let \alpha \neq 1. If X,Y are two independent random variable such that U=X+Y and V=X+\alpha Y are independent, then X and Y are normally distributed. In term of characteristic functions ... 18 votes 3 answers 4k views ### Is there a general solution for the differential equation f''(x) = f(f(x))? I'm currently an undergraduate studying differential equations and I've been fixated on the differential equation f''(x) = f(f(x)) for the past 2 days. I can't seem to crack it but it feels like it ... 10 votes 2 answers 452 views ### A functional equation involving the inverse function \newcommand\ep\epsilon\newcommand\R{\mathbb R}Let P denote the set of all continuous probability density functions (pdf's) p on \R vanishing at \pm\infty. Let us say that a pdf p\in P is ... 0 votes 1 answer 140 views ### Solve (x-a)^{\alpha +1} - \lambda*(b-x)^{\alpha + 1} = C(\frac{a+b}2 - x)^{\alpha} over \mathbb R [closed] I have been having trouble solving the following equation. Any help would be appreciated. Let a,b,C,\alpha,\lambda be real numbers with C < 0, 0 < \alpha < 1, \lambda > 1. We ... 0 votes 0 answers 192 views ### Functions that satisfy a reverse triangle inequality: do they have a name? Let f : \mathbb{R}^n \to \mathbb{R}_+ satisfy$$f(a) - f(b) \le C f(a - b)$$\forall a, b \in \mathbb{R}^n for some C \ge 0. Is there a name for such functions? (I would be happy to have a name ... 2 votes 1 answer 164 views ### Existence of a distinguished continuous version of the logarithm of a continuous function Let E be a \mathbb R-Banach space and \varphi\in C^0(E,\mathbb C\setminus\{0\}) with \varphi(0)=1. I want to show that there is an unique \psi\in C^0(E,\mathbb C) with \psi(0)=0 and$$\...
I am learning more about the theory of additive functions ($f(nm)=f(m)+f(n)$) and I am struck by how powerful the theorems given are. For example, we have a complete characterization of not only what ...