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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50 views

Largest value in the functional $\int_0^\infty e^{-rt}( x^2+2x+\dot x^2)dt$?

I am trying to understand the second order linear differential equation and the answer here (Finnish) that I have translated below. Translation Problem What is the value of $x(t)$ where the ...
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2answers
566 views

Which trigonometric identities involve trigonometric functions?

Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled functional equation gave the identity $$ \sin^2\theta+\cos^2\theta = 1 ...
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2answers
353 views

Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?

Do there exist functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y),$ but which aren't linear? I bet you they exist, but I can't think of any examples. Furthermore, what ...
5
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1answer
907 views

Examples of functions where $f(ab)=f(a)+f(b)$

What are some examples of continuous (on a certain interval) real or complex functions where $f(ab)=f(a)+f(b)$ (like $\ln x$?)
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180 views

$f(x+f(y))=f(x)+y^n$

Here is the problem: Fix $n\in\mathbb{N}$. Find all monotonic solutions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+y^n$. I've tried to show that $f(0)=0$ and derive some ...
4
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0answers
184 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...
3
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2answers
63 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
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4answers
496 views

Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly $4$ distinct real roots

Problem : Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly four distinct real roots, then which of the following options are correct : (a) $f(x) =0$ has all three real roots ...
3
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2answers
316 views

Implicit function $y = e^{(y-1)/x}$

I'd like to know if the function $ y = f(x) : [0,1] \rightarrow [0,1]$ defined implicitly by the transcendental equation $$\displaystyle y = e^{(y-1)/x}$$ is "well known" (name, properties) or is ...
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1answer
267 views

Finding a function that satisfies constraints numerically

I have the following system of equations for function $p(y)$ and I need help debugging my solution: $$\begin{align} 0&=\log(p(y))+1-\lambda-\gamma y^2-\eta ...
2
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1answer
356 views

Equation with a definite integral - can I differentiate it?

I have an equation like this: $$te^{t} = \int_0^t e^\tau u(\tau)d\tau$$ I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how can I do it ...
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3answers
239 views

Solving the functional equation $x[f(x+1)-f(x-1)]=1$ [duplicate]

Possible Duplicate: Solving the functional equation $f(x+1) - f(x-1) = g(x)$ How do I approach this problem $x[f(x+1)-f(x-1)]=1$.
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3answers
1k views

Function Satisfying $f(x)=f(2x+1)$

If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function and satisfies $f(x)=f(2x+1)$, then its not to hard to show that $f$ is a constant. My question is suppose $f$ is continuous and it satisfies ...
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1answer
58 views

Contour approach to Riemann zeta functional equation

I have a question regarding Riemann's first proof of the functional equation that was given in his paper on the Riemann zeta function. I am an undergraduate working on a fairly short undergraduate ...
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2answers
55 views

A scaling functional equation

I want to find a closed form of a function satisfying $$G(4z)=\frac{G(z)}{2z},$$ unfortunately I am not experienced with scaling problems, so I have no idea and would be thankful for any hints. ...
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1answer
64 views

Cauchy functional equation with non choice

Assume ZF+ not AC. Then how many solutions are there for Cauchy functional equation? Thank you
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1answer
79 views

Show that for this function the stated is true.

For the function $$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$ show that $$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$ Hey everyone, I'm very new to this kind of maths and would really ...
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0answers
108 views

where can I find a good Proof for Cauchy's functional equation

Do you know where can I find good proofs for Cauchy's functional equation?
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1answer
358 views

A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere

Let $g: \mathbf R \to \mathbf R$ be a function which is not identically zero and which satisfies the equation $$ g(x+y)=g(x)g(y) \quad\text{for all } x,y \in \mathbf{R}. $$ Show that $g(x)\gt0$ for ...
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3answers
119 views

Solving Series of equations

I have the following series of equations (n+2 equations n+2 variables): $k_0q_0+\lambda q_0 + c_0 = 0$ $k_1q_1+\lambda q_1 + c_1 = 0$ $k_nq_n+\lambda q_n + c_n = 0$ $q_1+q_2+....+q_n = 1$ ...
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1answer
176 views

A Dangerous Function [closed]

Recently, I have started solving many ques on functional equations. But, this ques for me was tough, $ f(y)^{x^2}+f(x)^{y^2}=f(y^2)^x+f(x^2)^y $ I've started substituting some values, trying to ...
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1answer
71 views

simplifying “$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}$”

Is this equality correct? For finite sets $A$ and $B_a$ (where $a\in A$), we have: $$\prod_{a\in A}\sum_{b\in B_a}{h(a,b)}=\sum_{f\in \prod_{a\in A}B_a}\quad \prod_{a\in A}{h(a,f(a))}$$
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0answers
61 views

properties about number of groups of linearly independent solutions in the general solutions of linear functional-differential equations

Are there any effective methods to determinate the number of groups of linearly independent solutions in the general solutions of linear functional-differential equations of the form ...
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1answer
110 views

$x^3[f(x+1)-f(x-1)]=1$

Given that $x^3[f(x+1)-f(x-1)]=1$, determine $\lim_{x\rightarrow \infty}(f(x)-f(x-1))$ explicitly.