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complex-part1 - Complex Variables Notes for Math 703 Part I...

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Complex Variables Notes for Math 703. Part I Updated Fall 2011 Anton R. Schep
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CHAPTER 1 Holomorphic (or Analytic) Functions 1. Definitions and elementary properties In complex analysis we study functions f : S C , where S C . When referring to open sets in C and continuity of functions f we will always consider C (and its subsets) as a metric space with respect to the metric d ( z 1 , z 2 ) = | z 1 - z 2 | , where | · | denotes the complex modulus, i.e., | z | = p x 2 + y 2 whenever z = x + iy with x, y R . An open ball with respect this metric will be also referred to as an open disc and denoted by D ( a ; r ) = { z C : | z - a | < r } , where a is the center and r > 0 is the radius of the open ball. The closed disc with center a and radius r is denoted by D ( a ; r ), so D ( a ; r ) = { z C : | z - a | ≤ r } . Recall that G C is called open if for all a G there exists r > 0 such that D ( a ; r ) G . If z = x + iy , then the conjugate z of z is defined by z = x - iy . Now z z = | z | 2 , so that 1 z = z | z | 2 for z 6 = 0. Elementary properties of complex numbers are given by: (1) The real part Re z of z satisfies Re z = 1 2 ( z + z ), while the imaginary part Im z of z is given by Im z = 1 2 i ( z - z ). (2) For all z 1 , z 2 C we have z 1 + z 2 = z 1 + z 2 and z 1 z 2 = z 1 z 2 . (3) For all z 1 , z 2 C we have | z 1 z 2 | = | z 1 | | z 2 | . 2. Elementary transcendental functions Recall also that if z = x + iy 6 = 0, then, using polar coordinates, we can write z = r cos θ + ir sin θ . In this case we write arg z = { θ + 2 : k Z } . By Arg z we will denote the principal value of the argument of z 6 = 0, i.e. θ = Arg z arg z if - π < θ π . Note that if z 1 = | z 1 | (cos θ 1 + i sin θ 1 ) and z 2 = | z 2 | (cos θ 2 + i sin θ 2 ), then we have z 1 z 2 = | z 1 || z 2 | (cos θ 1 cos θ 2 - sin θ 1 sin θ 2 + i (sin θ 1 cos θ 2 + cos θ 1 sin θ 2 )) = | z 1 z 2 | (cos( θ 1 + θ 2 ) + i (sin( θ 1 + θ 2 )). Hence we have arg ( z 1 z 2 ) = arg z 1 + arg z 2 . Define now e z = e x (cos y + i sin y ). Then | e z | = e x and arg e z = y + 2 . In particular e 2 πi = 1 and the function e z is 2 πi -periodic, i.e., e z +2 πi = e z e 2 πi = e z for all z C . We want now to define log w such that w = e z where z = log w , but we can not define it as just the inverse of e z as e z is not one-to-one. Consider therefore the equation w = e z for a given w . We must assume that w 6 = 0 as e z 6 = 0 (and thus log 0 is not defined). Then | w | = | e z | = e x and y = Arg w + 2 ( k Z ). Hence { log | w | + i (Arg w + 2 ) : k Z } is the set of all solutions z of w = e z . We write log w for any w in the set { log | w | + i (Arg w + 2 ) : k Z } . 3
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4 1. HOLOMORPHIC (OR ANALYTIC) FUNCTIONS Definition 2.1 . Let G C be an open connected set and f : G C a continuous function such that z = e f ( z ) for all z G . Then f is called a branch of the logarithm on G . It is clear that if f is a branch of the logarithm on G , then 0 / G and f ( z ) = log | z | + i (Arg z + 2 ) for some k Z , where k can depend on z . Also, if f is a branch of the logarithm on G , then for fixed k also g ( z ) = f ( z ) + 2 kπi is a branch of the logarithm on G . The converse also holds.
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