TAILIEUCHUNG - Book: Complex Analysis

In this section we will define what we mean by a complex func- tion. We will then generalise the definitions of the exponential, sine and cosine functions using complex power series. To deal with complex power series we define the notions of conver- gent and absolutely convergent, and see how to use the ratio test from real analysis to determine convergence and radius of convergence for these complex series. We start by defining domains in the complex plane. This requires the prelimary definition | Complex Analysis 2002-2003 K. Houston 2003 1 Complex Functions In this section we will define what we mean by a complex function. We will then generalise the definitions of the exponential sine and cosine functions using complex power series. To deal with complex power series we define the notions of convergent and absolutely convergent and see how to use the ratio test from real analysis to determine convergence and radius of convergence for these complex series. We start by defining domains in the complex plane. This requires the prelimary definition. Definition The e-neighbourhood of a complex number z is the set of complex numbers w G C z - w e where e is positive number. Thus the e-neigbourhood of a point z is just the set of points lying within the circle of radius e centred at z. Note that it doesn t contain the circle. Definition A domain is a non-empty subset D of C such that for every point in D there exists a e-neighbourhood contained in D. Examples The following are domains. i D C. Take c G C. Then any e 0 will do for an eneighbourhood of c. ii D C 0 . Take c G D and let e 2 c . This gives a e-neighbourhood of c in D. iii D z z a R for some R 0. Take c G C and let e 2 R c a . This gives a e-neighbourhood of c in D. Example The set of real numbers R is not a domain. Consider any real number then any e-neighbourhood must contain some complex numbers . the e-neighbourhood does not lie in the real numbers. We can now define the basic object of study. 1 Definition Let D be a domain in C. A complex function denoted f D C is a map which assigns to each z in D an element of C this value is denoted f z . Common Error Note that f is the function and f z is the value of the function at z. It is wrong to say f z is a function but sometimes people do. Examples i Let f z z2 for all z G C. ii Let f z z for all z G C. Note that here we have a complex function for which every value is real. iii Let f z 3z4 - 5 - 2i z2 z - 7 for all z G C.

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