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Review of complex numbers 4=R₁e 22- R₂e z=Re 2122 R₁ Re+82) 01+02 The complex conjugate of z = Reid = a + bi is Caz-Re(+2x/8) z= Rea-bi which is the reflection of z across the real axis. 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. Ca=2x/8 Delve into the intricacies of analytic continuation and advanced function theory. 1. Analytic Continuation Principles: ⚫a. State and prove the Identity Theorem for analytic functions, emphasizing its role in analytic continuation. ⚫b. Demonstrate how analytic continuation can be used to extend the domain of definition of the Riemann zeta function ((s) from R(s) >1 to the entire complex plane, excluding 8 = 1. 2. Branch Points and Multi-Valued Functions: ⚫ a. Define branch points and branch cuts for multi-valued functions. Provide examples with f(z) =√√ and f(z) = log(2). b. Construct the Riemann surface for f()=√ and describe its topological structure. 3. Meromorphic Continuation: a. Explain the concept of meromorphic continuation and its importance in complex analysis. b. Show how the function f()=sin() can be meromorphically continued to the entire complex plane and identify its poles. 4. Monodromy and Analytic Continuation: ⚫a. Define the monodromy group of a multi-valued function and explain its significance in analytic continuation. b. For the function f(x)=√, determine its monodromy group and discuss the implications for its analytic continuation around the origin. Note that Iz zz Re Re-i =R2e0 = R2 => |z|= √√zz = √√a²+ b² = R.

Review of complex numbers 4=R₁e 22- R₂e z=Re 2122 R₁ Re+82) 01+02 The complex conjugate of z = Reid = a + bi is Caz-Re(+2x/8) z= Rea-bi which is the reflection of z across the real axis. 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. Ca=2x/8 Delve into the intricacies of analytic continuation and advanced function theory. 1. Analytic Continuation Principles: ⚫a. State and prove the Identity Theorem for analytic functions, emphasizing its role in analytic continuation. ⚫b. Demonstrate how analytic continuation can be used to extend the domain of definition of the Riemann zeta function ((s) from R(s) >1 to the entire complex plane, excluding 8 = 1. 2. Branch Points and Multi-Valued Functions: ⚫ a. Define branch points and branch cuts for multi-valued functions. Provide examples with f(z) =√√ and f(z) = log(2). b. Construct the Riemann surface for f()=√ and describe its topological structure. 3. Meromorphic Continuation: a. Explain the concept of meromorphic continuation and its importance in complex analysis. b. Show how the function f()=sin() can be meromorphically continued to the entire complex plane and identify its poles. 4. Monodromy and Analytic Continuation: ⚫a. Define the monodromy group of a multi-valued function and explain its significance in analytic continuation. b. For the function f(x)=√, determine its monodromy group and discuss the implications for its analytic continuation around the origin. Note that Iz zz Re Re-i =R2e0 = R2 => |z|= √√zz = √√a²+ b² = R.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Complex Numbers
Section4.3: The Complex Plane
Problem 2ECP
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Review of complex numbers 4=R₁e 22- R₂e z=Re 2122 R₁ Re+82) 01+02 The complex conjugate of z = Reid = a + bi is Caz-Re(+2x/8) z= Rea-bi which is the reflection of z across the real axis. 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. Ca=2x/8 Delve into the intricacies of analytic continuation and advanced function theory. 1. Analytic Continuation Principles: ⚫a. State and prove the Identity Theorem for analytic functions, emphasizing its role in analytic continuation. ⚫b. Demonstrate how analytic continuation can be used to extend the domain of definition of the Riemann zeta function ((s) from R(s) >1 to the entire complex plane, excluding 8 = 1. 2. Branch Points and Multi-Valued Functions: ⚫ a. Define branch points and branch cuts for multi-valued functions. Provide examples with f(z) =√√ and f(z) = log(2). b. Construct the Riemann surface for f()=√ and describe its topological structure. 3. Meromorphic Continuation: a. Explain the concept of meromorphic continuation and its importance in complex analysis. b. Show how the function f()=sin() can be meromorphically continued to the entire complex plane and identify its poles. 4. Monodromy and Analytic Continuation: ⚫a. Define the monodromy group of a multi-valued function and explain its significance in analytic continuation. b. For the function f(x)=√, determine its monodromy group and discuss the implications for its analytic continuation around the origin. Note that Iz zz Re Re-i =R2e0 = R2 => |z|= √√zz = √√a²+ b² = R.
Transcribed Image Text:Review of complex numbers 4=R₁e 22- R₂e z=Re 2122 R₁ Re+82) 01+02 The complex conjugate of z = Reid = a + bi is Caz-Re(+2x/8) z= Rea-bi which is the reflection of z across the real axis. 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. Ca=2x/8 Delve into the intricacies of analytic continuation and advanced function theory. 1. Analytic Continuation Principles: ⚫a. State and prove the Identity Theorem for analytic functions, emphasizing its role in analytic continuation. ⚫b. Demonstrate how analytic continuation can be used to extend the domain of definition of the Riemann zeta function ((s) from R(s) >1 to the entire complex plane, excluding 8 = 1. 2. Branch Points and Multi-Valued Functions: ⚫ a. Define branch points and branch cuts for multi-valued functions. Provide examples with f(z) =√√ and f(z) = log(2). b. Construct the Riemann surface for f()=√ and describe its topological structure. 3. Meromorphic Continuation: a. Explain the concept of meromorphic continuation and its importance in complex analysis. b. Show how the function f()=sin() can be meromorphically continued to the entire complex plane and identify its poles. 4. Monodromy and Analytic Continuation: ⚫a. Define the monodromy group of a multi-valued function and explain its significance in analytic continuation. b. For the function f(x)=√, determine its monodromy group and discuss the implications for its analytic continuation around the origin. Note that Iz zz Re Re-i =R2e0 = R2 => |z|= √√zz = √√a²+ b² = R.
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