Reputation
2,407
Top tag
Next privilege 2,500 Rep.
Create tag synonyms
Badges
10 30
Newest
 Yearling
Impact
~32k people reached

Apr
25
comment Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.
@DerpMagoo You should get $f(\frac{1}{n+1})=0$ for all $n \in \mathbb{N}$. Also, note that $\frac{1}{n+1} \to 0 \in D(0,2)$ when $n \to \infty$ (so, you have sequence tending to element of $D(0,2)$ and on that sequence your function is zero, so...)
Apr
25
comment Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.
@DerpMagoo Yes. And if you take $a=\frac{1}{n+1}$ and look at your condition, you will get... Try it.
Apr
25
comment Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.
@DerpMagoo Do you know what is it Cauchy's integral formula?
Apr
25
answered Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.
Jan
9
awarded  Yearling
Nov
17
comment What does the theorem “If $\lim_{x\to a} f(x) = L$ and $f(x) = g(x) $ except at $a$ then $\lim_{x\to a} g(x) = L$” mean?
It means that limit is not depending on value of function in that point where we are taking limit. That is, for $\lim_{x\to a} f(x)$ it is not important what it is $f(a)$
Oct
11
awarded  Nice Answer
Sep
5
comment Proving that an operator $T$ on a Hilbert space is compact
@johny No. That is just arbitrary sequence of elements of our Hilbert space which is weakly convergent. I used that bounded operator is compact iff operator maps weakly convergent sequences into strongly convergent sequences.
Sep
5
answered Proving that an operator $T$ on a Hilbert space is compact
Sep
2
comment Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?
It's funny how you underline "Hint", but (at least on my monitor) it look like you overline "span".
Jul
29
accepted Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$
Jul
28
comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$
First, thanks. I don't see immediately why $f(M_g) = M_{f \circ g}$? Also, I don't know this version of spectral theorem. Where can I find it? (I know variant: for self-adjoint $A$ there exist spectral measure $E$ such that $A=\int_{\sigma(A)} \lambda \, dE(\lambda)$)
Jul
26
comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$
@PhoemueX Can you show me solution? Because I really don't know how to attack problem. Only I can see that from $\lambda_1(A) \geq \lambda_2(A) \geq \ldots \geq 0$ (if we arrange in that way) we get $f(\lambda_1(A)) \geq f(\lambda_2(A)) \geq \ldots \geq 0$ and I think that is very important.
Jul
25
comment Evaluate the Integral: $\int(x^5+5^x)\ dx$
You can split in sum of two integrals.
Jul
25
comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$
@PhoemueX I am mainly interested in eigenvalues (that is singular values (but that is the same here because $A$ is positive)). I would like (if that is possible) to see also why $\sigma(f(A)) \neq f(\sigma(A))$.
Jul
25
revised Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$
edited title
Jul
25
comment Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$
@DanielFischer Yes, you are right, I didn't thinking enough. And yes, we assume that $A$ is continuous linear operator.
Jul
25
asked Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$
Jul
22
comment Integration of $\int \frac{(1 + x)\sin x}{(x^2 +2 x)\cos^2 x-(1 + x)\sin2x}dx$
Hm, just for note: derivative of $x^2+2x$ is $2(1+x)$ and derivative for $\cos^2 x$ is $-\sin 2x$.
Jul
22
comment Serge Lang Never Explains Anything Round II
@D_S I don't know this things. But if you changed $K'$ with $M$, then, do you need to change it everywhere? (you have in text $K'/k'$....)