Review of complex numbers 4-R₁e 22= R₂e z=Re Do not use AI, I need real solution, attach required graph and code wherever needed. For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8-2/8 1=R1 R₂+2) 01+02 The complex conjugate of z = Rei = a + bi is Caz-Re(+2x/8) z= Rea- bi, which is the reflection of z across the real axis. Note that |z2=z-z= Re Re-i =R2e0 = R² = z=√√zz=√√√²+1²=R. Consider the multi-valued function f(z) = log(2), defined initially on C\(-0,0], where the branch cut is taken along the negative real axis. 1. Riemann Surface Construction: Construct the Riemann surface for log(z) by appropriately "gluing" copies of the complex plane cut along (-00, 0). Describe the topological and complex structure of this Riemann surface. 2. Analytic Continuation Along Different Paths: Define what it means to analytically continue log(2) along a path in C\{0}. Illustrate how different homotopy classes of paths affect the branches of log(). 3. Monodromy Theorem Application: Using the Monodromy Theorem, prove that the only possible analytic continuations of log(z) around a closed loop encircling the origin k times result in branches differing by 2rik. Conclude that the Riemann surface for log(=) is infinitely-sheeted. 4. Function Theory on the Riemann Surface: Discuss how meromorphic functions are defined on the Riemann surface of log(2). Provide examples of such functions and analyze their behavior. Requirements for Solution: Understand the concept of Riemann surfaces for multi-valued functions. Apply the Monodromy Theorem to determine the behavior of analytic continuations around branch points.
Review of complex numbers 4-R₁e 22= R₂e z=Re Do not use AI, I need real solution, attach required graph and code wherever needed. For reference I have attached the image, but if you need any reference then check out the book by Churchill only. 8-2/8 1=R1 R₂+2) 01+02 The complex conjugate of z = Rei = a + bi is Caz-Re(+2x/8) z= Rea- bi, which is the reflection of z across the real axis. Note that |z2=z-z= Re Re-i =R2e0 = R² = z=√√zz=√√√²+1²=R. Consider the multi-valued function f(z) = log(2), defined initially on C\(-0,0], where the branch cut is taken along the negative real axis. 1. Riemann Surface Construction: Construct the Riemann surface for log(z) by appropriately "gluing" copies of the complex plane cut along (-00, 0). Describe the topological and complex structure of this Riemann surface. 2. Analytic Continuation Along Different Paths: Define what it means to analytically continue log(2) along a path in C\{0}. Illustrate how different homotopy classes of paths affect the branches of log(). 3. Monodromy Theorem Application: Using the Monodromy Theorem, prove that the only possible analytic continuations of log(z) around a closed loop encircling the origin k times result in branches differing by 2rik. Conclude that the Riemann surface for log(=) is infinitely-sheeted. 4. Function Theory on the Riemann Surface: Discuss how meromorphic functions are defined on the Riemann surface of log(2). Provide examples of such functions and analyze their behavior. Requirements for Solution: Understand the concept of Riemann surfaces for multi-valued functions. Apply the Monodromy Theorem to determine the behavior of analytic continuations around branch points.
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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Question
![Review of complex numbers
4-R₁e
22=
R₂e
z=Re
Do not use AI, I need real solution, attach required graph and code wherever needed.
For reference I have attached the image, but if you need any reference then check out the book by
Churchill only.
8-2/8
1=R1 R₂+2)
01+02
The complex conjugate of z = Rei = a + bi is
Caz-Re(+2x/8)
z= Rea- bi,
which is the reflection of z across the real axis.
Note that
|z2=z-z= Re Re-i =R2e0 = R² = z=√√zz=√√√²+1²=R.
Consider the multi-valued function f(z) = log(2), defined initially on C\(-0,0], where the
branch cut is taken along the negative real axis.
1. Riemann Surface Construction:
Construct the Riemann surface for log(z) by appropriately "gluing" copies of the complex
plane cut along (-00, 0). Describe the topological and complex structure of this Riemann
surface.
2. Analytic Continuation Along Different Paths:
Define what it means to analytically continue log(2) along a path in C\{0}. Illustrate
how different homotopy classes of paths affect the branches of log().
3. Monodromy Theorem Application:
Using the Monodromy Theorem, prove that the only possible analytic continuations of
log(z) around a closed loop encircling the origin k times result in branches differing by
2rik. Conclude that the Riemann surface for log(=) is infinitely-sheeted.
4. Function Theory on the Riemann Surface:
Discuss how meromorphic functions are defined on the Riemann surface of log(2). Provide
examples of such functions and analyze their behavior.
Requirements for Solution:
Understand the concept of Riemann surfaces for multi-valued functions.
Apply the Monodromy Theorem to determine the behavior of analytic continuations around
branch points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28150985-1ea5-4755-95d4-f8af909a0667%2F7bf5b462-90eb-4670-a146-09872df7f836%2Fqvuqe1s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Review of complex numbers
4-R₁e
22=
R₂e
z=Re
Do not use AI, I need real solution, attach required graph and code wherever needed.
For reference I have attached the image, but if you need any reference then check out the book by
Churchill only.
8-2/8
1=R1 R₂+2)
01+02
The complex conjugate of z = Rei = a + bi is
Caz-Re(+2x/8)
z= Rea- bi,
which is the reflection of z across the real axis.
Note that
|z2=z-z= Re Re-i =R2e0 = R² = z=√√zz=√√√²+1²=R.
Consider the multi-valued function f(z) = log(2), defined initially on C\(-0,0], where the
branch cut is taken along the negative real axis.
1. Riemann Surface Construction:
Construct the Riemann surface for log(z) by appropriately "gluing" copies of the complex
plane cut along (-00, 0). Describe the topological and complex structure of this Riemann
surface.
2. Analytic Continuation Along Different Paths:
Define what it means to analytically continue log(2) along a path in C\{0}. Illustrate
how different homotopy classes of paths affect the branches of log().
3. Monodromy Theorem Application:
Using the Monodromy Theorem, prove that the only possible analytic continuations of
log(z) around a closed loop encircling the origin k times result in branches differing by
2rik. Conclude that the Riemann surface for log(=) is infinitely-sheeted.
4. Function Theory on the Riemann Surface:
Discuss how meromorphic functions are defined on the Riemann surface of log(2). Provide
examples of such functions and analyze their behavior.
Requirements for Solution:
Understand the concept of Riemann surfaces for multi-valued functions.
Apply the Monodromy Theorem to determine the behavior of analytic continuations around
branch points.
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