Continuity Properties for Complex Variables Analyticity

We see that we cannot take the usual properties of uniqueness and continuity for granted in dealing with complex variables. To make progress, then, we shall need to formalize the properties of continuity and single valuedness. In analysis of complex variables, these properties are called analytic. We discuss them next. 

As last illustrated, some peculiar behavior patterns arise with complex variables, so care must be taken to insure that functions are well behaved in some rational sense. This property is called analyticity, so any function w = f(s) is called analytic within some two-dimensional region R if at all arbitrary points, say Sq, in the region it satisfies the conditions  

It has a unique, finite derivative at Sq, which satisfies the Cauchy-Riemann conditions. 

The Cauchy-Riemann conditions are the essential properties for continuity of derivatives, quite apart from those encountered in real variables. To see these, write the general complex function 

Obviously, the total derivative of df/ds must be the same for the two cases, so multiplying Eq. 9.50 by (-/) gives  

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Continuity Properties for Complex Variables Analyticity 339 It is clear if this must equal w = u + iv, then... 

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