1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

424 CHAPTER 10 • CONFORMAL MAPPING Find the image of the vertical strip -t < x < 0 under the mapping w = cosz. Find th ...

cap p~Pat~J.: l 1 harmonic functions Overview A wide variety of problems in engineering and physics involve harmonic func- tions ...

426 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS Figure 11.1 The harmonic function u(x,y) = U 1 + U~ - Ui (x -a). a •EXAMPL ...

11.2 • INVARIANCE OF LAPLACE'S EQUATION ANO THE DIRICHLET PROBLEM 427 y ¢ (x, y) =constant ""'"'"~ ; (x, y) = K1 whenlzl = I F ...

428 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS 2x •EXAMPLE 11.4 Showthat,P(x,y)=Arctan 2 2 1 is harmonic in the x +y - disk ...

11.2 • INVARIANCE OF LAPLACE'S EQUATION AND THE· DIRICHLET PROBLEM 429 Substituting these equations into the formula for <I&g ...

480 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS v Figure 11. 4 The boundary conditions for the harmonic function <I> ( ...

11.2 • INVARIANCE OF LAPLACE'S EQUATION AND THE 0IR.ICHLET PROBLEM 431 •EXAMPLE 11.6 Find the function (x,y) that is harmonic in ...

432 CHAPTER 1 1 • APPLICATIONS OF HARMONIC FUNCTIONS 1.0 0.8 0.6 Figure 11.6 The graph of u = <P (x, y) with the boundary val ...

2 • INVARIANCE OF LAPLACE'S EQUATION AND THE DIRICHLET PROBLEM 433 y Figure ll.7 The boundary values for (x,y) for the Dirich ...

434 CHAPTER, 11 • APPLICATIONS OF HARMONIC FUNCTIONS y I w= i(l-z) l+z Figure 11.8 The Dirichlet problems for lzl < 1 a ...

11.2 • lNVARJANCE OF LAPLACE' S EQUATION AND THE DIRICHLET PROBLEM 435 y -1 ¢(x,0)=1 for-l<x<l v w=-i ( I --z) 1 +z <1& ...

436 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS 1.0 0.8 y Figure 11.10 2 1 -x^2 -y^2 2 l-r^2 The graph u = - Arctan = - Arct ...

1 1.2 • INVARIANCE OF LAPLACE'S EQUATION AND THE D IRICHLET PROBLEM 437 Figure 11.12 The graph 2 l- (x^2 +y^2 )^2 2 l-r^4 u - ...

438 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS Find the function <fi (x, y) that is harmonic in the upper half-plane Im ...

11.3 • POISSON'S INTEGRAL FOR.MULA FOR THE UPPER HALF-PLANE 439 Find the function <fi (x, y) that is harmonic in the quarter ...

440 CHAPTER. 11 • APPLICATIONS OF HARMONIC FUNCTIONS y (x, y) ----------,~--: -------E--------------------------- 11 I \ ;:. ,., ...

11.3 • POISSON'S INTEGRAL FORMULA FOR THE UPPER HALF-PLANE 441 •EXAMPLE 11.11 Find the function <P(x,y) that is harmonic in t ...

442 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS x Figure 11.14 The graph of u = 4> (x, y) with the boundary values 4> ...

0.8 0.4 0.0 --0.4 11.3 • POISSON'S INTEGRAL FORMULA FOR THE UPPER HALF-PLANE 443 -4 Figure 11 .15 The graph of ti = if> (x, y ...