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 1d answered Functions belonging to $L^p(\mathbb{R}^n ; \mathbb{R}^m)$ if and only of their norm belongs to $L^p(\mathbb{R}^n)$ 2d revised Inequality involving trace and operator norm added 115 characters in body 2d 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... Apr13 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! Apr11 comment Residue theorem why do you say that $g(z)=e^{itz}e^{-Im(z)}/(z+i)$ is holomorphic? Apr8 answered Chain rule in several complex variables: Wirtinger derivatives Apr8 answered Does in this case exist necessarely an eigenvalue equal to $0$? Apr7 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) Apr7 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 $s_+\geq1$. Apr6 revised Is an hypersurface uniquely determined by an equation? edited tags Apr6 comment Is an hypersurface uniquely determined by an equation? Well, $r^3=0$ defines the same hypersurface locally around $z_0$. As it does $r(z)(\|z\|^2+1)=0$... Apr2 answered integrating by parts on a manifold Mar21 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 $a_n=s_n-s_{n-1}\to (S-S)=0$. Mar21 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 Mar20 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? Mar19 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. Mar16 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... Mar15 answered Let $f$ be a meromorphic function on $\mathbb C$ such that $|f(z)|\ge |z|$ Mar15 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. Mar15 comment Unique continuous complex log of a function nowhere zero I added an equivalent approach, I hope more likely to suit your needs.