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  • 23 votes cast
May
16
comment $f$ convex: exists linear map $g$ s.t. $f\geq g$?
what are $x$ an $y$? Or better how are they quantified? Exists $x$? For every $y$?
Apr
16
answered Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$
Apr
15
revised Inequality involving trace and operator norm
added 115 characters in body
Apr
15
comment Inequality involving trace and operator norm
I don't understand what happened, but I never wrote the first 4 lines (which are false: eigenvalues of W are all real and positive, as the characteristic polynomial is $(t-1)^2$ no matter what $a$ is...
Apr
13
comment Residue theorem
Yes, that is what i was trying to point out in my comment! Sorry for my elapsed time of response, but this allowed you to figure it out by yourself!
Apr
11
comment Residue theorem
why do you say that $g(z)=e^{itz}e^{-Im(z)}/(z+i)$ is holomorphic?
Apr
8
answered Chain rule in several complex variables: Wirtinger derivatives
Apr
8
answered Does in this case exist necessarely an eigenvalue equal to $0$?
Apr
7
comment Does in this case exist necessarely an eigenvalue equal to $0$?
The Levi form is defined as the complex Hessian of a defining function restricted to the complex tangent hyperplane, i.e. to a space of dim $n-1$. So, if you start with a (Levi)convex hypersurface and you write it locally as $y_1=Q+hot$ around the origin, the Levi form in the origin will be restricted to the hyperplane $z_1=0$ (the vanishing direction you were complaining about)
Apr
7
comment Complex Hessian Signature
with $n=2$ and usual notation, $r_1(z)=|z_1|^2+|z_2|^2$ is a counterexample to $s_++s_-\leq n-1$ and $r_2=-r_1$ is a counterexample to $s_+\geq1$.
Apr
6
revised Is an hypersurface uniquely determined by an equation?
edited tags
Apr
6
comment Is an hypersurface uniquely determined by an equation?
Well, $r^3=0$ defines the same hypersurface locally around $z_0$. As it does $r(z)(\|z\|^2+1)=0$...
Apr
2
answered integrating by parts on a manifold
Mar
21
comment Is $\sum_{n=1}^\infty \sin\left(\frac{n\pi}{4}\right)$ convergent?
It is the first result that is proved to you in any calculus course just after defining what is a series of real numbers. If $s_n$ is the sequence of partial sums ($s_n=a_0+..+a_n$), the sum converges to $S$ if and only if $s_n\to S$, hence $a_n=s_n-s_{n-1}\to (S-S)=0$.
Mar
21
revised How to evaluate $\frac{1}{2\pi i}\int_C \frac{f(z) dz}{(z-z_1)^{m_1}(z-z_2)^{m_2}}$ and the other related contour?
added 4 characters in body
Mar
20
answered How to evaluate $\frac{1}{2\pi i}\int_C \frac{f(z) dz}{(z-z_1)^{m_1}(z-z_2)^{m_2}}$ and the other related contour?
Mar
19
comment Unique continuous complex log of a function nowhere zero
Probably in this book springer.com/birkhauser/mathematics/book/978-0-8176-4164-1 there is enough stuff for your needs.
Mar
16
comment Unique continuous complex log of a function nowhere zero
You don't select any branch. You just go on along the path defining the logarithm by analytic continuation...
Mar
15
comment Relation between being holomorphic in $\Delta\times\Delta$ and in every relatively compact polydisk.
If I guess correctly, this is a step in the proof of Hartogs theorem about separate analyticity. There you take a domain $U$ and prove that $f$ is holomorphic on every relatively compact polydisc. If this is the situation, you have that the hypotheses hold in a neighbourhood of the polydisc, so that you can enlarge the strip a bit.
Mar
15
comment Unique continuous complex log of a function nowhere zero
I added an equivalent approach, I hope more likely to suit your needs.