Recent content by Samwise1

Yes, as a matter of fact, I do need the proof of that fact. My "proof" doesn't seem to prove much. Could you tell me why the equality $(*)$ is true?

Thank you. I've come up with a proof of $(*)$. Here it is: Let $h_i : \Delta_3 \rightarrow \mathbb{R}$ be functions assigning $P \in \Delta_3$ its $i$-th barycentric coordinate. These maps are affine. Then we consider a plane, an affine subspace of $T$ containing the triangle and extend the...

I see. I didn't express myself very well. It was supposed to be a reformulation of the statement which I want to prove, that is that the barycentric coordinates of a point $p$ inside a triangle are distances of that point from the sides of the triangle = heights of respective smaller triangles...

I want to prove that the barycentric coordinates of a point $P$ inside the triangle with vertices in $(1,0,0), (0,1,0), (0,0,1)$ are distances from $P$ to the sides of the triangle. Let's denote the triangle by $ABC, \ A = (1,0,0), B=(0,1,0), C= (0,0,1)$. We consider triangles $ABP, \ BCP, \...

We are given $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in \mathcal{O} (ann (z_0, r, R)), \ \ 0<r<R< \infty $$. Prove that $$\frac{1}{\pi} \int _{ann (z_0, r, R)} |f(z)|^2 d \lambda(z) = \sum _{n \neq -1} \frac{R^{2n+2} - r^{2n+2}}{n+1}|a_n|^2 + 2 \log \frac{R}{r}|a_{-1}|^2$$. We know...