1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

444 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS Show that the function ¢> (:r;, y) given by Poisson's integral formula i ...

11.4 • TwO-DI MENSJONAL MATHEMATICAL MODELS 445 For convenience, we introduce the term complex potential for the analytic func- ...

446 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS equipotential curves and the heat flow lines as flux lines. This implies tha ...

11.5 • STEADY STA'l'E TEMPERATURES 447 y y V(x,y+&y) V(x+lix,y+&y} S(x, Heat y) = fJ flow Jines ~ V(x, y) V(x + lix, y) ...

448 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS is an analytic function. The curves T(x,y) = K 1 are called isothermals and ...

11.5 • STEADY STATE TEMPERATURES 449 Solu t io n Since T(x,y) is a harmonic function, this problem is an example of a Dirichlet ...

450 CHAPTER 11 • APPLI CATIONS OF HARMONIC FUNCTIONS y I T(x,0)=50 for - I <x< I Figure 11.19 The temperature T(x,y} in ...

y 11.5 • STEADY STATE TEMPERATURES 451 w =f(zl 0 v ar•_ 0 on - Figure 11 .20 Steady state temperatures with one boundary por ...

452 CHAPTER 11 • APPLICATIONS OF H ARMONIC F UNCTIONS T= -0.2 T = -0.4 T = -0.6 T = -1. 0 -1 y T =0.2 T=0.4 T= 1.0 Figure 11. 21 ...

11.5 • STEADY STATE TEMPERATURES 453 Find the temperature function T ( x, y) in the infinite strip bounded by the lines y = - x ...

464 CHAPTER 11 • APPLICATIONS OF H ARMONI C F UNCTIONS )' Figure 11.24 5. Find the temperature function T (x, y) in the semi-inf ...

T(x, O} = 0, T(x, O} = O, T(x, y) = 100, 11.5 • STEADY STATE TEMPERATURES 455 for x > l; for x < - 1; if z = e'^8 , 0 < ...

456 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS Find the temperature function T (x, y) in the first quadrant x > 0, y &g ...

11. 5 • STEADY STATE TEMPERATURES 457 Find the temperature function T (x, y) in the upper half-plane Im (z) > 0 that satisfi ...

458 CHAPTER 11 • APPLICATIONS OF HARMONIC F UNCTIONS in the upper half-disk lzl < 1, Im(z) > 0, show that the isothermals ...

11.6 • Two-DIMENSIONAL ELECTROSTATICS 459 Figure 11.35 11.6 Two-Dimensional Electrostatics A two-dimensional electrostatic field ...

460 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS EXAMPLE 11 .18 Consider two parallel conducting planes that pass per- pend ...

11.6 • Two-DIMENSIONAL ELECTROSTATICS 4 6 1 y w=logz v x Figure 11.36 The electrical field in a coaxial cylinder, where U2 < ...

462 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS ¢'=- 120 ?=-60 ?=0 ?=60 ?=120 Figure 11.37 The electric field produced by tw ...

11.6 • Two-DIMENSIONAL ELECTROSTATICS 463 y v w =S(t.) x " Figure 11.38 The potentials </> and 4'. -------~EXERCISES FOR S ...