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Review of complex numbers 22-R₂e 2122 R₁ R₂+82) 01+02 =Rje C32=Re(+2x/8) The complex conjugate of z = Rei= a + bi is z=Re-a-bi, which is the reflection of z across the real axis. Note that Iz²=zz Rei Re-i = R2e0=R2 => = Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8=2mi/8 Let D be a simply connected, proper open subset of the complex plane C (ie., D + C). 1. Riemann Mapping Theorem (Revisited): • Prove the Riemann Mapping Theorem: There exists a bijective holomorphic function (conformal map) f: DD, where D is the open unit disk in C. 2. Uniqueness up to Automorphism: ⚫ Show that such a conformal map f is unique up to post-composition with an automorphism of ID. Specifically, if g: DD is another bijective holomorphic map, then there exists an automorphism of D such that goof. 3. Caratheodory's Theorem on Boundary Extension: Assume that D has a boundary that is a Jordan curve. Prove that the conformal map f : D→ D extends to a homeomorphism between the closures D and D. 4. Hyperbolic Geometry Connection: • Explore the connection between the Riemann Mapping Theorem and hyperbolic geometry. Specifically, show how the conformal equivalence to D endows D with a hyperbolic metric and discuss its uniqueness. 5. Schwarz-Christoffel Mapping: ⚫ For a polygonal domain D, construct the Schwarz-Christoffel mapping from D onto D. Discuss the conditions under which such a mapping exists and its properties. Requirements for Solution: •Utilize normal families, Montel's Theorem, and other foundational results to establish the existence of conformal maps.

Review of complex numbers 22-R₂e 2122 R₁ R₂+82) 01+02 =Rje C32=Re(+2x/8) The complex conjugate of z = Rei= a + bi is z=Re-a-bi, which is the reflection of z across the real axis. Note that Iz²=zz Rei Re-i = R2e0=R2 => = Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8=2mi/8 Let D be a simply connected, proper open subset of the complex plane C (ie., D + C). 1. Riemann Mapping Theorem (Revisited): • Prove the Riemann Mapping Theorem: There exists a bijective holomorphic function (conformal map) f: DD, where D is the open unit disk in C. 2. Uniqueness up to Automorphism: ⚫ Show that such a conformal map f is unique up to post-composition with an automorphism of ID. Specifically, if g: DD is another bijective holomorphic map, then there exists an automorphism of D such that goof. 3. Caratheodory's Theorem on Boundary Extension: Assume that D has a boundary that is a Jordan curve. Prove that the conformal map f : D→ D extends to a homeomorphism between the closures D and D. 4. Hyperbolic Geometry Connection: • Explore the connection between the Riemann Mapping Theorem and hyperbolic geometry. Specifically, show how the conformal equivalence to D endows D with a hyperbolic metric and discuss its uniqueness. 5. Schwarz-Christoffel Mapping: ⚫ For a polygonal domain D, construct the Schwarz-Christoffel mapping from D onto D. Discuss the conditions under which such a mapping exists and its properties. Requirements for Solution: •Utilize normal families, Montel's Theorem, and other foundational results to establish the existence of conformal maps.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.4: Complex And Rational Zeros Of Polynomials
Problem 36E
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Review of complex numbers 22-R₂e 2122 R₁ R₂+82) 01+02 =Rje C32=Re(+2x/8) The complex conjugate of z = Rei= a + bi is z=Re-a-bi, which is the reflection of z across the real axis. Note that Iz²=zz Rei Re-i = R2e0=R2 => = Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8=2mi/8 Let D be a simply connected, proper open subset of the complex plane C (ie., D + C). 1. Riemann Mapping Theorem (Revisited): • Prove the Riemann Mapping Theorem: There exists a bijective holomorphic function (conformal map) f: DD, where D is the open unit disk in C. 2. Uniqueness up to Automorphism: ⚫ Show that such a conformal map f is unique up to post-composition with an automorphism of ID. Specifically, if g: DD is another bijective holomorphic map, then there exists an automorphism of D such that goof. 3. Caratheodory's Theorem on Boundary Extension: Assume that D has a boundary that is a Jordan curve. Prove that the conformal map f : D→ D extends to a homeomorphism between the closures D and D. 4. Hyperbolic Geometry Connection: • Explore the connection between the Riemann Mapping Theorem and hyperbolic geometry. Specifically, show how the conformal equivalence to D endows D with a hyperbolic metric and discuss its uniqueness. 5. Schwarz-Christoffel Mapping: ⚫ For a polygonal domain D, construct the Schwarz-Christoffel mapping from D onto D. Discuss the conditions under which such a mapping exists and its properties. Requirements for Solution: •Utilize normal families, Montel's Theorem, and other foundational results to establish the existence of conformal maps.
Transcribed Image Text:Review of complex numbers 22-R₂e 2122 R₁ R₂+82) 01+02 =Rje C32=Re(+2x/8) The complex conjugate of z = Rei= a + bi is z=Re-a-bi, which is the reflection of z across the real axis. Note that Iz²=zz Rei Re-i = R2e0=R2 => = Do not use AI, I need real solution, attach required graph and code wherever needed. 3For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8=2mi/8 Let D be a simply connected, proper open subset of the complex plane C (ie., D + C). 1. Riemann Mapping Theorem (Revisited): • Prove the Riemann Mapping Theorem: There exists a bijective holomorphic function (conformal map) f: DD, where D is the open unit disk in C. 2. Uniqueness up to Automorphism: ⚫ Show that such a conformal map f is unique up to post-composition with an automorphism of ID. Specifically, if g: DD is another bijective holomorphic map, then there exists an automorphism of D such that goof. 3. Caratheodory's Theorem on Boundary Extension: Assume that D has a boundary that is a Jordan curve. Prove that the conformal map f : D→ D extends to a homeomorphism between the closures D and D. 4. Hyperbolic Geometry Connection: • Explore the connection between the Riemann Mapping Theorem and hyperbolic geometry. Specifically, show how the conformal equivalence to D endows D with a hyperbolic metric and discuss its uniqueness. 5. Schwarz-Christoffel Mapping: ⚫ For a polygonal domain D, construct the Schwarz-Christoffel mapping from D onto D. Discuss the conditions under which such a mapping exists and its properties. Requirements for Solution: •Utilize normal families, Montel's Theorem, and other foundational results to establish the existence of conformal maps.
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