1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

404 CHAPTER 10 • CONFORMAL MAPPING y v w=--i(I -Z) l+z Figure 10.5 i (1 - z) The image of lzl < 1 under w = 1 . + z onto ...

10.2 • BILINEAR TRANSFORMATIONS 405 EXAMPLE 10.4 Construct the bilinear transformation w = S (z) that maps the points z 1 = -i ...

406 CHAPTER 10 • CONFORMAL MAPPING Using the fa.ct that 1 1:..ii = t• we rewrite this equation as z+2 l+w i;-=1- w· We now expan ...

10.2 • BILlNEAR TRANSFORMATIONS 407 y i w = S(z) v i Figure 10 .6 The bilinear mapping w = S (z) = [(l -i) z + 2] / [(1 ...

408 CHAPTER 10 • CONFORMAL MAPPING y v w = S(z) - x Figure 10.7 . -iz + 4i The mappmg w = S (z) =. z which determines a mapping ...

10.2 • BILINEAR TRANSFORMATIONS 409 y Figure 10.8 y aw Images of Arg ( w) = c under the mapping z = --. w-1 -------.-EXERCISES F ...

410 CHAPTER 10 • CONFORMAL MAPPING 1 2. If S1 (z) = ::;~ and S2 (z) = .~ 3 , find S1 (82 (z)) and 82 (81 (z)). Find the image o ...

10.3 • MAPPINGS INVOLVING ELEMENTARY FUNCTIONS 411 y ...... ..,.-........... -....... ~ ...... --· .... ,, in w=exp z z= - ...

412 CHAP'l'ER 10 • CONFORMAL MAPPING y v et-; w =--et+ j .t Z2=0 y \_ ./._ Z-i Z=exp z w=--Z+i Figure 1 0. 10 e• - i The compo ...

y Z=l+z\-._. 1- z 10 .3 • MAPPINGS INVOLVING E LEMENTARY FUNCTIONS 413 w=(~1- z ) 2 u Figure 10.12 The composite transformat i ...

414 CHAPTER 10 • CONFOR.MAL MAPPING y _____! -2 -I 0 Z=t2-J\ 2 "'=f(z) -- -2 y I ~-~ v 0 2 /=zt x Figure 10.13 The composite t ...

10.3 • MAPPINGS INVOLVING ELEMENTARY F UNCTIONS 4 1 5 where r1 = lz - l j and 81 = Arg (z - 1), and l ·~ (z + 1 )2 = .ji2e' • , ...

416 CHAPTER 10 • CONFORMAL MAPPING 1 10.3.2 The Riemann Surface for w = (z^2 - 1)2 Using the other branch of the square root, we ...

10.3 • MAPPINGS INVOLVING ELEMENTARY FUNCTIONS 417 (a) (b) I Figure 10. 15 The Riemann surfaces for the mapping w = (z^2 - 1)'. ...

418 CHAPTER 10 • CONFORMAL MAPPING 15. Find the image of the sector r > 0, 0 < IJ < i• under w = !:;:. Show that the f ...

10.4 • MAPPING BY TRIGONOMETRIC FUNCTIONS 419 )' 2 Z=exp(i2z~ Zi=-i w = 1an z v J. -iZ+ i / w =--Z+I x Figure 10. 16 The com ...

420 CHAPTER 10 • CONFORMAL MAPPING y 2 (a) w =sinz -----+ x t = Arcsin w ~ Figure 1 0.17 The transformation w = sin z. for -; &l ...

10.4 • MAPPING BY TRIGONOMETRIC FUNCTIONS 421 for u and v expressed as functions of x and y. To solve for u, we first equate the ...

422 CHAPTER 10 • CONFORMAL MAPPING v y w = Arcsin z ' (^4) ' ·~ ..,....., ' -~ 3 '1 2 ,... . x 2 3 4 0 Figuro 10.18 The mapping ...

10.4 • MAPPING BY TRIGONOMETR.IC FUNCTIONS 423 Therefore, we have Arcsin {l + i) ·;::;: 0.666239432 + il.061275062. Is there any ...