1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

104 CHAPTER 3 • ANALYTIC AND HARMONIC FUNC'tlONS EXAMPLE 3.6 Show that the function defined by is not differentiable at the po ...

3.2 • THE CAUCHY-RIEMANN EQUATIONS 105 " (3-l,7) . The parti~ -derivati-WS ~"' ai'i°d' ~ exist, so the Jn~·· ~ue t_l}eorem. for ...

106 CHAPTER 3 8 ANALYTI C ANO HARMONIC FUNCTIONS EXAMPLE 3.7 At the beginning of this section (Equation (3-13)) we defined the ...

3.2 • THE CAUCHY-RIEMANN EQUATIONS 107 u, v, Ux, uy, v.,, and Vy are continuous everywhere. By Theorem 3.4, f is differentiable ...

108 CHAPTER 3 • ANALYTIC ANO HARMONIC FUNCTIONS Here u.,, Uy, v.,, and Vy are continuous, and u., (x , y) = Vy (x, y) bolds for ...

3.2 • THE CAUCHY-RIEMANN EQUATIONS 109 where the domain is restricted to be { rei^8 : r > O and - 11 < B < 11}, then th ...

110 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTIONS -------~EXERCISES FOR SECTION 3.2 Use the Cauchy- Riemann conditions to determi ...

(e) f(z)=x^3 +i(l- y)^3. (f) f(z) = z^2 +z. (g) f (z) = x^2 + y^2 + i2xy. (h) f (z) = lz - (2 + i)l^2. 3.2 • THE CAUCHY-RIEMANN ...

112 CHAPTER 3 • ANALYTIC AND HARMONlC FUNCTIONS (b) Use the original Cauchy- Riemann equations for u and v and the results of pa ...

3.3 • HARMONIC FUNCTIONS 113 is known as Laplace's equation (sometimes referred to as the pot ential equa- tion). If</>, & ...

114 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTIONS EXAMPLE 3.11 If u (x, y) = x^2 - y^2 , then u.,,, (x,y) +Uvv (x,y) = 2-2 = O; h ...

3.3 • HARMONIC FUNCTIONS 115 are the real and imaginary parts of an analytic function. At the point ( x, y) = (2, -1), we have u ...

116 CHAP'l'ER 3 • ANALYTIC AND HARMONIC FUNCTI ONS Technically we should always specify the domain of function when defining it. ...

3.3 • HARMONIC FUNCTIONS 117 --~~~ ,.,,~-...... -... ___ ~ ~~ ---...... -...... -~ ---~ ~ -...._ - ..._ ...

ll8 CHAPTER 3 • ANALYTIC ANO HARMONIC FUNCTIONS Theorem 3.8 implies that </> (x, y) is a harmonic function. Using the vect ...

3.3 • HARMONIC FUNCTIONS 119 I I I Bquiporential ,. I I Streamline , , , , Figure 3.5 The families of orthogonal curves {(x,y): ...

120 Cl!APTE!l 3 • ANALYTIC ANO HARMONIC FUNCTIONS 2. Does an analytic function f (z) = u(x,y) +iv (x,y) exist for which v (x,y) ...

3.3 • HARMONIC FUNCTIONS 121 Suppose that ( xo , y 0 ) is a point common to the two curves <f> ( x, y) = c 1 and ..p ( x , ...

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chanter 4 sequ(nces, JUiia and mandelbrot sets, and power series Overview In 1980 Benoit Mandelbrot led a team of mathematicia ...