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Unformatted text preview: zz–, and x = (z+z–)/2, y = (z-z–)/2i. So the above equation can be rewritten
Now write this in terms of w = 1/z. Substituting z = 1/w, z– = 1/w– and multiplying by
ww– gives us
+ Dww– = 0
A + Bu - Cv + D(u + v ) = 0,
again the equation of a circle or straight line.
Theorem 6.5 Every map of the form f(z) = az + b
takes circles or straight lines to
cz + d circles or straight lines
Proof We can manipulate f(z) to write it in the form
f(z) = AÍ1 +
c + d/z˚
which is a composite affine maps and inversions.
Continuing with the examples..
2 (G) f(z) = z is conformal everywhere except at the origin. In fact, it doubles angles at
Some reverse ones:
(A) Find a complex function that maps the upper half plane into the wedge 0 ≤ Arg z ≤
(B) Ditto for the Strip 0 ≤ y ≤ π Æ Wedge 0 ≤ Arg w ≤ π/4. (Look at the exponential
Exercise set 6
p. 893 # 1–13 odd, , 21–27 odd
p. 900 #1, 3, 11, 13, 15, 17
1. Find an analytic complex function that maps the interior of the unit disc centered at (0,
0) onto the interior of the first quadrant. [Use composites of the conformal mappings in
the Appendix of the book.]...
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