Functions of One Complex Variable IThis book is intended as a textbook for a first course in the theory of functions of one complex variable for students who are mathematically mature enough to understand and execute E - 8 arguments. The actual pre requisites for reading this book are quite minimal; not much more than a stiff course in basic calculus and a few facts about partial derivatives. The topics from advanced calculus that are used (e.g., Leibniz's rule for differ entiating under the integral sign) are proved in detail. Complex Variables is a subject which has something for all mathematicians. In addition to having applications to other parts of analysis, it can rightly claim to be an ancestor of many areas of mathematics (e.g., homotopy theory, manifolds). This view of Complex Analysis as "An Introduction to Mathe matics" has influenced the writing and selection of subject matter for this book. The other guiding principle followed is that all definitions, theorems, etc. |
Contents
Chapter I | 1 |
Im 20 | 7 |
Chapter II | 11 |
Chapter III | 30 |
arctan | 56 |
Chapter IV | 58 |
frequently used in the later sections Let G | 74 |
G C is analytic | 100 |
equation will give that f and I are everywhere identical | 180 |
Chapter VIII | 195 |
G R is harmonic | 204 |
fz +1+2 | 207 |
f21exp | 208 |
Chapter IX | 210 |
T | 242 |
Plane 1 | 243 |
Chapter V | 103 |
y | 104 |
over it is not difficult to see that the | 106 |
exp 2+1 a + | 111 |
16 Determine the regions in which the functions f | 112 |
Also | 116 |
for any closed rectifiable curve y not passing through a₁ | 122 |
Chapter VI | 128 |
b a x u y | 138 |
Chapter VII | 142 |
水 | 159 |
for all z in K and n N But | 172 |
Chapter X | 252 |
r2 | 256 |
Moreover u is unique and is defined by the formula | 258 |
applying Dinis Theorem Exercise VII16 Another involves | 262 |
Chapter XI | 279 |
Chapter XII | 292 |
fS Bfa | 295 |
Appendix A | 303 |
Appendix B | 307 |
1 L V AHLFORS Complex Analysis | 311 |
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Common terms and phrases
a₁ Algebraic analytic continuation analytic function analytic on G assume bounded Cauchy sequence closed rectifiable curve compact subset complex numbers component constant contains continuous function convex Corollary covering space curve in G D₁ defined Definition differentiable disk entire function equation Exercise f and g f is analytic f is continuous f₁ fact finite number follows formula function element function f function on G G₁ gives harmonic function Hence homeomorphism implies infinite integer Lemma Let f Let G lim sup limit point logarithm meromorphic function metric space Möbius transformation open set open subset path plane point in G pole polynomial Proof properties Proposition prove R₁ reader real number rectifiable curve region G removable singularity Riemann Runge's Theorem satisfies show that f simply connected subharmonic Suppose f suppose that f t₁ Theory topological space x₁ y₁