Search
- Filter Results
- Location
- Classification
- Include attachments
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/00%3A_Front_Matter/01%3A_TitlePageOklahoma State University Tasty Bits of Several Complex Variables (Lebl) Jiří Lebl
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/07%3A_Appendix
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/06%3A_Complex_Analytic_Varieties/6.08%3A_Segre_Varieties_and_CR_GeometryFor every p∈M there exists a neighborhood U of p with the following property: If q∈M∩U and (X,q) is a germ of a complex subvariety such that (X,q)⊂(M,q), t...For every p∈M there exists a neighborhood U of p with the following property: If q∈M∩U and (X,q) is a germ of a complex subvariety such that (X,q)⊂(M,q), then there exists a complex subvariety Y⊂U (in particular a closed subset of U) such that Y⊂M and (X,q)⊂(Y,q).
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/06%3A_Complex_Analytic_Varieties/6.05%3A_VarietiesWe define the dimension of X at p to be \[\dim_p X \overset{\text{def}}{=} \max \bigl\{ k \in \mathbb{N}_0 : \text{ $\forall$ neighbhds. $W$ of $p$, $\exists \, q \in W \cap X_{\mathit{reg}}$ ...We define the dimension of X at p to be dimpXdef=max If (X,p) is a germ and X a representative, the dimension of (X,p) is the dimension of X at p.
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/06%3A_Complex_Analytic_Varieties/6.02%3A_Weierstrass_Preparation_and_Division_TheoremsThe Weierstrass division theorem is a generalization of the division algorithm for polynomials with coefficients in a field, such as the complex numbers: If f(\zeta) is a polynomial, and \(P(\zeta...The Weierstrass division theorem is a generalization of the division algorithm for polynomials with coefficients in a field, such as the complex numbers: If f(\zeta) is a polynomial, and P(\zeta) is a nonzero polynomial of degree k, then there exist polynomials q(\zeta) and r(\zeta) with degree of r less than k such that f = qP + r.
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/01%3A_Holomorphic_Functions_in_Several_Variables/1.01%3A_Onto_Several_VariablesAs in one variable, we define the Wirtinger operators \[\frac{\partial}{\partial z_k} \overset{\text{def}}{=} \frac{1}{2} \left( \frac{\partial}{\partial x_k} - i \frac{\partial}{\partial y_k} \right)...As in one variable, we define the Wirtinger operators \frac{\partial}{\partial z_k} \overset{\text{def}}{=} \frac{1}{2} \left( \frac{\partial}{\partial x_k} - i \frac{\partial}{\partial y_k} \right) , \qquad \frac{\partial}{\partial \bar{z}_k} \overset{\text{def}}{=} \frac{1}{2} \left( \frac{\partial}{\partial x_k} + i \frac{\partial}{\partial y_k} \right) . An alternative definition is to say that a continuously differentiable function f \colon U \to \mathbb{C} is holomorphic if it sat…
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/01%3A_Holomorphic_Functions_in_Several_Variables/1.05%3A_Cartan's_Uniqueness_TheoremCartan's uniqueness theorem is another analogue of Schwarz’s lemma to several variables. It says that for a bounded domain, it is enough to know that a self mapping is the identity at a single point t...Cartan's uniqueness theorem is another analogue of Schwarz’s lemma to several variables. It says that for a bounded domain, it is enough to know that a self mapping is the identity at a single point to show that it is the identity everywhere. As there are quite a few theorems named for Cartan, this one is often referred to as the Cartan's uniqueness theorem. It is useful in computing the automorphism groups of certain domains.
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/01%3A_Holomorphic_Functions_in_Several_Variables/1.03%3A_DerivativesUsing the standard chain rule, \[\begin{align}\begin{aligned} \frac{\partial}{\partial z_j} \left[ g \circ f \right] & = \frac{1}{2} \left( \frac{\partial}{\partial x_j} - i \frac{\partial}{\partial y...Using the standard chain rule, \[\begin{align}\begin{aligned} \frac{\partial}{\partial z_j} \left[ g \circ f \right] & = \frac{1}{2} \left( \frac{\partial}{\partial x_j} - i \frac{\partial}{\partial y_j} \right) \left[ g \circ f \right] \\ & = \frac{1}{2} \sum_{\ell=1}^m \left( \frac{\partial g}{\partial s_\ell} \frac{\partial u_\ell}{\partial x_j} + \frac{\partial g}{\partial t_\ell} \frac{\partial v_\ell}{\partial x_j} - i \left( \frac{\partial g}{\partial s_\ell} \frac{\partial u_\ell}{\part…
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/05%3A_Integral_Kernels
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/04%3A_The_-Problem
- https://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/03%3A_CR_Functions
