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

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Unformatted text preview: Complex Variables Notes for Math 703. Part I Updated Fall 2011 Anton R. Schep 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 + ir sin . In this case we write arg z = { + 2 k : k Z } . By Arg z we will denote the principal value of the argument of z 6 = 0, i.e. = Arg z arg z if- < . Note that if z 1 = | z 1 | (cos 1 + i sin 1 ) and z 2 = | z 2 | (cos 2 + i sin 2 ), then we have z 1 z 2 = | z 1 || z 2 | (cos 1 cos 2- sin 1 sin 2 + i (sin 1 cos 2 + cos 1 sin 2 )) = | z 1 z 2 | (cos( 1 + 2 ) + i (sin( 1 + 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 k . In particular e 2 i = 1 and the function e z is 2 i-periodic, i.e., e z +2 i = e z e 2 i = 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 ( k Z ). Hence { log | w | + i (Arg w + 2 k ) : 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 ) : k Z } ....
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complex-part1 - Complex Variables Notes for Math 703. Part...

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