query: A functional equation over a circle: $f(r \cos \phi)+f(r\sin \phi)=f(r)$
positive_passages.docid: 146719
positive_passages.text: Functional equation $f(r \cos \varphi)+f(r \sin \varphi)=f(r)$
negative_passages.docid: 111556
negative_passages.text: Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?
negative_passages.docid: 48925
negative_passages.text: BMO1 2009/10 Q5 functional equation: $f(x)f(y) = f(x + y) + xy$
negative_passages.docid: 49794
negative_passages.text: Solve $f(x^2+y+f(y))=2y+f(x)^2$ over $\mathbb{R}$
negative_passages.docid: 82597
negative_passages.text: How to find $ f(x)$ if $f(1-f(x))=x$ for all $x$ $\in \mathbb{R}$
negative_passages.docid: 134043
negative_passages.text: Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $
negative_passages.docid: 68409
negative_passages.text: Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
negative_passages.docid: 144042
negative_passages.text: On the functional square root of $x^2+1$
negative_passages.docid: 26687
negative_passages.text: Show that $f$ maps the unit circle to itself
negative_passages.docid: 82448
negative_passages.text: Why is $f(x) = x\phi(x)$ one-to-one?
negative_passages.docid: 69203
negative_passages.text: Calculus proof for the area of a circle
negative_passages.docid: 157744
negative_passages.text: Cauchy's functional equation with composite function: $gf(x+y)=gf(x)+gf(y)$
negative_passages.docid: 11806
negative_passages.text: Functional equation.
negative_passages.docid: 103019
negative_passages.text: Find all functoins $f: \mathbb R \rightarrow \mathbb R$ such that $\forall x,y \in \mathbb R f(xy+f(x))=xf(y)+f(x)$
negative_passages.docid: 143200
negative_passages.text: $f(r,\theta)=(r\cos\theta,r\sin\theta)$ Then
negative_passages.docid: 25953
negative_passages.text: Solving a functional relation $f\left( x \cdot f(y)\right)=x^2 \cdot y^a$
negative_passages.docid: 41226
negative_passages.text: Find $f:f(xf(x)+f(y))=f(x)^2+y$