1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

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ANSWERS 619

Section 11.9. The Schwarz-Christoffel Transformation: page 492



  1. f'(z) = A(z-a)-(or-kor)/" = A(z-a)k-i, integrate and get f(z ) = ~(z-a)k,
    then choose A = k.

  2. f'(z) = A(z + l)~(z - 1)~ = A[ z! +^1 !), in tegration and the
    (z2-l) (z2-l)


boundary conditions f(-1) = 0 and f(l) = -1 produces w = f(z) =

~{(z^2 - 1)~ + Log(z + (z^2 -1)~)] - i.

5. f'(z) = A(z+ 1)-^1 z(z - l)-^1 , and w = f(z) = Log(z^2 -1)!.


  1. f'(z) = A(z + 1)^1 z-^1 = A(l +~),integrate and get f(z) = z + Logz.

  2. Select X1 = -^1 -;;"', xz = , X3 = 1, then form f' (z) = A(z +^1 -;;"')-<> (z)


(z-1)"'-^1.

Computation reveals that A = (1-;;")"'-^1 , which is used to construct the
desired function
w = f (z) = J A(z +^1 -;;"')-"' (z) (z - 1)"'-
1
dz= (z - 1)"' (1+ 1 ~'..)^1 -".

1 1. f'(z) = Azf(z - 1)^1 = A(z~ - zf), integrate and get

f(z) = 2i z! (z - 3).


Section 11.10. Image of a Fluid Flow: page 497


1. f'(z) = A(z + 1)-z' z(z - 1) 2

1
= A~, integration and the boundary
(z--1)~

conditions f(-1) =0 and f(O) =i produce w = f(z) = (z^2 -1)~.


  1. w = f(z) = (z- 1)" (1+ 1 ':.',,.)1-"'.


5. w = f(z) = -1 + f~ 1 «~j)i d~.

w = f(z) = i + ~[4(z - l)tz~ - 2Arctan(l - ~)L
+ Log(l - (1- ~) ~) - Log(l + (1-~)t)].

Section 11.11. Sources and Sinks: page 507


  1. F 1 (w) =log::;~~ is the complex potential for a source at w 1 =1 and sink at


wz = - 1. The function w = S(z) = z^2 maps z 1 = 1 and z2 = i onto W1 and

wz, respectively. Therefore, t he composition F2 (z) = F 1 (S(z)) = F 1 (z^2 ) =

2
log = 2 ~! is the desired complex potential.

3. F(z) = Jog(sinz).

5. F(z) = log(z^2 -1).
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