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Chapter2 - Chapter 2 Dierentiation Mathematical study and...

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Chapter 2 Differentiation Mathematical study and research are very suggestive of mountaineering. Whymper made several efforts before he climbed the Matterhorn in the 1860’s and even then it cost the life of four of his party. Now, however, any tourist can be hauled up for a small cost, and perhaps does not appreciate the difficulty of the original ascent. So in mathematics, it may be found hard to realise the great initial difficulty of making a little step which now seems so natural and obvious, and it may not be surprising if such a step has been found and lost again. Louis Joel Mordell (1888–1972) 2.1 First Steps A (complex) function f is a mapping from a subset G C to C (in this situation we will write f : G C and call G the domain of f ). This means that each element z G gets mapped to exactly one complex number, called the image of z and usually denoted by f ( z ). So far there is nothing that makes complex functions any more special than, say, functions from R m to R n . In fact, we can construct many familiar looking functions from the standard calculus repertoire, such as f ( z ) = z (the identity map ), f ( z ) = 2 z + i , f ( z ) = z 3 , or f ( z ) = 1 z . The former three could be defined on all of C , whereas for the latter we have to exclude the origin z = 0. On the other hand, we could construct some functions which make use of a certain representation of z , for example, f ( x, y ) = x - 2 iy , f ( x, y ) = y 2 - ix , or f ( r, φ ) = 2 re i ( φ + π ) . Maybe the fundamental principle of analysis is that of a limit. The philosophy of the following definition is not restricted to complex functions, but for sake of simplicity we only state it for those functions. Definition 2.1. Suppose f is a complex function with domain G and z 0 is an accumulation point of G . Suppose there is a complex number w 0 such that for every > 0, we can find δ > 0 so that for all z G satisfying 0 < | z - z 0 | < δ we have | f ( z ) - w 0 | < . Then w 0 is the limit of f as z approaches z 0 , in short lim z z 0 f ( z ) = w 0 . This definition is the same as is found in most calculus texts. The reason we require that z 0 is an accumulation point of the domain is just that we need to be sure that there are points z of the domain which are arbitrarily close to z 0 . Just as in the real case, the definition does not require 13
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CHAPTER 2. DIFFERENTIATION 14 that z 0 is in the domain of f and, if z 0 is in the domain of f , the definition explicitly ignores the value of f ( z 0 ). That is why we require 0 < | z - z 0 | . Just as in the real case the limit w 0 is unique if it exists. It is often useful to investigate limits by restricting the way the point z “approaches” z 0 . The following is a easy consequence of the definition. Lemma 2.2. Suppose lim z z 0 f ( z ) exists and has the value w 0 , as above. Suppose G 0 G , and suppose z 0 is an accumulation point of G 0 . If f 0 is the restriction of f to G 0 then lim z z 0 f 0 ( z ) exists and has the value w 0 .
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