Complex Variables

By Steven G. Krantz. 2007

306 citations recorded by the source corpus.

PREFACE BASIC IDEAS Complex Arithmetic Algebraic and Geometric Properties The Exponential and Applications HOLOMORPHIC AND HARMONIC FUNCTIONS Holomorphic Functions Holomorphic and Harmonic Functions Real and Complex Line Integrals Complex Differentiability The Logarithm THE CAUCHY THEORY The Cauchy Integral Theorem Variants of the Cauchy Formula The Limitations of the Cauchy Formula APPLICATIONS OF THE CAUCHY THEORY The Derivatives of a Holomorphic Function The Zeros of a Holomorphic Function ISOLATED SINGULARITIES Behavior near an Isolated Singularity Expansion around Singular Points Examples of Laurent Expansions The Calculus of Residues Applications to the Calculation of Integrals Meromorphic Functions THE ARGUMENT PRINCIPLE Counting Zeros and Poles Local Geometry of Functions Further Results on Zeros The Maximum Principle The Schwarz Lemma THE GEOMETRIC THEORY The Idea of a Conformal Mapping Mappings of the Disc Linear Fractional Transformations The Riemann Mapping Theorem Conformal Mappings of Annuli A Compendium of Useful Conformal Mappings APPLICATIONS OF CONFORMAL MAPPING Conformal Mapping The Dirichlet Problem Physical Examples Numerical Techniques HARMONIC FUNCTIONS Basic Properties of Harmonic Functions The Mean Value Property The Poisson Integral Formula TRANSFORM THEORY Introductory Remarks Fourier Series The Fourier Transform The Laplace Transform A Table of Laplace Transforms The z-Transform PDES AND BOUNDARY VALUE PROBLEMS Fourier Methods COMPUTER PACKAGES Introductory Remarks The Software Packages APPENDICES Solutions to Odd-Numbered Exercises Glossary of Terms List of Notation A Guide to the Literature BIBLIOGRAPHY INDEX

Subfields: Applied Mathematics, Mathematical Physics, Modeling and Simulation

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Citations in the corpus

Notes on conformal invisibility devices (2006, unknown): “Therefore, the distortions of images composed of various spatial Fourier components are uniform, which reduces the risk of detection.”
Transformation Optics and the Geometry of Light (2008, background): “…this formalism provides the most transparent way of describing non-Cartesian coordinate systems: even something as apparently familiar as electromagnetism in spherical polar coordinates is much simpler in the language of differential geometry than in the standard treatment found in the textbooks.1”
Structured H∞‐control of infinite‐dimensional systems (2017, methodology): “As these examples (1-28) are finite dimensional, systune based on our previous work1 and on Apkarian et al62 was used to compare with a standard structured H∞-synthesis.”
Some results related to a conjecture of R. Brück concerning meromorphic functions sharing one small function with their derivatives (2007, unknown): “L.A. Rubel and C.C. Yang [9], E. Mues and N. Steinmetz [8], G. Gundersen [3] and L.- Z. Yang [10], J.-H. Zheng and S.P. Wang [18], and many other authors have obtained elegant results on the uniqueness problems of entire functions that share values CM or IM with their first or k-th derivatives.”
Möbius transformations of matrix polynomials (2015, background): “Such functions are variously known as fractional linear rational functions [1, 17, 58], bilinear transformations [7, 54, 55, 56] or more commonly, Möbius functions.”