# 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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### $f(xy) = f(x+y)$ : A variation of an old problem

I have seen the following problem in various places, such as Toppr and Quora. Suppose that the function $f:\mathbb{N}\to\mathbb{N}$ satisfies $f(x+y)=f(xy)$ for all $x,y\in\mathbb{N}$. Show that $f$ ...
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### Solutions to $f(x^2) = 2f(x)^2-1$

I'm looking for solutions over $\mathbb{R}$ to the functional equation: $$f(x^2) = 2f(x)^2 -1$$ The left hand side is very similar to the expansion for $\cos(2\theta)$. But the right hand side ...
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### Doubt in an alternate solution to an IMO problem from 1987

Consider the following problem from 1987 IMO Show that there is no function $f:\mathbb N→\mathbb N$ Such that $f(f(n))=n+1987$ In the book that I was referring the solution is given as follows: " ...
1 vote
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### Alternate solution to an IMO 1993 problem

Let $\mathbb N=\{1,2,3,…\}$. Determine if there exists a strictly increasing function $f:\mathbb N→\mathbb N$ such that $f(1)=2$ $f(f(n))=f(n)+n$ for all $n$. Recently I posted the following solution ...
1 vote
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### Let ℝ+ be the set of positive numbers. Determine all functions $f:\mathbb R^+\to\mathbb R^+$ such that $f(x^2+xf(y)) = f(f(x))(x+y)$ [closed]

Let $\mathbb R^+$ be the set of positive numbers. Determine all functions $f:\mathbb R^+\to\mathbb R^+$ such that $$f(x^2+xf(y)) = f(f(x))(x+y)$$ for all positive real numbers $x$ and $y$. I think ...
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### Functional equation of complex numbers

If $f:\Bbb C\to\Bbb C$ satisfies $f(f(x))=\left(x\overline x-x-\overline x\right)^2$ and $f(1)=0$, find $f(\mathrm i)$. If $f(a+b\mathrm i)=a^\prime+b^\prime\mathrm i$, define $g:\Bbb R^2\to\Bbb R^2$ ...
1 vote
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### Find all injective functions from reals to reals that satisfy $f(x + f(y)) = f(x) - f(y)$

Find all injective functions from reals to reals that satisfy $f(x + f(y)) = f(x) - f(y)$, from reals to reals. I tried subbing in values to cancel some variables out, like $y = x$ and $y=-f(x)$, but ...
1 vote
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### What field should I search to solve delay like equations?

I want to solve equations like: $$t f(t) + u(t-2) f(t-2) = \sin(t)$$ It's like delay differential equations but without the derivative. What field should I search in to solve this kind of equation?
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### Proving a Functional Equation for the JacobiTheta Function

Let $\Theta(t) = \sum_{k = - \infty}^{\infty} e^{-\pi k^{2} t}$. How can it be proved that $\Theta(\frac{1}{t}) = \sqrt{t}\Theta(t)$? I have read a proof here https://scholarship.claremont.edu/cgi/...
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### Find all $f:\mathbb{Q^+}\rightarrow \mathbb{Q^+}$ so that $\forall x,y\in\mathbb{Q^+}$ : $f(f(x)^2y)=x^3f(xy)$ [duplicate]

Find all $f:\mathbb{Q^+}\rightarrow \mathbb{Q^+}$ so that $\forall x,y\in\mathbb{Q^+}$ : $f(f(x)^2y)=x^3f(xy)$ so $f(0)=0$ when $x=y=0$ $f(f(1)^2)=f(1)$ when $x=y=1$ $f(f(x)^2)=x^3f(x)$ when $y=1$ and ... 61 views

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### Existence of Positive Integer Solution for a Polynomial Equation [closed]

Let $f(x)$ be a polynomial of degree n, and it is known that $f(x)$ has no positive integers as its roots. In other words, there are no positive integer values of x for which $f(x)=0$. Consider the ...
1 vote
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### Linear functional equations

Linear functional equations like $(x + 1)P (x) = (x − 10)P (x + 1)$ are fairly common in competition math, and there are some general techniques for proving things about the solutions, such as degree ...
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### IMC 2023 problem 1

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that have continuous second derivative and for which the equality $f(7x+1)=49f(x)$ holds for all $x \in \mathbb{R}$. I found a solution online to ...
1 vote
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### A periodic functional equation

I am not experienced with functional equations, and I have been wondering about equations of the form $$f(x) = f(mx+c)$$ For positve $m,c$, where we further restrict the solutions to be continuous. ...
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### Solve continuous function: $f(bx)-f(by)=f(x)-f(y)$

Solve for continuous function $f$ where $f(bx)-f(by)=f(x)-f(y)$ for all positive number $b$. $f$ is defined on positive numbers. Example: $f(x)=\log x$ Check: $f(bx)-f(by)=\log x+\log b-\log b-\log y$ ...
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### About a counterexample for an integral-functional equation in number theory.

I was reading http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf How did the counterexample for the equation on page 8 look like ?? Specificly : (quote) “Tur´an’s lecture (probably a quite ...
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### Sequences $(a_n)$ such that $i^2+j^2=k^2+\ell^2\Longrightarrow a_i+a_j=a_k+a_{\ell}$

In trying to solve the problem posed in this old message, I asked myself the following question : Which sequences $(a_n)_{n\in\mathbb{N}}$ (with $\forall n\in\mathbb{N}\,,\,a_n\in\mathbb{Z}$) satisfy ...
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### Existence of solution of functional equation on $S^1$.

I want to know if there exists a non-trivial ($h \neq 0$) continuous function $h:S^1 \rightarrow \mathbb{C}$ such that for all $z \in S^1$, $$h(z) = h(\sqrt z) + h(- \sqrt z).$$ My suspicion is that ...
1 vote
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### How to solve this functional equation: $f(x^2) = f(x) f^{-1}(x)$?

We know that the equation admits the particular solutions: $f(x) = x$, $f(x) = x - 1$ and $f(x) = \frac{1}{x}$ But how to get these results? The method of guessing a function and substituting it into ...
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### Function on the real plane that can be expressed in terms of a function on the real line

Let $f:\mathbb{R}^2\to\mathbb{R}$ such that $f(x,y)+f(y,z)+f(z,x)=0 \text{ } \forall \text{ } x,y,z \in \mathbb{R}.$ Show that there exists a $g:\mathbb{R}\to \mathbb{R}$ such that $f(x,y)=g(x)-g(y).$ ...
### Find all functions over $\mathbb{Z}^+$ satisfying this divisibility criterion
The problem is to find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $[f(m)]^2 + f(n)$ divides $(m^2+n)^2$ for all $m$,$n\in\mathbb{Z}^+$. I already have my claim and attempted ...
Let $X = 2\Bbb{Z} + 1$ or $2 \Bbb{N} + 1$ where $0 \in \Bbb{N}$, this approach will probably play well with both forms. See Extending the Collatz function to larger domains. Define the shortcut ...