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appendixf

# appendixf - F Complex Analysis In this appendix we collect...

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F Complex Analysis In this appendix, we collect a few basic definitions and facts from complex analysis. Complex analysis is a vast and beautiful subject. Although we make only limited use of complex analysis in this volume, there is a rich interaction between harmonic analysis and complex analysis, some of which can be seen in the texts by Dym and McKean [DM72] and Young [You01]. Some basic texts on complex analysis include Conway [Con78], Marsden and Hoffman [MH87], and Stein and Shakarchi [SS03b]. F.1 Analytic Functions Complex analysis is concerned with functions that map the complex plane to itself. An analytic function is one that has a complex derivative. This is a very strong requirement — no matter what “path” to the origin that we take, the limit in the definition of the derivative must exist. Definition F.1 (Analytic Function). Let Ω C be open. Then a function f : Ω C is analytic or holomorphic on Ω if the limit f ( z ) = lim h 0 f ( z + h ) f ( z ) h (F.1) exists for all z Ω . The function f is called the derivative or complex deriva- tive of f. If f : C C is analytic on C , then we say that f is entire . Note that the variable h in equation (F.1) is a complex variable. The meaning of the limit is that for every ε > 0 , there exists a δ > 0 such that 0 < | h | < δ, z + h Ω = vextendsingle vextendsingle vextendsingle f ( z ) f ( z + h ) f ( z ) h vextendsingle vextendsingle vextendsingle < ε. For example, any polynomial p ( z ) = a 0 + a 1 z + · · · + a n z n is entire, as is the exponential function e z . The function f ( z ) = 1 /z is analytic on C \{ 0 } . If r > 0 , then we set r z = e z ln r .

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416 F Complex Analysis There are many equivalent formulations of analyticity, which we will not present. For us, the important fact is that analytic functions are very highly constrained. Some of the properties of analytic functions are laid out in the next theorem. In particular, an analytic function is uniquely determined by its values on any infinite set that has an accumulation point. Theorem F.2 (Properties of Analytic Functions). If f is analytic on an open set Ω C , then the following statements hold. (a) f is analytic on Ω . (b) f has infinitely many complex derivatives on Ω . (c) Let S be any infinite subset of Ω that has an accumulation point in Ω . If g is also analytic on Ω and f ( z ) = g ( z ) for all z S, then f ( z ) = g ( z ) for all z Ω . We also have the following two important properties of analytic functions. Theorem F.3 (Liouville’s Theorem). If f is both bounded and analytic on all of C , then f is constant. Theorem F.4 (Maximum Modulus Principle). Let Ω be a bounded, open, and connected subset of C . Suppose that f is analytic on Ω and continu- ous on Ω , and let M = sup z Ω | f ( z ) | be the maximum of | f | on the boundary of Ω . Then | f ( z ) | ≤ M for all z Ω . Moreover, if | f ( z ) | = M for some z Ω then f is constant.
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