Complex Analysis in one Variable by NARASIMHAN.
This book is based on a firstyear graduate course I gave three times at the University of Chicago. As it was addressed to graduate students who intended to specialize in mathematics, I tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of v...
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Format:  eBook 
Language:  English 
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Boston, MA :
Birkhäuser Boston : Imprint: Birkhäuser,
1985.

Edition:  1st ed. 1985. 
Series:  Springer eBook Collection.

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505  0  a 1 Elementary Theory of Holomorphic Functions  1 Some basic properties of ?differentiable and holomorphic functions  2 Integration along curves  3 Fundamental properties of holomorphic functions  4 The theorems of Weierstrass and Montel  5 Meromorphic functions  6 The LoomanMenchoff theorem  Notes on Chapter 1  References : Chapter 1  2 Covering Spaces and the Monodromy Theorem  1 Covering spaces and the lifting of curves  2 The sheaf of germs of holomorphic functions  3 Covering spaces and integration along curves  4 The monodromy theorem and the homotopy form of Cauchy’s theorem  5 Applications of the monodromy theorem  Notes on Chapter 2  References : Chapter 2  3 The Winding Number and the Residue Theorem  1 The winding number  2 The residue theorem  3 Applications of the residue theorem  Notes on Chapter 3  References : Chapter 3  4 Picard’s Theorem  Notes on Chapter 4  References : Chapter 4  5 The Inhomogeneous CauchyRiemann Equation and Runge’s Theorem  1 Partitions of unity  2 The equation % MathType!MTEF!2!1!+ % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqJc9 % vqaqpepm0xbba9pwe9Q8fs0yqaqpepae9pg0FirpepeKkFr0xfrx % frxb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq % GHciITcaWG1baabaGaeyOaIyRabmOEayaaraaaaiabg2da9iabew9a % Mbaa!3DAD!$$ \[\frac{{\partial u}} {{\partial \bar z}} = \phi \]$$  3 Runge’s theorem  4 The homology form of Cauchy’s theorem  Notes on Chapter 5  References : Chapter 5  6 Applications of Runge’s Theorem  1 The MittagLeffler theorem  2 The cohomology form of Cauchy’s theorem  3 The theorem of Weierstrass  4 Ideals in ? (?)  Notes on Chapter 6  References : Chapter 6  7 The Riemann Mapping Theorem and Simple Connectedness in the Plane  1 Analytic automorphisms of the disc and of the annulus  2 The Riemann mapping theorem  3 Simply connected plane domains  Notes on Chapter 7  References : Chapter 7  8 Functions of Several Complex Variables  Notes on Chapter 8  References : Chapter 8  9 Compact Riemann Surfaces  1 Definitions and basic theorems  2 Meromorphic functions  3 The cohomology group H1(.  
520  a This book is based on a firstyear graduate course I gave three times at the University of Chicago. As it was addressed to graduate students who intended to specialize in mathematics, I tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of view which relate the subject to other branches of mathematics. Complex analysis in one variable is ideally suited to this attempt. Of course, the branches of mathema tics one chooses, and the connections one makes, must depend on personal taste and knowledge. My own leaning towards several complex variables will be apparent, especially in the notes at the end of the different chapters. The first three chapters deal largely with classical material which is avai lable in the many books on the subject. I have tried to present this material as efficiently as I could, and, even here, to show the relationship with other branches of mathematics. Chapter 4 contains a proof of Picard's theorem; the method of proof I have chosen has farreaching generalizations in several complex variables and in differential geometry. The next two chapters deal with the Runge approximation theorem and its many applications. The presentation here has been strongly influenced by work on several complex variables.  
590  a Loaded electronically.  
590  a Electronic access restricted to members of the Holy Cross Community.  
650  0  a Functions of complex variables.  
650  0  a Mathematical analysis.  
650  0  a Analysis (Mathematics).  
650  0  a Applied mathematics.  
650  0  a Engineering mathematics.  
650  0  a Topology.  
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