7 more on conformal mappings and harmonic functions

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Unformatted text preview: zz–, and x = (z+z–)/2, y = (z-z–)/2i. So the above equation can be rewritten as B(z+z–) C(z-z–) Azz– + + +D=0 2 2i Now write this in terms of w = 1/z. Substituting z = 1/w, z– = 1/w– and multiplying by ww– gives us B(w+w–) C(w-w–) A+ + Dww– = 0 2 2i or 2 2 A + Bu - Cv + D(u + v ) = 0, again the equation of a circle or straight line. More generally: 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 È B˘ ˙ f(z) = AÍ1 + Î 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 the origin. Some reverse ones: Examples (A) Find a complex function that maps the upper half plane into the wedge 0 ≤ Arg z ≤ π/4. (B) Ditto for the Strip 0 ≤ y ≤ π Æ Wedge 0 ≤ Arg w ≤ π/4. (Look at the exponential map.) Exercise set 6 p. 893 # 1–13 odd, , 21–27 odd p. 900 #1, 3, 11, 13, 15, 17 Hand-In: 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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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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