By Carmen Medina
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Extra info for Complex Variables With Aplications
4-4), and using the definition of the derivative, we have the desired result: + + The equations contained in Eq. 4-1) are of no use in establishing the differentiability or in determining the derivative of any expression involving 1 ~ or 2. We can rewrite such functions in the form u ( x , y) iv(x, y) and then apply the Cauchy-Riemann equations to investigate differentiability. If the derivative exists, it can then be found from Eq. 3-6) or Eq. 3-8). These ideas are illustrated in the following.
3-1) + US and still not have a derivative. I n Eq. 3-1 obtained from the right side of Eq. 3-1) for both the positive and negative choices. If two different numbers are obtained, f' ( x o )does not exist. As an example of how this can occur, consider f ( x ) ='21x1 plotted in Fig. 3-1. It is not hard to show that f ( x ) is continuous for all x. Let us try to compute f' ( 0 ) by means of Eq. 3-1). With xo = 0 , f(xo) = 0 , and f(xo Ax) = 2lAx1, we have + of a complex variable must be continuous at a point to possess a derivative there (see Exercise 19), but continuity by itself does not guarantee the existence of a derivative.
EXAMPLE 7 In Chapter 3, we will define the function of a complex variable eZ, show that it is entire and never zero, and will see that its derivative (as in the case of the corresponding real function) is ez. Show that ei is nowhere analytic. Solution. Assuming that ei is differentiable, we follow the chain rule and get @ dz - dez d i The first term on the right is ei, while the second does not exist. zl-z. ) Thus ei is not differentiable and is nowhere analytic. 4-1 ---,. v ------m-=-- + times instead of expressing a function of z in the form f ( z ) = u ( x , Y) Y), it is convenient to change to the polar system r, 0 so that x = r cos 0, 76 Chapter 2 ~h~ complex Function and Its Derivative + y = r sin 0.