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Shukla, and Fatemeh YarahmadiMelissa HallingCover PageyesTOC OnlyCompile but don't publishLicensePublic DomainCC BYCC BY-SACC BY-NC-SACC BY-NDCC BY-NC-NDGNU GPLAll Rights ReservedCC BY-NCGNU FDLCK-12Show Page TOCyesnoTranscludedyesPrintOptionsNo Header/FooterNo TitleNo Header/Footer/TitleOER program or PublisherCollege of the Canyons - Zero Textbook Cost ProgramASCCC OERI ProgramCK-12OpenStaxGALILEOOpenSUNYMIT OpenCourseWareVirginia Tech Libraries' Open Education InitiativeThe Publisher Who Must Not Be NamedeCampusOntarioWAC ClearinghouseBC CampusLumenOpen OregonOpenStax CNXPDXOpenEvergreen Valley CollegeAutonumber Section Headingstitle with space delimiterstitle with colon delimiterstitle with dash delimitersLicense Version1.02.02.53.04.01.31.2Print CSSDefaultDenseVirginia TechScreen CSSDefaultVirginia TechCosmologyNumber of Print ColumnsTwoThreeInclude attachmentsContent TypeDocumentImageOther Searching inAll resultsAbout 2 results2.3: Holomorphic Vectors, the Levi Form, and Pseudoconvexityhttps://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/02%3A_Convexity_and_Pseudoconvexity/2.03%3A_Holomorphic_Vectors_the_Levi_Form_and_Pseudoconvexity\[ \begin{align}\begin{aligned} \frac{\partial^2 (r \circ f)}{\partial z_j \partial z_k} \bigg|_{(z,\bar{z})} & = \frac{\partial}{\partial z_j } \sum_{\ell=1}^n \biggl( \frac{\partial r}{\partial \zet...\[ \begin{align}\begin{aligned} \frac{\partial^2 (r \circ f)}{\partial z_j \partial z_k} \bigg|_{(z,\bar{z})} & = \frac{\partial}{\partial z_j } \sum_{\ell=1}^n \biggl( \frac{\partial r}{\partial \zeta_\ell} \bigg|_{(f(z),\bar{f}(\bar{z}))} \frac{\partial f_\ell}{\partial z_k} \bigg|_{z} + \frac{\partial r}{\partial \bar{\zeta}_\ell} \bigg|_{(f(z),\bar{f}(\bar{z}))} \cancelto{0}{\frac{\partial \bar{f}_\ell}{\partial z_k} \bigg|_{\bar{z}}} \biggr) \\ & = \sum_{\ell,m=1}^n \biggl( \frac{\partial^…More2.5: Hartogs Pseudoconvexityhttps://math.libretexts.org/Bookshelves/Analysis/Tasty_Bits_of_Several_Complex_Variables_(Lebl)/02%3A_Convexity_and_Pseudoconvexity/2.05%3A_Hartogs_PseudoconvexityAgain using Cauchy–Schwarz \[\begin{align}\begin{aligned} \sum_{j=1,\ell=1}^n \bar{c}_j c_\ell \frac{\partial^2 r}{\partial \bar{z}_j \partial z_\ell} \Big|_q & = \sum_{j=1,\ell=1}^n ( \bar{a}_j + \ba...Again using Cauchy–Schwarz \[\begin{align}\begin{aligned} \sum_{j=1,\ell=1}^n \bar{c}_j c_\ell \frac{\partial^2 r}{\partial \bar{z}_j \partial z_\ell} \Big|_q & = \sum_{j=1,\ell=1}^n ( \bar{a}_j + \bar{b}_j ) (a_\ell + b_\ell) \frac{\partial^2 r}{\partial \bar{z}_j \partial z_\ell} \Big|_q \\ & = \sum_{j=1,\ell=1}^n \bar{a}_j a_\ell \frac{\partial^2 r}{\partial \bar{z}_j \partial z_\ell} \Big|_q \\ & \phantom{=} \quad + \sum_{j=1,\ell=1}^n \bar{b}_j c_\ell \frac{\partial^2 r}{\partial \bar{z}_j…MoreShow more results
