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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?

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
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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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 a domain, why does this not mean that the function is one-to-one in the domain? 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 the real 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 a function f(2) analytic at zo with f'(20) + 0 and such that f(2) is not one-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 contained in 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) is not conformal. 4. Prove that two smooth curves intersect at an angle a at oo if and only if their images under stereographic projection (see Section 2.4) intersect at an angle a at the north pole. 5. Show that f(z) and f(2) are both isogonal at points where f(2) is analytic with nonzero derivative. 6. If two straight lines are mapped by a bilinear transformation onto circles tangent to each other, show that the two lines must be parallel. Is the converse true? 7. Find the radius of the largest disk centered at the origin in which w = e is one-to-one. Is the radius different if the disk is centered at an arbitrary point 20? 8. For f(z) = e-, find arg f'(2). Use this to verify that lines parallel to the y 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 whose area is approximately |f'(zo)|²A. 10. Either directly or by making use of Theorem 11.5, show that the function w = z" maps the ray arg z = 0 (0 <0 < 27/n) onto the ray arg z = no. 390 Il Conformal Mapping and the Riemann Mapping Theorem
Transcribed Image Text: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 a domain, why does this not mean that the function is one-to-one in the domain? 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 the real 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 a function f(2) analytic at zo with f'(20) + 0 and such that f(2) is not one-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 contained in 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) is not conformal. 4. Prove that two smooth curves intersect at an angle a at oo if and only if their images under stereographic projection (see Section 2.4) intersect at an angle a at the north pole. 5. Show that f(z) and f(2) are both isogonal at points where f(2) is analytic with nonzero derivative. 6. If two straight lines are mapped by a bilinear transformation onto circles tangent to each other, show that the two lines must be parallel. Is the converse true? 7. Find the radius of the largest disk centered at the origin in which w = e is one-to-one. Is the radius different if the disk is centered at an arbitrary point 20? 8. For f(z) = e-, find arg f'(2). Use this to verify that lines parallel to the y 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 whose area is approximately |f'(zo)|²A. 10. Either directly or by making use of Theorem 11.5, show that the function w = z" maps the ray arg z = 0 (0 <0 < 27/n) onto the ray arg z = no. 390 Il Conformal Mapping and the Riemann Mapping Theorem
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