ANSWERS 619Section 11.9. The Schwarz-Christoffel Transformation: page 492
- f'(z) = A(z-a)-(or-kor)/" = A(z-a)k-i, integrate and get f(z ) = ~(z-a)k,
then choose A = k. - 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)!.
- f'(z) = A(z + 1)^1 z-^1 = A(l +~),integrate and get f(z) = z + Logz.
- 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)~.
- 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- 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.