q 8 Ponnusamy S., S...vvas 10 necessai y w Tequine uie cui ves nl Theorem 11.1 to De SIIOOLII:3. Can nonanalytic functions be conformal?4. What kind of functions are isogonal?5. Why does the derivative play such a central role?6. If a function is one-to-one in some neighborhood of each point in adomain, why does this not mean that the function is one-to-one in thedomain?7. If f is conformal on a domain D, is f always one-to-one on D?8. If f is conformal on a domain D which is symmetric with respect to thereal axis, is f(z) conformal on D?9. What is the relationship between conformal and one-to-one?10. At what angle do parallel lines intersect at oo?11. How might we define a function to be analytic at o?12. Is the sum of conformal maps conformal? The product? The composi-tion?Exercises 11.8.1. Given a complex number zo and an e > 0, show that there exists afunction f(2) analytic at zo with f'(20) + 0 and such that f(2) is notone-to-one for |z – z0| < e. Does this contradict Theorem 11.2?2. Show that z? is one-to-one in a domain D if and only if D is containedin a half-plane whose boundary passes through the origin.3. Find points at which the mapping defined by f(z) = nz + z" (n E N) isnot conformal.4. Prove that two smooth curves intersect at an angle a at oo if and onlyif their images under stereographic projection (see Section 2.4) intersectat an angle a at the north pole.5. Show that f(z) and f(2) are both isogonal at points where f(2) isanalytic with nonzero derivative.6. If two straight lines are mapped by a bilinear transformation onto circlestangent to each other, show that the two lines must be parallel. Is theconverse true?7. Find the radius of the largest disk centered at the origin in which w = eis one-to-one. Is the radius different if the disk is centered at an arbitrarypoint 20?8. For f(z) = e-, find arg f'(2). Use this to verify that lines parallel to they axis and a axis map, respectively, onto circles and rays.9. Suppose f(2) is analytic at zo with f'(zo) # 0. Prove that a "small"rectangle containing zo and having area A is mapped onto a figure whosearea is approximately |f'(zo)|²A.10. Either directly or by making use of Theorem 11.5, show that the functionw = z" maps the ray arg z = 0 (0 <0 < 27/n) onto the ray arg z = no.390Il Conformal Mapping and the Riemann Mapping Theorem

Answered: 8. If f is conformal on a domain D…

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8. If f is conformal on a domain D which is symmetric with respect to the real axis, is f(z) conformal on D?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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