1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

84 CHAPTER 2 • COMPLEX FUNCTIONS The beauty of this structure is that it makes this "full square root func- tion" continuous for ...

2.5 • THE RECIPROCAL TRANSFORMATION w = ~ 85 (a) Show that f is, in general, an n -valued function. (b) Write the principal nth ...

86 CHAPTER 2 • COMPLEX FUNCTIONS We can extend the system of complex numbers by joining to it an "ideal" point denoted by oo and ...

2.5 • THE RECIPROCAL TRANSFORMATION w = ~ 8 7 continuous mapping from the extended z plane onto the extended w plane. We leave t ...

88 CHAPTER 2 • COMPLEX FUNCTIONS but not conversely. That is, Equation (2-34) makes sense when (u, v) = (O, 0), whereas Equation ...

2.5 • THE RECIPROCAL TRANSFORMATION W = ~ 89 y 1 w=-I z -- v -2 Figure 2.24 The mapping w = ~ discussed in Example 2.23. To st ...

90 CHAPTER 2 • COMPLEX FUNCTIONS i' II " -·~ I II " y II " ···· -b= I ..... ·~ .... b=~ ........................... ~ .. b=-~ ...

2.5 • THE RECIPROCAL TR.ANSFOR.MA'I'ION W = ~ 91 Limits involving oo. The function f (z) is said to have the limit L as z appro ...

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cba~yp.t~l 3 harmonic functions Overview Does the notion of a derivative of a complex function make sense? If so, how should it ...

94 CHAPTER. 3 • ANALYTIC AND HARMONIC FUNCTIONS If we let w = J (z) and t:.w = J (z) - J (zo), then we can use the Leibniz notat ...

3.1 • DIFFERENTIABLE AND ANALYTIC FUNCTIONS 95 form z = xo + iy. ) . f(z)-f(zo) 1 . f(xo+ iy)- f(xo+iyo) 1m = un z-->zo z -zo ...

96 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTIONS Points of nonanalyticity for a function are called singular points. They are impo ...

3.1 • DIFFERENTIABLE AND ANALYTIC FUNCTIONS 97 We can establish Equation (3-8) from Theorem 3.1. Letting h (z) = f (z) g (z) and ...

98 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTIONS -------~EXERCISES FOR SECTION 3.1 Find the derivatives of the following function ...

3.1 • DIFFERENTIABLE AND ANALYTIC FUNCTIONS 99 (a) Show that P' (z) = a1 + 2a2z + .. · + na,.z"- \ (b) Show that, for k = 0, 1, ...

100 CHAPTER 3 • ANALYTIC AND H A RMONIC FUNCTIONS (e) Identity (3-12). 13. Consider the differentiable function f ( z) = z^3 and ...

3.2 • THE CAUCHY-RIEMANN EQUATIONS 101 y v I.OJ t~~---······················•Z) 4.06 .... .... 4 .04 1.005. ~ zplane w plane w = ...

102 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTI ONS (3-14). ...

3.2 • THE CAUCHY-RIEMANN EQUATIONS 103 Note some of the important implications of this theorem. If f is differentiable at zo, t ...